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

David W. Pravica, Njinasoa Randriampiry, and Michael J. Spurr

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 [4–6], 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.

(a)

(b)

(c)

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 [4–6, 8–12].

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

where consolidating powers of gives (111), a reindexing on the subtracted summation in (111) gives (113), and the recursion relation obtained from setting (106) equal to (108) gives (114). Thus, (109) holds by induction.

Remark 12. The utility of representing by (109) is that there are no nested integrals in (109), whereas the previous expressions (22) for higher order involve nested integrals. Thus, we have gained computational efficiency.

Replacing each in (109) by the corresponding integral expression in (21) yields the following.

Corollary 13. For , the nth order -Legendre polynomial is given by

9. Convergence Results for

On closed sets not containing , one has uniform convergence of to as . We will see that, via expression (109), one can obtain this result by relying on the fact that converges uniformly to away from . These results, coupled with an application of the reciprocal symmetry (71), will let us obtain pointwise convergence on all of .

9.1. Uniform Convergence Away from

First, we obtain a uniform convergence for on subsets of the form .

Proposition 14. Given , one has that converges uniformly to on the interval as .

Proof. From (21), it follows that

from which we see that is increasing on and decreasing on with a maximum at of

From (21), along with the oddness of , we have that is even and . Thus,

for all . Furthermore, on the interval , the function assumes its maximum value at an endpoint or .
From Proposition 19, we have that, given , there is a such that for all

From Proposition 20, we have that, given , there is a with

for all . From the increasing property of on , we have

for all with . By evenness, (121) also gives uniform convergence on the interval , and the proposition is proven.

Next, we obtain uniform convergence on closed sets not containing .

Theorem 15. Given with , let . Then converges uniformly to on as .

Proof. The proof breaks into two parts: (1) uniform convergence on the middle, that is on , and (2) uniform convergence on the tails, that is on .
We first handle the middle case. From Proposition 14, converges uniformly to on the interval as . From (104) and (105), one has that on

Thus, given with , there is such that

for all with and for all with . Then, for each and , one has , given that

for all . Thus, for and , by relying on (109) and (104), we have

When

we see that (126) is less than . However, (127) holds when

which is true for all with

Thus, given , for all with from (126) and (127), one has

for all . Thus, uniform convergence is obtained on , and part 1 of the proof is shown.
We proceed to part 2 of the proof, uniform convergence in the tails . Let be given. Then, for , one has

where (132) follows from (143) of Proposition 17, with , and holds for and with

as obtained in (154).
From (17), one has

from which it follows via (136) that there is a with such that for all

for all . Since

there is a with such that

for all and all . Thus, since , from (137), we have uniform convergence in the tails, finishing the proof of part 2 and concluding the proof of the theorem.

Remark 16. The decay expressed in (134) gives a stronger result than uniform convergence, as is shown in Section 10.

The following proposition was utilized in showing part 2 of Theorem 15.

Proposition 17. Given and given , then for each with there exists such that

for all and all .

Proof. From (21), one has

Here, (146) follows from oddness of the integrand . Then, (147) follows from the bound (27). Also, (148) follows from the bound

for , from [4, 14], with , , and . Finally, (150) follows by a completion of squares. Note that the requirement that in (151) becomes or in (148). However, in order for (150) to be decreasing in , we require the slightly stronger condition that

or equivalently

Now, given , and as in the hypotheses of the proposition, first choose with and then choose with so that

Then, for and , one has that (153) holds, from which (150) is decreasing in . This then gives (143) and (144) and completes the proof of the proposition.

9.2. Pointwise Convergence on

It is now possible to give a pointwise convergence result.

Theorem 18. converges pointwise to as .

Proof. By virtue of Theorem 15, one automatically has pointwise convergence for . Thus, we concentrate on to obtain the result. Observe first that for any one has

This follows from the estimate

where (27) was used to obtain (157) and where the integrand in (157) being at most was used to obtain (158). Now, by L’Hopital’s rule, one has, for each , that

By the -Wallis limit (17), one also has

Applying (159) and (160) to (158) gives (155).
Next, observe from the reciprocal identity (71) with set equal to , that

from which one obtains

Proposition 20 gives that . Applying this to (162) gives

Setting in (155) gives

Letting be arbitrary in (155) gives

for all .
Next, from (109), we have

From (165), one has that . Taking limits of (166) with this in mind yields

where (168) follows from (165) and (169) follows from (104). By evenness/oddness, respectively, one has

Equations (169) and (170) give pointwise convergence at and the theorem is proven.

