Abstract
For a wide class of solutions to multiplicatively advanced differential equations (MADEs), a comprehensive set of relations is established between their Fourier transforms and Jacobi theta functions. In demonstrating this set of relations, the current study forges a systematic connection between the theory of MADEs and that of special functions. In a large subset of the general case, we introduce a new family of Schwartz wavelet MADE solutions for and rational with . These have all moments vanishing and have a Fourier transform related to theta functions. For low parameter values derived from , the connection of the to the theory of wavelet frames is begun. For a second set of low parameter values derived from , the notion of a canonical extension is introduced. A number of examples are discussed. The study of convergence of the MADE solution to the solution of its analogous ODE is begun via an in depth analysis of a normalized example . A useful set of generalized -Wallis formulas are developed that play a key role in this study of convergence.
1. Introduction
This paper expands the study of a class of solutions of multiplicatively advanced differential equations (MADEs) by determining the relationship of their Fourier transforms to Jacobi theta functions. The class of solutions under consideration consists of the Dirichlet-like [1] series:
Each of the in (1) satisfies the MADE. where with in a reduced form and ; see [2]. Note that (2) is multiplicatively advanced in that the argument is an advancing of the parameter by , as is taken to be greater than and .
In general, as is shown in [2], there are nonunique ways to extend the from to the negative reals, obtaining on with also satisfying the MADE (2). To overcome this issue of nonuniqueness, we first extend (perhaps discontinuously) to the negative reals by defining to be.
Then, we utilize only to obtain a new function in which is naturally generated by , and we further observe that for low parameters (namely, and ) is indeed an extension of satisfying (2). Here, is obtained as with in a reduced form.
In attempting to compute the Fourier transform of , we are led, instead, into discovering a relation between a weighted average of the of form where depend on and is an root of unity to a similar weighted average of where are determined by and is a root of unity, while is the Jacobi theta function given by (10) below, is a scaled root of , and . See equations (98) and (99) of Theorem 13 for further details.
One then applies the inverse Fourier transform to (5) to obtain the new functions which are central to this study. These new comprehensively generalize each of the main examples we have previously studied, including in [3] and in [4, 2]. For certain low values of , we show that the give unique, canonical extensions of the associated which satisfy the MADE (2). In general, for higher values of , the are not extensions of ; however, they are all Schwartz wavelets with connections not only to wavelet theory but also to special function theory in that is expressed in terms of (6) and thus in terms of the Jacobi theta function. See (98) and (99) of Theorem 13 for specifics.
We next determine a simple criteria for the to not identically vanish, whereby the relationship of their Fourier transforms to the theta function remains substantive, as described in Theorem 23. This relationship is seen to greatly extend and generalize each of the special cases of Fourier transform computations that have been computed in all of our previous collected work, including [2–6]. Furthermore, even in those cases where vanishes identically, we are still able to provide a (more technical) relation between the Fourier transform of and the Jacobi theta function in Theorem 28. We then begin applying the above discoveries to wavelet theory, producing numerous examples of new Schwartz wavelets generating frames for .
Furthermore, considering the MADE (2) to be a perturbation of the classical ODE with in (2) considered to be the perturbation parameter as , we then initiate a study of convergence of the normalized solution of (2) to the classical solution of (8). In particular, we prove convergence for the normalized to the classical solution of (8) with convergence being uniform on any compact set of ; see Figure 1. We also exhibit graphical evidence for such convergence of other normalized .
The convergence seen in Figure 1 mirrors earlier convergence results [4] we have been previously able to obtain in the canonical extensions for the special cases: (1) which normalizes to discussed in Section 5; and (2) which normalizes to , also discussed in Section 5. The convergence of to and the convergence of to are illustrated in Figure 2.
(a)
(b)
We conclude the paper with a set of generalizations of Wallis’ formula for that we call generalized -Wallis formulas, and we demonstrate their utility in the study of convergence of normalized solutions of MADEs to their classical analogue ODEs.
We mention that the current work falls in the area of functional differential equations of multiplicatively advanced type. Studies in functional differential equations include for instance [7–9]. More precisely, the current work falls under the area of -difference differential equations, where the multiplicative advancement is seen as a dilation that is denoted . There is a robust study within the area of -difference differential equations with dilations involving . This is highlighted by works of L. Di Vizio [10–12]; C. Hardouin [11]; T. Dreyfus [13, 14]; A. Lastra [14], [15–20], [21–23]; S. Malek [14], [15–20], [21–23], [24–27]; J. Sanz [21–23]; H. Tahara [28]; and C. Zhang [12, 29], along with further references by these researchers and others. Also, for good background references to the current work, consult [2–6, 30–34] (especially [2, 4]). These last references also exhibit a number of various applications of global solutions of MADEs.
1.1. Preliminaries and Salient Properties of the Jacobi Theta Function
We shall need to extend the definition of to the case that the argument is complex and lying in the right half plane.
Definition 1. Let , with . Then for and the function given by (1), one defines for (that is, for the real part of nonnegative)
which is analytic for .
Next, recall that for the Jacobi theta function is given by
where
Two properties of the Jacobi theta function of interest are that which are proven in [5] and [6], respectively. From the product formula in (10), one sees that
As indicated earlier, the Jacobi theta function plays a major role in this study in the computation of Fourier transforms.
2. Proof of the Relation of Fourier Transforms to Jacobi Theta Functions
We proceed immediately to the computation of Fourier transforms. For with and given is as in (1), we define
We now restrict to be rational with , and let . One then has the following computation of the Fourier transform of : where in (15) and (16) and for conciseness in moving from (15) to (16), one has and , where and in are taken to be in a reduced form with and . We extend to the complex plane by setting for where denotes the set of where the denominator in (17) vanishes. Note that is defined at by virtue of the quadratic exponent of in the denominator counteracting any growth of the linear exponent of in the numerator of (17). Also for , note that, if is any open region with compact closure , one has that for the distance from to is positive, as the only cluster point of is . Furthermore, the distance from to is also positive. Hence, we have
Hence, the truncated sums which are analytic on , approach uniformly on as . Thus, is analytic on [35] and therefore analytic on .
We have seen in [2] that the “alternating -combinatoric” in (16) can be given by the residue of at a simple pole under a computation of an appropriate contour integral about a region containing . Here, is the Jacobi theta function given by (10). Observe that the term in (16) would then be obtained from evaluation of at . Therefore, as a starting point, we would be interested in integrating the expression around an appropriate closed contour in the complex -plane, where we set later. However, since there is in general a multivalued issue with expression (20) if or are not integers, we set in (20), where is the least integer such that and are both integers, and we integrate
Note that, since and have no common factors and since , one has that divides . Similarly, since and have no common factors and since , also divides . Since is the least such integer, is the least common multiple of and . Thus, we set
We then divide (21) by and integrate the following simplified version of (21)
around a closed contour in the complex -plane, where the exponents and are now integers (avoiding any multivalued issue in a would-be contour integral involving (20) by instead using the integration in (24)). The contour will later be taken to be the oriented boundary of an annulus centered at the origin. This key step in avoiding multivalued issues in moving away from (20) to the integral in (24) allows us to overcome the limiting assumptions in earlier work (that is odd and is even in Theorem 6.3 of [2] or that is an integer and is in Theorem 6.5 of [2]) to now handle the general case in this study (where and are allowed to be rational, with ).
In anticipation of a residue computation of the expression (24), we begin by examining the product representation of the Jacobi theta function in (10) and removing one appropriate factor from the product corresponding to the vanishing of when . That is, note that from (10), one has that for where for , the expression in (26) is defined by the bracketed expression in (25), namely,
Similarly, for one has that where for , the expression in (29) is defined by the bracketed expression in (28), namely,
Thus, via (27) and (30), the expression is defined for each .
