Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms
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.
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  series:
Each of the in (1) satisfies the MADE. where with in a reduced form and ; see . 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 , 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  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  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.
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 ; T. Dreyfus [13, 14]; A. Lastra , [15–20], [21–23]; S. Malek , [15–20], [21–23], [24–27]; J. Sanz [21–23]; H. Tahara ; 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
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  and therefore analytic on .
We have seen in  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
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  or that is an integer and is in Theorem 6.5 of ) 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,
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.”
Proof. The proof is given in Lemma 5.1 of .
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
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.
The previous discussion allows us to reach the following conclusion:
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.
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