9.3. Estimates Giving Proposition 14

The propositions in this subsection provide the estimates on which Proposition 14 is based.

Proposition 19. For , given , there is a such that

for all .

Proof.

By virtue of Proposition 20 below, which bounds , we only need bound the first term, , in the right-hand side of inequality (172).
Evenness of and the setting of in the reciprocal identity (70) together give

for all . Now, since and is decreasing for , we have that , and this applied to (173) yields

Next, pick with , and apply Proposition 17 with and to obtain

By (17), one has

and one sees directly that

Thus, the right-hand side of (175) approaches as . So, from (175) and (174), given , there is a such that, for all ,

This gives the proposition.

Next, by (29), we have

from which one obtains that

Proposition 20. For each , there exists a such that for all

Proof. Subtracting (180) from (117), one has

The change of variables is made on the first integral in (182), and the algebraic identity is used to obtain

Now (183) is used to reexpress (182) as

from which one sees that for all . Deploying the bound (27) within the integral in (185) gives

where

It follows from the -Wallis formula (17) that

We now show that can be made arbitrarily small for all sufficiently close to . In the light of (188), this is accomplished by first showing that the corresponding statement holds for the bracketed expression in (186).
Let be arbitrary, with being specified later. The integral in over the interval in (186) is now subdivided into two integrals, the first over and the second over . First, on , one computes

Now, the function is increasing on , and it assumes its maximum value of at the right endpoint . Thus, we bound the integral in (189) by the length of the interval times the bound on the numerator. This gives

An application of L’Hopital’s rule gives that

which implies that in (191) we have

Combining (193) with (191) gives that

We next estimate the portion of the integral in (186) over the interval . First, since on this interval, one has

We next bound the right-hand side of (195), using the following estimate from [4, 14]:

which holds for . Setting , , and as in (195) yields that

for , or equivalently for . Since the final expression in (197) approaches as , by choosing sufficiently large one can make (197) and hence (195), arbitrarily small for and sufficiently close to .
Now, let be given. By (186), we have for all

One has from (188) and (197) that

Fix such that . By (197) and (200), there exists such that for all one has

For this value of , by virtue of (194), one has

which in turn says that there is a such that for all one has

Thus, for and for all , applying (204) and (202) to (198) and (199) yields

This gives the proposition.

10. Convergence in

We turn next to convergence in .

Theorem 21. converges to in as for each . That is,

Proof. We first handle the case , which, by boundedness of the functions under study, turns out to be sufficient to handle the remaining cases .
Let be given. First note that, by (118) and (120), there is a such that

for all . Thus, is uniformly bounded for . Next, note that, for with and , one has from (109) that

where . Observe that, since , , and are all greater than in (209), one has . Note also that, from (141), one has the existence of a such that for all

for all such that . From (209) and (210) and the fact that , one has that, for , we have

for all . Thus, is uniformly bounded in . Letting , we also have that . Thus, our two functions of interest are bounded on for .
Next, choose with . Then, by oddness/evenness, respectively, one has

and we proceed to bound the three resulting integrals in (213) and (214).
From (130), there is a such that one obtains for all and . This gives

for all .
Next, by choice of , one has

for all .
Finally, from (134), one has the existence of a such that

for all . Now, from (151), one has

for . Thus, one uses (221) to bound (219) for , obtaining

Now, from (17), we have that (223) approaches as . Thus, there is a such that, for ,

Finally, using (215), (217), and (224) to bound (213) and (214) gives that

for all . Thus, (206) holds in the case .
Next, observe that if and are both bounded on , then, for , one has

Thus, in this bounded setting, convergence is sufficient for convergence.
We have observed that is bounded by for . Also . By the above remarks, it follows that (206) now also holds in the cases , and the theorem is proven.

11. Approximating

The goal of this section is to provide and analyze the two approximations for given by (228) and (229) below. These two expressions are related to the exact expression (109) for , but they are computationally simpler approximations of . We will also show that as the difference of with each of the approximations (228) and (229) is converging uniformly to on compact sets of .

The first expression, (228), is obtained by removing all delays in (109):

The second expression, (229), is obtained by setting in the summation for (228):

The convergence is as follows. Given any compact set , there is an with . Given such and corresponding and given , there is a such that for all one has

for all and therefore for all . Note first that, for there is equality in (228) and (229) for all and all . So, we assume that and proceed as follows.