We pause the discussion on representing in terms of in order to record a series of useful computational lemmas. The first such lemma evaluates as an “alternating combinatoric.”
Lemma 2. For and as in (26), one has And for and as in (29), one has
Proof. The proof is given in Lemma 5.1 of [2].
The second lemma will provide a structure for the proof of the third lemma, and it will be utilized in a subsequent residue computation.
Lemma 3. For an integer , let be an root of unity. Then, Hence,
Proof. Upon expansion of the middle expression in (33), the left-most equality is self-evident. The right-most equality in (33) follows from the fact that for each one has is a root of , and hence, is a factor. To obtain (34), one divides the right two expressions in (33) by . The lemma is now proven.
The third lemma will simplify the computation in (50) below.
Lemma 4. For an integer let be an root of unity. One has
Proof. Set in (34) to obtain (35). The lemma is shown.
We record three further lemmas on the behavior of roots of unity for later computational use.
Lemma 5. Let be an root of unity, and let for some have order . Then,
Proof. If , then . If , then let . Observe that since we have which gives . Since , we conclude . In particular, if , then and the second equality in (36) holds. The lemma is demonstrated.
The next lemma generalizes the previous lemma.
Lemma 6. Let be an root of unity. Let be fixed. Then,
Proof. If is a multiple of then and then . If is not divisible by , then is a root of unity with order, say, , with . Then, where the vanishing in (39) follows from the vanishing of each summand in (38), which in turn follows from an application of Lemma 5. This proves the lemma.
The following is a refinement of Lemma 6.
Lemma 7. Let be an root of unity, with , where . Let be fixed. Then if , one has and if , one has
Proof. Let . Then, if one has , giving (41). If , one has Hence, If , , giving the first case in (42). If and , then is a root of unity and the right hand side of (44) vanishes while , giving for the second case. Finally, if and , then is a root of unity and the right hand side of (44) does not vanish while , resulting in . This shows the third case and finishes the proof. Note that the third case is the only case with , because in the first case if with , then . This gives the lemma.
We return now to the discussion in (25)–(30). Let . In (25) and (26), we let and be the root of unity; one then has that, for each with , where (27) was used to move from (45) to (46) and (33) in Lemma 3 was used to move from (46) to (47). Thus, for , the residue of at is given by where we have factored out from each factor in in the denominator of (48) to obtain (49); and the first equality in (50) follows from Lemma 4 if (and is automatic if ); and the second equality in (50) follows from (31) of Lemma 2.
Let . In (28) and (29), we let and ; one then has that, for each with , where (30) was used to move from (52) to (53) and (33) in Lemma 3 was used to move from (53) to (54). Thus, for , the residue of at is given by where we have factored out from each factor in in the denominator of (55) to obtain (56); and the first equality in (57) follows from Lemma 4 if (and is automatic if ); and the second equality in (57) follows from (32) of Lemma 2.
Note that the form of the final expression in (51) agrees with the form of the final expression in (58). Hence, we have that for all the residue of at is given by
The previous discussion allows us to reach the following conclusion:
Proposition 8. Let and be as in (22) and (23). Let , , and satisfying all be fixed. Let be fixed. Then the residue of at is given by
Proof. Referring to (24) and (59) and the discussion above, equality in (61) follows immediately, once one determines that is analytic in at . Thus, we must require that , which is equivalent to , as seen in the next sentence. Equality in (62) follows directly from the facts that and . The proposition is now shown.
Note that when and in (62), the resulting expression matches the summand in (16) up to the constant factors and .
Having found the residues at , the next proposition allows for the determination of the residues of in (60) at the roots of . First, observe that precisely when , namely, when where is a root of , is the root of unity, and .
Proposition 9. Let where . For , , a fixed root of , and , the residue of at is given by
Proof. Equation (65) holds for upon setting in the expression . Observe that for the expression can be factored as follows: where one uses (33) to move from (66) to (67) and one uses the definition to simplify (67) into (68). Notice next that evaluation of the bracketed expression in (69) at yields where equality in (70) follows from (35) in Lemma 4.
To compute the residue of in (64) at one observes where (71) follows from (69) and (72) follows from (70), while (73) follows from a consolidation of the factors and (74) follows from the facts that is a root of unity and . Of course, we must have that is analytic at , and hence, for any , which is seen to be equivalent to for any , and any . Furthermore, we must have , which is equivalent to . Thus, we require that , as hypothesized. The proposition is now proven.
Lemma 10. Let , and let . Set be the positively oriented boundary of the annular region in enclosed by the circular paths and , where increases from to . Let , where and .
Let for denote the roots of , where , is a root of , and . Choose sufficiently large so that for one has , the interior of . Then, for as in (62), one has
Proof. The integral over yields times the enclosed residues, which occur at for in the interior of , as well as at the zeroes of in , which by construction of are for and . So the residue theorem gives where in (77) has been replaced by (62) to obtain the summation in (78) and where in (77) has been replaced by (65) to obtain the summation in (79). The expressions in (78) and (79) now give (75) and (76), and the lemma is proven.
Lemma 11. Let the assumptions in the first paragraph of Lemma 10 hold. Then, and hence,
Proof. We show (80) and (81), which implies immediately that (82) and (83) hold. Now, the vanishing of each of the limits in (80) and (81) holds because grows sufficiently rapidly as approaches infinity for or . This rapid growth will follow directly from the identity (12). Designate as a reference circle of radius which by construction satisfies that never vanishes. This nonvanishing is the result of the fact that , whereby , along with the fact that all zeros of occur for for some , from (13). By continuity of on the compact set , there exist constants and , each depending on , such that for all
Note that implies with , and by (12), one has with . Then for such that and all , one has which approaches as approaches infinity, as is the dominant term. Similarly, implies with , and by (12), one has with
Thus, for such that and all , one has which also vanishes as approaches infinity, as is the dominant term. The lemma is now shown.
With Lemma 11 in mind, we record the following corollary which will be utilized later.
Corollary 12. Let be the sector in emanating from the origin with argument falling in the interval with . Let and , where and are as in Lemma 10. Then
Proof. One has
Since the limit as approaches infinity of (93) vanishes, by (80), the vanishing in (89) holds. The vanishing in (90) holds by replacing with and with in (91)–(93), taking limits as approaches infinity, and relying on (81). This proves the corollary.
We have now arrived at a preliminary version of the main theorem of this study. It is “preliminary” in that in certain cases each side of the main equality (98) and (99) in Theorem 13 immediately below reduces to an identically function, and it will be necessary later to give nonidentically vanishing criteria in our main result Theorem 23, where we will also relate the expression in (99) to the Fourier transform of a function naturally generated by . First, we pause to catalogue the current set of results.
Theorem 13. For , let be as in (1) and be as in (14), respectively, where we assume and . Let where . Then, where is given by (11), where is the Jacobi theta function, where and where and with and in reduced form. is taken to be the least common multiple of and . Also, for any fixed root of , one has that in (95) is given by Setting (94) for and requiring that give the following relation of the weighted average of the rotations of the Fourier transforms with the average of the rotations of : Here, is given by (17). Also, for any fixed root of , one has that in (99) is given by
Proof. Integrating the integrand (64) over the oriented boundary of the annular region as in Lemma 10 gives expressions (75) and (76). Taking the limit of (75) and (76) as approaches infinity, and relying on Lemma 11 gives where is given by (97). Dividing (101) and (102) by and moving the double summation to the left side of the equality give (94) and (95). Now, multiplying (94) and (95) by gives
Setting in (103) and (104) gives that (97) becomes (100), and we then also replace in (104) by to obtain (105)–(107) below. Factoring out the powers of from (103) now yields
One now recognizes that in (106) where (109) follows from (17).
Relying on (109) one sees that (106)-(107) becomes which is equivalent to (98) and (99) when one observes that for in (110) one has .
