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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 890456, 24 pages
http://dx.doi.org/10.1155/2014/890456
Research Article

Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

Mathematics Department, East Carolina University, Greenville, NC 27858, USA

Received 7 March 2014; Accepted 14 July 2014; Published 16 October 2014

Academic Editor: Cristina Pignotti

Copyright © 2014 David W. Pravica et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The family of nth order q-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by the nth degree Legendre polynomials. The nth order q-Legendre polynomials are shown to have vanishing kth moments for , as does the nth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

1. Introduction

The Legendre polynomials of degree can be obtained by starting with , and then relying on the recursive relation for to obtain Legendre polynomials of higher degree. For instance, take in (1) and solve for to find . Proceeding with (1), one obtains the remaining .

The have many interesting properties. In addition to satisfying (1), the satisfy Legendre’s ordinary differential equation where denotes differentiation in the variable ; see [1]. Furthermore, the restricted to the interval form an orthogonal complete set for the square integrable functions in the norm . The can also be constructed by a Gram-Schmidt orthogonalization on the polynomials on the interval scaling so that , as required by (1). This implies the vanishing of the following moments: on for ; see [2]. From [3], we also have that where denotes the inverse Fourier transform where denotes the th spherical Bessel function of the first kind where, for denoting the characteristic function of the set , we have We refer to the in (4) as the truncated th degree Legendre polynomials.

In [4], the th order -advanced spherical Bessel functions of the first kind are introduced. Paralleling (6), one has that, for ,  , where is the -advanced sine function Since is defined to be odd, is then even, and (8) reveals to be even in when the order is even and odd when the order is odd. Many further interesting properties of and and other related functions are developed in [46], which are good background references. For our purposes here we only note a few facts about the . First, the belong to the class of Schwartz functions and they are solutions to the multiplicatively advanced differential equation (MADE) as is proven in [4]. Note that (10) is a MADE from the fact that the argument in the right-hand side of (10) is a multiple of by . The inverse Fourier transforms of are developed in [4] and given there as where the integral operator appearing in (11) and (12) acts on and is defined by In (11), one has that is the Jacobi theta function for , where for . From [4], one has the -Wallis limit which relates from (15) to from (16) asymptotically as :

Since most of the functions studied here will exhibit wavelet properties, we mention that function is considered to be a wavelet if See [7] for further background on wavelets.

Solving (4) for yields In analogy to (19), we make the following definition.

Definition 1. For and , the th order -Legendre polynomials are given by

See Figure 1 for graphical representations of . A main purpose of this paper is to study the functions . These -Legendre polynomials are Schwartz approximations to the truncated Legendre polynomials , as the next theorem shows.

fig1
Figure 1: (a) is shown in solid for and is compared with which is dashed. (b) is shown in solid for and is compared with which is dashed. (c) is shown in solid for and is compared with which is dashed.

Theorem 2. The -Legendre polynomials are Schwartz functions and are expressible in terms of the Jacobi theta function as follows: where is as in (13). Furthermore, for each , one has convergence in norm In addition, converges pointwise to on . For , the are wavelets. Finally, is even in for even and odd for odd.

Proof. To obtain the 0th order case (21), one substitutes (11) into (20). To obtain the higher order cases (22), one substitutes (12) into the case of (20) and then one substitutes (11) into the result to give
Examining (20), one has that for the are Schwartz from the fact that the are Schwartz, which in turn follows from the fact that the are Schwartz and that preserves the Schwartz property. Similarly, for , the fact that the are wavelets follows from the fact that the are wavelets, because the order of vanishing at of is as is observed from (8) using Taylor’s remainder theorem. See Theorem 8 for further discussion. The convergence in (23) follows from Theorem 21 in Section 10 below. Pointwise convergence follows from Theorem 18 in Section 9 below. Finally, since the remarks following (8) give as even in when is even and odd when is odd, and since preserves evenness or oddness of a function, one sees from (20) that is even in when is even and odd when is odd.

Remark 3. Equation (22) in the case reduces to (21) if one interprets . Also, Theorem 2 gives genesis to the title of this paper.