At this point, we have shown (98) and (99) only for , so we next handle the case that . Notice that (108) and (109) are defined at via where (114) follows from (10). Now, at , from (114) one observes that (110) and (111) still hold in the sense that it becomes where the vanishing of (115) follows from the vanishing of (159) which is proven in the paragraph containing (164) in Proposition 20 below (with set to ) and the vanishing of the limit in (116) follows from Corollary 32 below. Finally the requirement that is equivalent to the condition that does not belong to for any by Lemma 14 below. The theorem is now proven.
The following lemma gives a simple characterization for having some with (which is to be avoided in Theorem 13).
Lemma 14. In the setting of Theorem 13, with the notation as in (96), one has
Proof. Observe that the existence of an with is equivalent to the existence of a with either (in which case for some ) or (in which case for some ).
Since , one has
where (120) follows from the fact that . Now from (121), one has that must be divisible by . Thus,
Conversely, if , then with . Since and have no common factors, one concludes that is not divisible by or . Thus, for some and some . Hence,
Setting gives that (121) holds. We conclude
Equations (122) and (124) now give the equivalence (117) and (118), and the lemma is proven.
3. The Functions Naturally Generated by the
In this section, we again assume the notation of the previous section. In particular, the notation in (96) holds. Namely, are rational with , with . Also and with and , where both and are in a reduced form. Finally, is the least common multiple of and , , , and . Let , and let . With this notation in mind, we are able to make the following definitions.
Definition 15. Assume that . Then by Lemma 14, one sees from (117), (118), and (119) that . Hence, the real part . For the real part define and for real part define while for we also define where is the characteristic function of the interval . We emphasize the coefficient in (125) versus the coefficient in (126) and (128).
Note that for and for
Definition 16. Assume . The function naturally generated by is given by where the leftmost summation over the negative index in (132) stands for summation over the indices with and the rightmost summation over the positive index in (125) stands for summation over the indices with .
While the are constructed here in Definition 16, they were originally obtained as the inverse Fourier transform of the expression in (98) in Theorem 13 (scaled by ). In this sense, the are natural. We point out that the may vanish identically, as discussed in Proposition 21 below. However, there are a wealth of that are not vanishing identically, as characterized in Proposition 21. In any case, the have useful properties that are now recorded as a series of propositions. The first observation is that is real valued.
Proposition 17. For all , .
Proof. Observe that for one has , and hence, conjugation preserves the sign of . Thus, conjugation also preserves the sets and . This allows us to conclude that independently of the factor in (128) versus the factor in (125), where . Hence, from (132), one has where the last equation in (134) follows since conjugation permutes each of the sets , , and . The proposition now follows.
The second result gives smoothness.
Proposition 18. is at , and hence, is in . Furthermore, is Schwartz.
Proof. Away from , is via (125)-(126). From (132), note that for , one has and for one has . Smoothness will follow at , by showing that for all . From (125) and (126), one has where (135) follows directly from differentiating (126) and (136) also follows from (126). Also, where (135) follows directly from differentiating (125) and (138) follows from (1). Equality of (136) with (138) would be equivalent with
Now if , then from Lemma 6. Hence (139) holds. On the other hand, if , then from Lemma 6. In this case, one has that for some where we have used (96) to rewrite in the second equality of (142). Now from Lemma 2.4 of [2], one has , where from (96) we have used . From (13), it follows that if and only if there is a with . Now, where (96) was used to rewrite and . Observing equality of the rightmost expression in (144) with the rightmost expression (143), one has that in (139)
Thus, in all cases, (139) holds, whence equality of (136) and (138) holds. We conclude that is at and thus on .
Finally, the fact that is Schwartz follows from (1) the fact that it is in and (2) from the fact that for sufficiently large one has for constants and , which in turn follows from the expressions (125) and (126) and from Proposition 8.1 of [2]. The proposition is now proven.
One consequence of Proposition 18 is that has a Fourier transform which is Schwartz. The following proposition allows us to observe that was defined so that its Fourier transform would be given by (98) (up to the constant factor ).
Proposition 19. The Fourier transform of is given by
Proof. From (131), one has We next evaluate in (147). Observe that for one has Similarly, for , one has From the matching forms of (148) and (149), one sees that (147) becomes Noticing that reexpresses (150) as where (153) follows from (17). Now, (153) gives (146) and the proposition is proven.
Another property of is that all of its moments vanish.
Proposition 20. All moments of vanish. That is, Equivalently, all the derivatives of satisfy
Proof. We proceed by showing (155). Differentiating (152) yields Evaluating at gives Here, movement from (157) to (158) is justified as follows: where (160) follows from as in (96); (161) follows from cancelling out ; (162) follows from multiplying up and down by ; and (163) follows from (10).
Examining (159), one sees that if in (159), then by Lemma 6, one has and (159) vanishes. On the other hand, if , then there is a with whence by (13). Thus, (159) again vanishes. Since (159) now vanishes in all cases, we have vanishing of every derivative of at , giving (155). Thus, (154) now holds, and we have vanishing of all moments. This completes the proof of the proposition.
Notice the similarity of the argument in Proposition 20 with that of Proposition 18. In each case, a factor formed from the sum of powers of roots of unity fails to vanish precisely when the remaining theta function factor vanishes.
The next proposition gives a simple characterization for to vanish (and to not vanish) identically.
Proposition 21. For all notation as in (96), the following equivalences hold: Thus, occurs precisely when does not vanish identically, which in turn occurs precisely when the identity (98) and (99) in Theorem 13 is not an identically zero tautology.
Proof. The right-most equivalence in (165) holds by linearity and injectivity of the Fourier transform; and the equivalences in (166) and (167) hold by Proposition 19 and Theorem 13, respectively. It remains to show the following equivalence: Observe that in the argument of the theta function in (167), the expression holds precisely when (and then ). One might expect that the previous statement would be that ; however, is an integral power of precisely when is a multiple of , resulting in canceling of terms.
Next, summing over indices with like values of first gives
By Proposition 22 below, the expression in (170) does not vanish identically. Thus, (169) and (170) vanish identically if and only if vanishes. However, this summation of roots of unity to the power vanishes precisely when by Lemma 6. Thus, is equivalent to identically vanishing in (169) and (170) and therefore equivalent to identically vanishing in (167). Now, where the last equivalence holds from the fact that is in a reduced form. Thus, the leftmost equivalence in (165) holds. From this left-most equivalence, one deduces that does not vanish identically precisely when . The proposition is now proven.
The following proposition, utilized in Proposition 21, relies on properties of the Jacobi theta function to obtain a nonidentically vanishing condition.
Proposition 22. The function is not identically in the argument . Hence, given in (170) is not identically in , where is as in (100) and (184).
Proof. If , then (172) becomes , which does not vanish identically. For instance when , it becomes . If , we show that the function is also not identically . This in turn is equivalent to the numerator in the right hand side of (174), namely, not being identically in . Setting in (175) and (176) gives that and then only the summand in (175) survives: where the nonvanishing in (178) is obtained from (13) along with the fact that does not lie on the negative real axis for . To see this latter point, if , one has , from which one has that is divisible by (as is assumed to be in reduced form). However, each is not divisible by . Thus, (172) does not vanish identically. By the identity theorem, (172) does not vanish on any subset of having a limit point. One concludes that (173) is not identically in , as the set of as in (184) ranges over a set with limit point as varies in . The proposition is now shown.
In light of Proposition 21 and Lemma 14, we refine and sharpen Theorem 13 by (1)removing all identically tautologies in (98) and (99) via making the additional assumption that (2)guaranteeing that the expressions in (98) and (99) are well-defined via making the easily checked assumption that (3)incorporating into the theorem while including the additional properties of garnered from Propositions 17–20
In doing so, we arrive at the main theorem of this study.
Theorem 23. Let , and for , let be defined as in (1), with . For , let , as in Definition 16, be the function naturally generated by . Let the notation of (96) hold. In particular, let and be in a reduced form; let with (which gives and ); and let with . Then, is a real valued Schwartz wavelet with all moments vanishing with Fourier transform given by satisfying where is the Jacobi theta function, where and where for any fixed root of , one has that in (182) is given by Also, is given by (17). Furthermore, given , one has that is uniquely defined by Definition 16, and it satisfies the same multiplicatively advanced differential equation (MADE) on as does on , namely,
Proof. is real valued and Schwartz via Propositions 17 and 18. From Proposition 19, one has that equality in (187) below holds, and, from equation (98) and (99) in Theorem 13, one has that equality in (188) below holds. From Lemma 14, the hypothesis that is equivalent to the condition that (186)–(188) are defined for all . By Proposition 21, is identically in precisely when ; thus, the hypothesis that in the theorem gives precisely all the nonidentically zero examples of (186)–(188). Furthermore, from the hypothesis that , one has that , from which we conclude that . Setting and in (186)–(188) yields equations (180)–(182). Similarly, setting and in (96) (respectively (100)) yields (183) (respectively (184)). Next, all moments vanish by Proposition 20. It remains to show that is a wavelet solving the MADE (185). The wavelet property follows from the following three criteria: (1) because is Schwartz(2) because each moment, including the -moment, of vanishes(3) because is Schwartz and decays rapidly in the tails and because vanishes to infinite order at by Proposition 20
Now, from [2], on satisfies the MADE
This follows, since
Setting and in equations (16)–(18) in Theorem 2.2 of [2] gives the MADE (189). The MADE for now follows from (189) as is a linear combination of expressions involving (by Definition 16 and by (125)–(127)). This proves the theorem.
Remark 24. While the assumption that in Theorem 23 may at first seem to be limiting, in fact, there remains a wealth of nonidentically-zero cases (with ). The examples below in Sections 5 and 6 demonstrate that Theorem 23 is quite general in nature. For now, observe that if is a multiple of then . Hence, if is the prime factorization of , the nonidentically zero cases for and its Fourier transform are handled by those with for all and with the parameters and satisfying and in a reduced form.
Remark 25. We also point out that even when , we are able to relate the Fourier transform of to the Jacobi theta function. However, the Fourier relation in this setting comes from an analogue of that is generically noncontinuous at and is not a wavelet. See Theorem 28 and the related discussion below.
Remark 26. In a development similar to (169)-(170), it is also possible to show that
However, the development in (169) and (170) is preferred in order to harness properties of the Jacobi theta function to gain a nonidentically vanishing criterion, as in Proposition 22. Nonetheless, in comparing the expression in (170) with that in (192), observing that they share a common factor , and contemplating equation (98) - (99), one is led to consider the relation between
including when . This will be done in Theorem 28 below, where it will allow for the recovery of Fourier transform information for in the cases that .
Remark 27. At this juncture, we pause to convey a surprising consequence of expressing the Fourier transform of in terms of Jacobi theta functions, as in Theorem 23. Knowledge of the Fourier transform of in terms of the Jacobi theta function allows for the proof of nonvanishing results in the cases of in Corollary 34 and Proposition 36 below. This pair of nonvanishing results serves to underpin a main connection between the solutions of MADEs of type to the theory of wavelet frames in harmonic analysis. That is, in these cases, is a mother wavelet generating a wavelet frame for the square integrable functions , as is demonstrated in Theorem 37 below. Thus, the current study lays the foundation for a set of strong connections between solutions of MADEs, special function theory, and the theory of wavelet frames. Next, we recover information on the Fourier transform of when in the second main theorem of this work, again relating the Fourier transform to the Jacobi theta function.
Theorem 28. For , let be as in (1) and be as in (14), respectively, where we assume and , with notation as in (96), in particular and are in reduced form. Let be the Jacobi theta function. Let and let . Then,
Here, for a fixed root of , one has that in (195) (and (201) below) is given by
Also, is the ray emanating from the origin given by (210) and (206), and
Furthermore, is given by (11), with
where is taken to be the least common multiple of and .
One also has the following relation of the weighted partial average of the rotations of the Fourier transforms with a partial average of rotations of :
Again, in (201) is given by (197). The expression in (202) is referred to as the defect.
Proof. Instead of integrating the integrand in (64) over the oriented boundary of the annulus as in Theorems 13 and 23, we shall integrate over the oriented boundary of the intersection of the sector with the annulus , where is chosen as in (206) below and . Thus, . Here, and , where as before, and , but in the current setting, one has and that increases from to . Also, and are portions of two rays, as given by
Referring to (117) and (118) in Lemma 14, one has and this in turn is equivalent to the fact that for all the roots (in ) of do not lie among the zeroes of by the remarks at the ends of Propositions 8 and 9. Hence, for all and each associated as in (197) and for each integer , one has which does not fall among the zeroes of . Namely, for all and each associated , and for all values of one has . In particular, setting , then, for all and each associated and for all values of and , one has that . We now choose where is chosen sufficiently small so that none of the falls in the sector . Then, the corresponding sector gives the intersection which has oriented boundary as describe above. Define . Choose sufficiently large such that all are contained in . Then integrating in (64) over the contour gives times the enclosed residues. Namely, where the evaluation of the enclosed residues in (207) occurs via Propositions 8 and 9, similar to the analogous computation in the setting of Theorem 13. Also, comparing (208) with (75), notice the summation (over ) in (208) has terms (as compared to terms in (75)) as the sector sweeps through an argument of in (208) as opposed to sweeping through an argument of in (75). The choice of is made in order to have start at in (208), whereby the first term (at ) in the double summation in (208) will have , matching the expression for in (16) after letting approach infinity. Again, comparing (209) with (76), notice the summation (over ) in (209) has terms (as compared to terms in (76)) as the sector sweeps through an argument of in (209) as opposed to sweeping through an argument of in (76). The choice of is made to give as the first () root of falling in , that is with smallest argument such that .
Now, defining the rays one has that integration along approaches integration along for as . Hence, one has the following limit: where in moving from (212) to (213), one can drop the integrals over and in the limit by Corollary 12; in moving from (213) to (214), we have relied on the remarks following (211); in moving from (214) to (215) we used ; in moving from (215) to (216), we have simplified powers of by utilizing and ; and in moving from (216) to (217), we have used and . Relying on (217) and taking the limit of (207)–(209) as approaches infinity yield
Rearranging terms in (218) and (219) and multiplying through by give (194)–(196). Referring to (16), multiplying (194)-(196) by , and factoring out the powers from the numerator and from the denominator of (194) give (200)–(202). The theorem is now demonstrated.
Remark 29. When , we will consider the expression (202) (the integral together with its coefficient) to be the defect from having the expression in (200) vanish at , as the expression in (201) vanishes as approaches by Corollary 32 below. As a consequence, one concludes that if the defect (202) (when ) does not vanish, then expression (200) cannot be the Fourier transform of a wavelet. Observe that when , the term in (202) does not vanish, and vanishing of the defect when is equivalent to the vanishing of the integral expression in (202).
Remark 30. If one were to allow for the case in Theorem 28, one would then have and in this case. Furthermore, in this case, Theorem 28 would reduce to Theorem 23, as the coefficient would be in (202). Thus, if one were to remove the hypothesis that in Theorem 28, Theorem 23 could be subsumed into Theorem 28. We choose to keep the cases and separate in order to highlight the special properties of when .
The following proposition gives a pair of flatness conditions, and it forms the technical basis for the proof of Corollary 32.