To conclude this section, we mention some useful results here. First, from [4, 6], the following bound holds on the reciprocal of : for . This bound will be especially useful in analyzing the decay rate of the functions of interest for in the tails .

Second, there is also a -advanced cosine function From [4, 5], we have the Fourier transforms will be utilized to obtain Proposition 20, which in turn helps in yielding the uniform convergence results.

2. Main Results

Specific properties of the -Legendre polynomials are established. First, we show that the satisfy an analogue of the recursion relation (1), namely, (34) below. Next, also satisfy a multiplicatively advanced analogue of the Legendre ordinary differential equation (2), namely, the -Legendre multiplicatively advanced differential equation (MADE) given by (56) below. Moment vanishing properties of the exactly analogous to those of the truncated in (3) are shown in (68). We obtain a reciprocal symmetry for in (71). The are used to generate a nearly orthonormal frame for in Section 7. Alternative expressions for the are obtained in Section 8. In Section 9, we obtain uniform convergence of the to as on all closed sets of not containing . This result combined with the reciprocal symmetry property then gives pointwise convergence of to on . In Section 10, convergence of these functions is demonstrated. Approximations to the are provided in Section 11. Finally, as encountered in the process of showing the -Legendre MADE, we give a more general condition under which a MADE remains a MADE under inverse Fourier transform. This is used to provide new wavelet solutions of MADEs.

It is worth mentioning that the study of MADEs and related topics has seen recent growth. See, for instance, contributions from [46, 812].

3. Recursive Relations for the

In this section, we obtain a -version of the recursion formula (1) for -Legendre polynomials, namely, (34) below. This follows from a recursion relation on the given by (33). We begin with a lemma describing the derivative of .

Lemma 4. For and , one has

Proof. First, recall that and take . Then, is well defined. Differentiation now yields giving the lemma.

Lemma 4 is the starting point in proving the following recursion relations.

Theorem 5. For , Or, equivalently,

Proof. The case is handled directly. Namely, from (8), one has giving (33) for . Taking inverse Fourier transforms of both sides of (35) and multiplying the resulting equation by yield which is (34) for . Thus, we assume from this point on and begin first by showing (33). Setting the index equal to in (30) and solving the result for yield From (10), with the index set to be , one obtains Differentiating (37) yields Replacing the second derivative term in (39) with that in (38) yields where (37) was used to replace the bracketed expression in (41) and obtain (42). Continuing yields where (30) was used to replace the bracketed expression in (43) and obtain (44). Multiplying (43) and (44) through by and combining terms gives (33). Note the multiplicative advance in the argument of the term in (33) and (44).
To obtain (34), one takes the inverse Fourier transform of (33), relying on the fact that to obtain One next utilizes the fact that to reexpress in (45), obtaining Multiplying (47) through by gives Relying on (20) from Definition 1 gives Solving for gives (34) and finishes the proof. Note the multiplicative delay in the argument of the term in (34).

As is done at the beginning of the paper for the Legendre polynomials, we utilize the new recursion relation (34) to generate the first few -Legendre polynomials. Observe that is given directly by (21). Next, from (22), with set to , we obtain From (34), with , one obtains From (34), with , one obtains From (34), with , one obtains Proceeding on, one obtains the general th order -Legendre polynomial by multiplying each term of the th degree Legendre polynomial by a power of and by a multiplicative delay of by a power of and then summing. The expression extending (50)–(55) to general will be given in Theorem 11 and Corollary 13 in Section 8 below.

4. MADEs for the

In this section the -Legendre polynomials are shown to satisfy a -version of Legendre’s ODE (2), namely, the multiplicatively advanced differential equation (MADE) given by (56) below.

Theorem 6. The -Legendre polynomial satisfies the multiplicatively advanced differential equation This reduces to the Legendre differential equation (2) as with for .