Proposition 31. Let and be fixed with . Then for and for any power , one has
Due to its length, we leave the proof of Proposition 31 until the end of this section, where the interested reader can peruse it. Instead, we immediately proceed to Corollary 32 and its proof, as Corollary 32 is used throughout this study.
Corollary 32. Let , and let be as in (184) and (197). Namely, where is any root of . As a consequence, . Let . Then for
Proof. Note that . Hence, One concludes that Applying Proposition 31 with as gives (221). Note that the assumption that assures that does not fall along the negative real axis. This gives the corollary.
We finish this section with the proof of Proposition 31.
Proof of Proposition 31. Set , where ranges from to . Then, from (10), one has where
From (224) and (10), one has that, for , where the inequality in (228) holds for all . Hence, inverting (227) and (228), taking square roots, and multiplying by , one obtains that for each
To handle the limit as , pick . To handle the limit as , pick . Thus, (220) holds when .
The remaining case is . Here, the possibility that is excluded, since if were to equal then , which has an infinite number of zeroes (for with ). It will be shown below (see (239) through (241)) that we have the following two limits:
It will also be shown (see (242) and (259) below) that the following rate of growth holds: where denotes the greatest integer function evaluated at . Hence, from (231), we have the related rate of growth
For the time being, we assume (230)–(232) and use (225) to show that (220) holds as (equivalently as ) via where one moves to (233) via (230) and (231). The vanishing in (234) is seen from the fact that when the exponent in (234) is increasingly negative as , while remains bounded.
Similarly, we use (225) to show next that (220) holds as (equivalently as ). In this setting, where one moves to (235) and (236) via (230) and (232). The vanishing in (237) is seen from the fact that when the exponent in (237) is increasingly negative as , while remains bounded.
All that remains is to show (230)–(232). We first show , from which it will immediately follow that . Note that where the reindexing occurs in moving to (238). For satisfying and , one traps in (238) by where we have further assumed that is sufficiently large so that , (that is falls in the interval ). Since
hold, the trapping in (239) gives that . An immediate consequence is that , giving (230). We conclude that there is an so that for one has
We next show (231). From (226), observe that for one has where a reindexing by moves one from (243) to (244); one obtains (245) from (244) by (226); and one factors out a from each product in the left most product expression in (245) to proceed forward.
Now, is nonvanishing for all , as has vanishing points if and only if , up to full revolutions, and (by (10)). Thus, the compact interval under maps to a compact interval (with ). Since , one has and is bounded for all .
It remains to show that in (246) is bounded as approaches infinity (again, in the setting that ). An upper bound independent of follows via where the reindexing is used to obtain (249).
Starting with (248)–(249) to obtain (252), a lower bound independent of for (with determined below) follows via where appearing in (254) is chosen sufficiently large so that (that is, falls in the interval ). This justifies movement from (255) to (256).
Hence, we have from (251) and (257) that
From (242), (247), and (258) applied to (225) and (246), one obtains for , where is fixed. Thus, (231) is demonstrated. As a consequence, we also have that (232) now holds. The proposition is proven.
4. Connection to the Theory of Wavelet Frames
We have seen in the previous section that expressing the Fourier transform of in terms of Jacobi theta functions, as in Theorem 23, provides a strong connection to the theory of special functions. However, more can be concluded. Namely, from the relation of the to the Jacobi theta function, we also demonstrate in this section the connection to the theory of wavelets and wavelet frames. In particular, for low values of , we establish that each as in Theorem 23 is a Schwartz mother wavelet for a wavelet frame generating all of .
Recall that is a wavelet if it belongs to , has first moment , and satisfies the admissibility condition that . Furthermore, such a wavelet is a mother wavelet for a frame of form
if generates , where is the scale factor, is the translation parameter, and is the norm of in . One defines the diagonal term by and the off-diagonal term by which together give the frame condition sufficient for in (260) to be a frame. As is shown in [36] and [37], for sufficiently small, (263) is in turn implied by the conditions (264) immediately below:
In Proposition 36 below, we see that there are natural scale factors that allow us to compute for the wavelet for low values of , where knowledge of properties of Jacobi theta functions lets us compute exactly, which in turn will establish the nonvanishing of in the left hand criteria of (264).
But first, we need to obtain an even stronger nonvanishing analogue of Proposition 22 in the setting of Theorem 23 (in particular, with ) under the additional assumption that .
Proposition 33. Under the notation and assumptions of Theorem 23 (in particular and ), along with the additional assumption that , one has that (172) becomes which never vanishes for if and for and (265) vanishes precisely when Equivalently, (265) vanishes precisely when for some , where if is even, and if is odd.
Proof. If , then and (265) becomes , which never vanishes for . If , then and (265) becomes where equality in (267) follows from the fact that must be odd, as is in reduced form and . Now from (10) the numerator in the bracketed expression in (267) becomes where (1)if is even in (270) the odd cases cancel, leaving the in the even case of (271)(2)if is odd in (270) the even cases cancel, leaving the in the odd case of (271)Now, in the even case, where the last equality in (272) follows from (10). And in the odd case, where the last equality in (273) again follows from (10). Thus, (268)–(271) reduces to Thus, (267) reduces to Then, (276) vanishes for precisely when , which by (10) occurs precisely when for some . This last statement is equivalent to (266). Equivalently, we have such vanishing when for some . This gives the proposition.
Corollary 34. In the setting of, and under the assumptions of, Theorem 23, with and , with or , one has that where and is any fixed root of .
Proof. By Proposition 33, when the nonvanishing of (277) holds automatically for . Also by Proposition 33, equation (266), when , the vanishing of (277) holds if and only if for one has for some and for if is even or for if is odd, where in (278)–(279) one has if and if . From (279) and the fact that is odd (since is even), one sees that for any values of and , because is imaginary while is real. Thus, (277) never vanishes for when , and the corollary is proven.
Remark 35. Although, for by Corollary 34 one has (277) never vanishes for , one has that (277) vanishes to infinite order at by Corollary 32.
Proposition 36. Let . Under the assumptions and notation of Theorem 23, with as in Definition 16, for and with whereby , one has that
which is and never vanishes for , in particular for .
And for and with whereby , one has that
where in (283) if is even and if is odd. Expressions (283)–(284) are and never vanish for , in particular for .
Proof. First, we handle the case with . Under this assumption, along with the assumptions of Theorem 23, one has that (180)–(182) become From (184), one has for , whence for in the theta function expression in (285). One concludes that where equality in (286) follows from (12). From (286) and the fact that , we have (288) below: where the move from (289) to (290) is justified by (10). Now (290) is equal to (281), which by (285), is equal to (282). From its definition in (287), is invariant under multiplication of the argument by . This -invariance can also be checked via properties of the theta functions in expression (281). From (277) in Corollary 34 coupled with (285), one has nonvanishing of for . Since only vanishes for for by (10), we have that never vanishes for any as its argument is positive. From these two nonvanishing results applied to (282) one has that is and never vanishes for . The case is now shown.
We turn next to the case, with . In this setting, from Theorem 23 one has that (180)–(182) becomes and where (292) and (293) follow from (275) and (276). From (184), one has for , whence and for in the theta function expressions in (292). From (292), along with the fact that with , one then has that where is as in (293); (12) was used to move from (295) to the subsequent line; all terms involving a power of have been factored out in (296); and in moving to (297), the expressions involving powers of have been multiplied by while the bracketed expression in (296) was recognized as the Fourier transform of via (292). One then uses (297) to compute the diagonal term as follows: where (297) is used to move from (298) to (299) (after taking absolute value and squaring) and (10) is used to move from (299) to (300). Finally, (300) is seen to be equivalent to (284), which is in turn equivalent to (283). From its definition in (298), is invariant under multiplication of the argument by . This -invariance can also be checked via properties of the theta functions in expression (300). From (277) in Corollary 34 coupled with (291), one has non-vanishing of for . Since only vanishes for for by (10), we have that never vanishes for any as its argument is positive. From these two nonvanishing results applied to (300), one has that is and never vanishes for . The case is now shown, and the proposition is proven.
At this point, at least for low values of , we are prepared to explicitly make the connection to wavelet frame theory that follows from knowledge of the Fourier transform of in terms of the theta function.
Theorem 37. Let . Under the assumptions and notation of Theorem 23, in particular , and with as in Definition 16, one has, for with , and also for with , that for sufficiently small
Proof. The theorem will follow by establishing (264) above. Now, for by Proposition 36. Next, from Theorem 23, we have that is a Schwartz wavelet with all moments vanishing. Since all moments vanish, is flat at . Since is Schwartz, is also Schwartz and therefore decays faster than for any for near . Hence, for choice of sufficiently large, one has . Thus, (264) is satisfied, and the theorem is demonstrated.
5. Canonical Extensions
From Theorem 3.2 in [2], a given on has, in general, infinitely many Schwartz wavelet extensions to all of with satisfying the same MADE as . In this section, we shall demonstrate, in the setting of Theorem 23 for low values of , that there is a natural uniquely determined extension of to , namely, the canonical extension of given as follows. We remark that for these values one has as in Theorem 23 automatically.
Definition 38. Under the assumptions and notation of Theorem 23 (in particular ), with (and ), the canonical extension of from to all of is defined to be , where is the function naturally generated by as given by Definition 16, with .
We clarify the above definition by emphasizing that canonical extensions of as the vary (as given in Definition 38) form a strict subset of the set of functions naturally generated by (as in Definition 16) . This follows from the fact that, for , in a large number of cases one has that for and is therefore not an extension (see Propositions 42 and 43 below). We next show that for is indeed an extension of .
Proposition 39. Under the assumptions and notation of Theorem 23, for with one has that for , where is given by (1).
Proof. We handle each value of separately. First, let . In this setting, one has . From (131) and (132), one has that and hence, on holds for because for . This last equality follows from (125) in conjunction with (1) after noting that and so . We remark that, from Case in Section 6.1 below, the Case under consideration here consists of precisely the that are flat at the origin and the canonical extension is extended to be on the negative real axis, equivalently the canonical extension is .
Next, let . Since is in reduced form, one concludes that is odd. Now, . From (131)-(132), one has that where one moves from (303) to (304) via (125) and (128). From (304), one sees that when . Also note from (304) that the canonical extension is an even function if is odd, and it is an odd function if is even when .
Finally, let . Since is in reduced form, one concludes that for with . In this setting, which gives that
For each case that one has that and . From (306), has negative real part for and for . Then for and for , (128) gives that . Combining these results with (131) and (132) and (125) gives for that
Hence, in each of the cases one has , and the proposition is proven.
In the next theorem, we assume all of the hypotheses of both Theorems 13 and 23 to determine a family of with canonical extensions having optimal properties.
Theorem 40. Let . Assume that all the hypotheses and the notation of Theorems 13 and 23 hold. Then for equivalently, for the associated have canonical extensions which (1)are Schwartz wavelets on (2)are mother wavelets for a frame generating , as in (301), where is sufficiently small, and when and when (3)have all moments vanishing(4)have Fourier transforms given by (180)-(182) which relate the canonical extension of to the Jacobi theta function(5)satisfy the MADE (185)(6)have nonvanishing diagonal frame terms on : given by (281)-(282) for and , and given by (283)-(284) for and , each expressible in terms of the Jacobi theta function
Proof. Since one has is a canonical extension of to by Proposition 39 and Definition 38. Properties (1), (3), (4), and (5) follow from Theorem 23. Property (2) follows from Theorem 37. Since , Property (6) follows from Proposition 36. It remains to show the equivalence of (297) with (310).
We first show (297) (310). Assuming (297), one has where the case that is ruled out for not being in reduced form. From this, one concludes . Since , one concludes that when ; when ; and when .
When , which happens for each value of , one has
Thus, the case gives that falls in one of , , , , or .
When , which happens only when (for ), equivalently, when , equivalently, when , one has
Hence, is even when , and in this case, falls in .
When , which happens only when (for ), equivalently, when , equivalently, when , one has
Hence, for when , and in this case, falls in or .
All of the above cases combine to require that satisfies (310).
Conversely, we show (310) (309). So assume (310). Within this setting, examine the case that is odd, with . Then,
from which we deduce that and . This implies , none of which is ruled out as each such is divisible by . Furthermore, from the possible , one sees that .
Next, examine the case that is even with . Then, . Hence,
One concludes that and , in a reduced form. Thus, (which is divisible by ), and .
Finally, examine the case that with and . Then,
One concludes that which does divide for , all in a reduced form. Thus, and .
All cases combine to give and , which is (309). Thus, (309) (310), and the theorem is now proven.
Selected examples of canonical extensions are now provided.
[Example odd and ] In this case,
One sees that and . Thus, and , while , consistent with Examples [] in Section 6.1 and [, , and ] in Section 6.2 below, where the current case that and is seen to be flat at the origin. One extends to be identically for . This yields , which equals the canonical extension by (302) as . This particular canonical extension was first introduced in [3] as the function . From Theorem 40, one has that satisfies properties (1) through (6), including that is a Schwartz wavelet with all moments vanishing, satisfying the MADE , generating a frame for , and having Fourier transform , as was previously proven in [3] but is now seen as a special case of Theorem 40.
[Example even and ] In this case,
One sees that and . Thus, , while , consistent with Examples [,] (with the prime taken to be ) in Section 6.1 and [, ] in Section 6.2 below. Also, we record that . Since , has canonical extension given by (304). That is, which is the extension of to be an even function. This canonical extension was first introduced in [4] and denoted there by . From Theorem 40, one has that satisfies properties (1) through (6), including that is a Schwartz wavelet with all moments vanishing, satisfying the MADE , generating a frame for , and having Fourier transform , as was previously proven in [4] but is now seen as a special case of Theorem 40. If one normalizes by , one obtains , which converges uniformly [4] to on compact subsets of as , as is illustrated in Figure 2(a). Furthermore, satisfies properties (1) through (6), including satisfying the same MADE above (by linearity) and having Fourier transform . See [4] and [6] for further details.
[Example odd and ] In this case,
One sees that and . Thus, , while , consistent with Examples [,] in Section 6.1 (with the prime taken to be ) and [, ] in Section 6.2 below. Also, we record that . Since , has canonical extension given by (290). That is, which is the extension of to be an odd function. This canonical extension was first introduced in [4] and denoted there by . From Theorem 40, one has that satisfies properties (1) through (6), including that is a Schwartz wavelet with all moments vanishing, satisfying the MADE , generating a frame for , and having Fourier transform , as was previously proven in [4] but is now seen as a special case of Theorem 40. If one normalizes by , one obtains , which converges uniformly to [4] on compact subsets of as , as is illustrated in Figure 2(b). Furthermore, satisfies properties (1) through (6), including satisfying the same MADE above (by linearity) and having Fourier transform . Also, while . See [4] and [6] for further details.
While the above examples are consistent with earlier examples in our previous work, the examples for are all not previously seen. To illustrate an example from (322), we choose for and develop it here.
[Example odd and ] In this case,
One sees that and . Thus, , while , and , consistent with Examples [,] (with the prime taken to be ) in Section 6.1 and [, ] in Section 6.2 below. Also, we record that . Since , has canonical extension given by (307). That is, where (324) follows from Definition 15 and (325) follows from (323) and (9). From Theorem 40, one has that satisfies properties (1) through (6), including that is a Schwartz wavelet with all moments vanishing, satisfying the MADE , generating a frame for , and having Fourier transform .