Proof. Multiplying (10) by yields Relying on the facts that one applies the inverse Fourier transform to (57) to obtain Simplifying the left-hand side of (59) and relying on (46) to simplify the right-hand side of (59) yields Using the derivation property of on the left-hand side of (61) gives Simplifying (62)-(63) and multiplying through by yield Letting with yields Thus, we solve for the right-hand side of (65) scaled by to obtain which simplifies to (56) after a final substitution . The theorem is now proven.

Remark 7. Equation (56) is appropriately considered to be a MADE over the apparent delayed differential equation (64) in that the term with the highest order derivative with constant coefficient should be the dominant term for small and thus expressed in terms of the unscaled variable. This will be further addressed in Section 12.

5. Vanishing of Moments for the

Let . In light of (19), (3) can be rewritten as which tells us that the 0th through th moments of the truncated th degree Legendre polynomial vanish. In light of (20) in Definition 1, the statement analogous to (67) is given by (68) in the next theorem.

Theorem 8. Let . The 0th through th moments of the th order -Legendre polynomial vanish. Consider

The proof is outlined here. Recall that the th moment of vanishing is equivalent to the th derivative of vanishing at . From (20), one has immediately that, for all , where the last equality follows from (8). Now, the factor and the outer factor in (69) guarantee that the first derivatives of vanish at , after noting that the derivatives of are bounded for all . This gives (68). In contrast, it is shown in [4] that the th derivative of does not vanish at .

6. A Reciprocal Symmetry for

There is an interesting reciprocal symmetry satisfied by , and this will help produce a pointwise convergence result in Section 9.

Theorem 9. For all . one has or equivalently

Proof. From (21), we have, for , Here, the change of variables was used to obtain (74), the identity was used to obtain (76), the integrand in (76) was multiplied by to obtain (77), the identity was used to obtain (78), the change of variables was used to obtain (79), and (21) was used to obtain (80) above. Thus, (70) holds for . By replacing by in the expression (70), one obtains (71) for . Note that, by evenness of , one has that both the identities (70) and (71) hold for all .

7. Nearly Orthonormal Frames from the

As in [13], a countable set of functions is a frame for if there are constants with The frame condition (81) is equivalent to We construct a frame from the in the following manner. For each and , let where normalizes to give , see [1].

From (23), it follows that Thus, for each , there is a such that for all with For conciseness, by suppressing and , set

Theorem 10. Given , the associated set of functions , with , as in (87), is a nearly orthonormal frame for in the sense that where the Kronecker delta function satisfies for and .

Proof. We start by noting that This follows since the are orthonormal and complete on , as in [2]. Thus the translates of by multiples , namely the , are orthonormal and complete in . Let be given. We bootstrap on the fact that is an orthonormal frame to show that smooth approximations given by (87) are also a frame for . For all functions , one computes that
Now, one uses Cauchy-Schwarz and (86) to obtain the bound Thus, one can bound (90) from below by discarding the last term of (92) and relying on (94) to obtain (96) as follows: Similarly, one bounds (92) from the above by bounding the last two terms in (92) via (94) as follows: Thus, combining (98) with (100) yields that, for all functions , Hence, given , the associated set is a frame for .
Next, we show near orthonormality of the frame . Observe that, given , one has Thus, where is the Kronecker delta function. We conclude that is a nearly orthonormal frame.

8. Alternative Expressions for

The goal of this section is to provide alternative expressions for that extend equations (50)–(55). This will be done in Theorem 11 and Corollary 13 below. We obtain this extension by consulting [2] and expressing the th degree Legendre polynomial as where and denotes the greatest integer function. For , the recursion relation (1) in this notation takes the form after reindexing in the rightmost summation in (107) to obtain (108). This implies a recursion relation on the coefficients of like powers of obtained in setting (106) equal to (108).

We are now prepared to state the next theorem generalizing (50)–(55).

Theorem 11. For , the th order -Legendre polynomial is given by where is the coefficient of in the th degree Legendre polynomial , as given by (104) and (105).

Proof. Note that (109) is true in the case as it is a tautology, and it has been shown to hold for via (50)–(55). Assume that (109) has been established up through order . Then, the recursion relation (34) expressed in terms of (109) gives that