One computes while the first derivative at the origin is where the penultimate equality in each of (326) and (328) follows from (10), the second equality in (327) follows from the fact that (as is proven in equation (11) of [2]), and the inequality giving nonvanishing in (328) follows from (13). While we are here, we also compute the second derivative at the origin by
Normalize by to obtain , where satisfies (1) through (6) of Theorem 40, while having Fourier transform and satisfying the same MADE as , namely, with initial conditions
This second derivative reducing to in (331) follows from Lemma 41 below. For small , as , (330) and (331) can be considered to be a perturbation of the ODE initial value problem (332) and (333), where with initial conditions which is solved by . Figure 3 provides graphical evidence that converges to near . We have scaled Figure 3(b) by to better visualize the graph on the negative axis.
Lemma 41. For the Jacobi theta function (10) satisfies
Proof. We show both equalities in (334) together by requiring that each below be always simultaneously or always simultaneously . One has where the second equality in (335) is given by the left hand equation in (12) and the last equality in (336) is given by the right hand equation in (12). Dividing (335) and (336) through by gives (334), and the lemma is shown.
(a)
(b)
In the next two propositions, we see for that there are numerous cases for which is not an extension of a given .
Proposition 42. Let the notation of Theorem 23 hold. Let with and be as in (337). Recall that for given one has an integer and natural numbers with while , , and . We observe that there are an infinite number of such that Furthermore, recall that If for any such as in (338), one has that then That is, the function naturally generated by does not restrict to and is not a canonical extension of .
Proof. Note first that there are no two consecutive integers and with for if there were two such consecutive integers, by subtracting one equation from the other in (342), one would conclude that also. However, this is impossible as is in a reduced form with , while being both an integer and in a reduced form would require . We conclude that there is an infinite number of with (338) holding.
Fix any satisfying (338). For this , from (138) and from the discussion between (138) through (144), one concludes
Furthermore, for our same fixed , from the fact that , [2], one has
Since (338) holds by (10). If (343) were to equal (344), then dividing through by and recalling that in our setting would require that which is disallowed by the hypothesis (340). Thus, if satisfies (338) and (340), one cannot have equality of (343) and (344), whence does not restrict to on . The proposition is proven.
We next harness Proposition 42 to show that for with none of the functions restricts to on .
Proposition 43. Under the same setting, notation, and assumptions as Proposition 42, for with as in (337) but with the restriction that or , one has does not restrict to on . Thus, on does not have a canonical extension for these values of .
Proof. Let or . For given , define to be the unique value with and . Observe that in (339) for fixed one has . Thus, the set associated to is identical to the set associated to .
In Table 1, for the given value of and associated possible one has, by inspection, the unique values with that determine the set . We denote the values by . This last is the same value giving , where .
For instance, to obtain the second row of the above Table 1, when and has associated value , one has
precisely when the in (346) assume the values or , that is, when
In this case, one has and then ; the remaining values of form .
Now note that, in all cases in Table 1, which follows from the fact that . We conclude that for all and with in a reduced form, and for all integers one has
The inequality in (349) follows from the fact that no integer multiple of for is an odd multiple of .
Now, we have seen in Proposition 42 that for each of there are infinitely many with (338) holding. Fix such an , and set = in (349) to obtain from which we see that both (338) and now (340) hold. We conclude from Proposition 42 that does not restrict to on for any with . Therfore, such do not have canonical extensions. This completes the proof of the proposition.
We remark that for and not divisible by , results similar to Proposition 43, with more tables analogous to Table 1 (but having many more cases), and with inequalities analogous to (349) (but with multiple cosine terms), should give that canonical extensions are at least rare, if not nonexistent.
6. Examples
The goals of this section are twofold: (1)To classify those and for low values of such that and meet the assumptions of Theorem 23 and satisfy (180)-(182);(2)To fill out selections from the examples in goal 1 in detail
Throughout the discussion, the notation given in (96) holds, and .
6.1. Classifying and in Theorem 23 with Low Values
Recall that the fractions and in (96) are in reduced form.
[]: since , we conclude . This is equivalent to
Now (351) holds if and only if is an odd integer and is an even positive integer. To match notation in previous work [2], we relabeled and . Such are precisely the that are flat at (as is shown in Proposition 2.2 of [2]). We further analyze this case in detail below and record now that
: since , we conclude and . Thus, there are two subcases.
[, ]:
[, ]:
Thus, we have
[]: this case is disallowed.
[]: since , we conclude and . There are now four subcases.
[, ]:
Please see Section 7 for a detailed convergence study involving the current example (356) with and .
[, ]:
[, ]:
[, ]:
Thus, we have
Other and with meeting the criteria of Theorem 23 and not among the above cases can be determined similarly via the following proposition.
Proposition 44. Let be the prime factorization of and . The and with meeting the criteria of Theorem 23 are that satisfy the conditions
Proof. From Proposition 21, one sees that the hypothesis that in Theorem 23 is required in order to have a non-identically vanishing of (180)–(182). From this, we conclude , giving (363). Since is a divisor of , one obtains (364). Equations (365) and (366) follow from the fact that and are in a reduced form. (361) and (362) follow from (96). The proposition is shown.
6.2. Selected Examples from Section 6.1 in Further Depth: Canonical Extensions
[, , ]: we expand on [Example ] in Section 6.1, where it was shown that , , and , which specifies (96) as
Inserting these values in (131) gives (368), while insertion in (180)–(182) gives (369) and (370) below. That is,
Furthermore, where (371) follows from moving from the numerator to the denominator; (372) is obtained by multiplying numerator and denominator by ; and (373) follows from (12). Also, from (184), in (373) is any fixed root of . This example recovers Theorem 6.3 in [2]. Also, if one sets and , one has which is discussed earlier in [Example odd and ] in Section 5, and which was denoted in [3]. This recovers the inaugural wavelet in the first paper that inspired the current direction of study.
Finally, satisfies the same multiplied advanced differential equation on as does on . That is, (185) becomes
[]: we expand on [Example ] in the previous section for the case . There are two subcases, when or .
[, ]:
Note that in this case, since and must be odd, is even, and is odd, so we set and . Then, and . Inserting these values in (131) gives (376)–(378), namely,
Note that (378) and (379) show that is an even function. Also, from (180)-(182) along with (184), one has with where . Finally, the MADE (185) becomes
When , the above results recover the even case of Theorem 6.5 of [2].
When and , the above results give
with
In this setting (383)–(385) we recover [Example even and ] from Section 5, with normalization .
[, ]:
Note that in this case is odd, and is odd, so we set and . Then, and . Inserting these values in (131) gives (387)–(390), namely,
Note that (390) and (391) show that is an odd function. Also, from (180)–(182) along with (184), one has with where . Finally, the MADE (185) becomes
When , the above results recover the odd case of Theorem 6.5 of [2].
When and , the above results give
In this setting, (395) recovers [Example odd and ] from Section 5, with normalization .
[]: we expand on [Example ] in the previous section for the case . There are two subcases.
[, ]:
Note that with and with . Thus, we set and where . Hence, and . Then, and where . Inserting these values in (131) gives (397)–(400), namely,
Also, from (180)–(182) along with (184), one has: with where . Finally, the MADE (185) becomes
[, ]:
Note that is odd and with . Thus, we set with where and . Then, and where . Inserting these values in (131) gives (405)–(408), namely, where each of the three summations in (408) vanishes at as one can compute that with as in (404), where the vanishing follows from (13).
Also, from (180)–(182) along with (184) one has with where . Finally, the MADE (185) becomes
When and , equations (410) and (412) recover, respectively, the Fourier transform and the MADE satisfied by as seen in [Example odd and ] in Section 5.
7. Convergence of MADEs to Classical Solutions, an Example
The purpose of this section is to provide an overview of an example of a satisfying a MADE where the normalization converges to the corresponding solution of its classical analogue (the ODE obtained when the parameter in the original MADE is set to ). Our example is , which is not a canonical extensions as will be seen below. As such, it represents a new phenomenon with corresponding new challenges. First, can be obtained as the output of a reproducing kernel computation for an input wavelet , in the same manner of computation as in [4]. Second, since it satisfies a MADE of order , there are derivatives to determine and use to compute initial conditions at the origin. Third, determining the initial conditions for the analogous ODE will be accomplished with the aide of a significant new result, namely the generalized -Wallis formulas in Theorem 46. This theorem is also crucial in our proof of convergence of the normalized solution of the MADE to the solution of the analogous ODE. The normalization will be our main object of study as the parameter . From (425), the parameters and show our example to be from [, ] in Section 6.
7.1. Preliminaries
For , the reproducing kernel computation relevant to our setting is
Moving from (413) to (414) is accomplished via Plancherel’s theorem where the expressions for the Fourier transform of are given by Theorem 23. The resulting integral of Fourier transforms is evaluated with a residue computation in the upper half-plane (for ) similar in nature to the computation of reproducing kernels for in [4]. This computation is lengthy, and the details are left as part of a more general set of reproducing kernel computations in an upcoming work. A direct computation moves one from line (414) to (415), while equation (132) of Definition 16 gives for that which justifies movement from (415) to (416).
Setting in (413)–(416) yields which in turn gives the functional identity
Normalizing (413)–(416) by the squared norm and applying the functional identity in (419), one observes that
Recalling that is real valued, and applying Cauchy-Schwartz to (420) yields which gives a unit global bound on the normalization independent of for all . Now, equation (132) of Definition 16 also gives for that where comparison of (422) with (417) allows movement to (423). So is an even Schwartz function and consequently the bound in (421) extends to a global bound: for all and for each , one has
From (415) and (416), one sees that satisfies the same MADE as does , where and . From these values, we deduce that
by which one sees that . From these parameter values, one sees that falls in the example class [] with (356) holding. Since , we conclude from Proposition 43 that is not a canonical extension of . See also Table 1 where fall in the category of row , and . From (2), one determines the MADE for to be
And hence,
while
Each of (426)–(428) we take to be our MADE under study for this example. We take the classical analogue of our MADE to be the ODE obtained by setting in the original MADE. In our case, the classical analogue of (426)–(428) in this example is
Next, we turn to the computation of our derivatives, which will in turn lead to the initial conditions for our MADE and (by taking the limit as ) the initial conditions of our analogous ODE (429). Relying on (417) and the fact that
(which can be directly computed from (1)), one sees that
When , we have a direct check that the MADE given in (427) holds, after noting that which follows from (441) below with . Evaluating (431) at yields where we have used the fact that
to move to (432). Note that the last equality in (433) follows from (10).
Setting with in (432) and relying on (12), it follows that
From (434), we see that for the theta factor by (13). Also from (414), for one has . We conclude that for odd. Setting in (434) gives that . Thus, the derivatives of all orders of the normalized function at are expressed via where moving to the case in (437) is facilitated by the fact that which is obtained from the right hand equation in (12) when is replaced by and is replaced by . We remark that, from (430) and (433), the derivatives of all orders of all normalized (and functions constructed from the such as ) at are expressible in terms of ratios of theta functions, as is seen overtly in (436) above for the normalized .
We next determine the derivatives of all orders for the ODE in (429) analogous to our MADE in (428) by taking the limits as of the derivatives obtained in (437). That is where we have relied on the power of the generalized -Wallis formulas for ratios of theta functions obtained in Section 7.3 and given by (470), (482), and (485) below to evaluate the limit in the case of (438) above. More precisely, by (482) in the case at hand, we have
One final observation regarding higher order derivatives of order follows. From (430), one has
7.2. Convergence
We now are prepared to prove that satisfying the MADE converges as to the solution of the analogous ODE having initial conditions given by (439) with . That is, the initial conditions are given by
The unique ODE solution satisfying (442) is given by , by inspection. Convergence of to will be in the sense of uniform convergence on all compact subsets of . The proof will hinge on three factors (for sufficiently large): (1) proximity of to the Taylor polynomial on ; (2) proximity of to the Taylor polynomial on ; and (3) proximity of to on . These in turn then force proximity of to .
Proposition 45. For any compact set in , converges uniformly to on as .
Proof. First, the -degree Taylor polynomials of and , respectively, expanded about are given by where (443) follows from (439), and (444)–(446) follow from (437). These have respective remainder terms for some between and , and for some between and , where (448) follows from (441). Given a compact set in there is a such that , and thus it is sufficient to prove uniform convergence on . For , from the triangle inequality one has Now for and relying on (448), one sees where moving from (450) to (451) is obtained from the global bound (424). Similarly, from (447), one has as . Also, from (443) and (444)–(446) and for , we have which, after rearranging and factoring, equals Applying (451), (454), and (452), respectively, to the terms in (449), one has that for each Given , choose sufficiently large such that one has to begin bounding the second term in (456). To handle the terms in (455), note that and pick so that Then, for , one has Hence, for and where (462)–(463) control the terms in (455) as well as the last term in (456). To handle the remaining term in (456), pick with such that for all one has which follows from the fact that via (440) above and/or the generalized -Wallis formula (482) below (in the case that and ). Then for , it follows that Applying (462) and (463) along with (466) to (455) and (456) with taken to be one has that for for . So approaches uniformly on as , and the proposition is proven.
7.3. Generalized -Wallis Formulas
In [6], we have proven a -Wallis formula given by the first equality in (468) where the last two equalities of (468) are Wallis’ formula for . We now finish the paper by generalizing the above result in order to provide a number of related generalized -Wallis formulas.
Theorem 46. Let with , and let with . Then the following families of generalized -Wallis formulas hold:
Proof. The proof relies on the following factorization of : From (10) and (11), one has where we have reindexed replacing by in the denominator to obtain (472). Relying on the facts that and , one sees (since ) that giving (469), where (471) was used to obtain the first equality in (476). Now when (470) holds since (which in turn follows from (13)). For , one now has from (469) that giving (470). This finishes the proof of the theorem.
Remark 47. We call the above results generalized -Wallis formulas via the following reasoning. First, note that the left-most infinite product in (474) generalizes the infinite product in the Wallis formula for in (468). In particular note that when and the infinite product in (478) becomes the infinite product in (479), and the Wallis product for is then duplicated in (474)–(476). Second, note that the product in (473) is the -analogue of the product (478). Hence, we are introducing generalized -Wallis formulas.
Corollary 48. Let satisfy . Let with and with then
Proof. Relying on (12), one has giving (481), where the last equality follows from (469). If then (482) holds since then by (13). If on the other hand then from (481) giving (482). Observe that when and , then setting , , , and in (481) recovers the -Wallis limit in (468). This proves the corollary.
We point out that in the case of rational exponents can be evaluated by obtaining a common denominator for the fractions in the exponents and reexpressing them as and to obtain:
Corollary 49. In the case of rational exponents where and and with and .
Proof. One applies Corollary 48 with and with and to obtain the last equation in (485). This proves the corollary.
We point out that, in our setting, important and useful applications of these generalized -Wallis formulas (especially those of type (470), (482), and (485)) occur in proving (1)initial conditions for a classical ODE analogous to a given MADE as seen earlier in (439) and (440)(2)convergence of MADE solutions to their classical analogous ODE as seen in (464) and (465) in Proposition 45
Data Availability
There are no further data beyond that submitted in the paper.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this article.
Acknowledgments
This work was supported by the ECU Thomas Harriot College of Arts and Sciences, the Open Access Publishing Support Fund sponsored by ECU Libraries, and the ECU Mathematics Department.