To obtain convergence for (228), note first that, for each , one has

where the maximum of occurring at is shown in [4]. From (26), we bound in (232) from the above to give

for all with .

From L’Hopital’s rule, one has that

which combines with (138) to imply that the coefficients in (234) have the following limit:

Thus, from (234) and (236), one has that there is a such that, for all and all ,

Thus, for , one has

for all and all .

Now, from (109) coupled with (238), one has that

for all and all . Placing (240) below and solving the resulting inequality for gives that, given , there is a given by

such that

for all and . This gives convergence for (228).

We turn next to (229). Observe that, for , one has

where (244) follows from (118) and (245) holds for with chosen as in Proposition 20 (for set equal to ). Placing (245) below and solving the resulting inequality for gives that, given , there is a , given by

such that

for and . Thus, by setting in (242) and (247) and relying on the triangle inequality, one has

for and . This gives convergence for (229). Now let and the result holds.

We remark here that in (229) is a smooth Schwartz analogue of the truncation function in (19). Note also that the approximations (228) and (229) need not be wavelets, while each for is a wavelet. Furthermore, neither approximation (228) nor (229) satisfies any naturally apparent differential equation. Thus, one may be giving up useful properties if one relies on the somewhat simpler approximations (228) or (229) in place of (109).

12. Remaining a MADE under Inverse Fourier Transform

In this section, the goal is to give a generalization of the techniques used in Section 4. This generalization gives a condition on an original MADE sufficient to conclude that its resulting inverse Fourier transform remains a MADE. This motivates the search for a global solution for a MADE with value at , because its inverse Fourier transform will be a wavelet solution to a corresponding MADE.

Consider a differential equation of the form

Here and are polynomials in the two variables and as in (250). More explicitly, (249) is given in expanded form by

Now, suppose that there are nonnegative integers and so that as

where the degrees and are not both (i.e., or ). Then, letting in the operators in (249) (but not in the arguments for the function ), one obtains or equivalently . That is, as in the operators, (249) and (250) formally converge to the equation

which is equivalent to

We refer to (249) as a MADE if it formally converges to (253) where as (i.e., ). In other words, (249) is said to be a MADE when the terms with the highest order derivative having constant coefficients in (249) have one term with argument of not multiplicatively advanced or delayed by and some lower or equal order term with constant coefficient does contain an argument of form .

Next, take the inverse Fourier transform of (250) and denote . Then, from (46) and (58), one obtains

where the polynomials and are identical to those in (249). Now, using the derivation property of to reorder the terms to precede the terms in (254), collecting powers of , and absorbing powers of into coefficients, one obtains two resulting polynomials and in the two variables and with

Now, suppose that as in the operators of (255) one obtains

where and are not both . (i.e., or ). Then, similar to the process for (253), one has (255) that formally converges to the equation

Now, when (i.e., ), (257) will not have its highest order term with constant coefficients satisfying that its argument is neither advanced nor delayed by . Thus, when (i.e., ), we change variables so that to give

yielding a MADE. Similarly, one now relies on the change of variable to convert (255) into a MADE. Thus, (249) starts as a MADE when (i.e., ) and converts to a MADE under inverse Fourier transform when (i.e., ).

Example 22. Equation (57) in the proof of Theorem 6 is

which can be rewritten as

where

Note that substituting into in (260) gives in (259). For the case , taking limits as gives

Thus, (259) is appropriately considered a MADE as . That is the highest order term in (259) with constant coefficients, namely, , does not have its argument advanced or delayed by .
Taking inverse Fourier transforms of both sides of (259), relying on (46) and (58), and multiplying through by yields (64); namely,

This is equivalent to

where

Taking limits as gives

with . Since acts on a term with a delayed argument in (264), one makes the substitution in (263) to obtain the MADE equivalent to (56)

Example 23. For , consider the MADE

Equation (268) is appropriately considered a MADE as follows. After using derivation properties to have the variables precede the operators, (268) becomes

With the corresponding polynomials

Letting yields and . Since , (269) and (268) are MADEs.
Taking inverse Fourier transform of (268) and relying on (46) and (58), one obtains

which simplifies to

After using derivation properties to have the variables precede the operators, (272) becomes

with the corresponding polynomials

Letting yields and . Since , we convert (273) by letting , which results in placing our equation in MADE form. For our purposes here, it will be more convenient to make this substitution in (272) rather than (273). Thus, since the highest order term with constant coefficients is , we make the substitution with to reexpress (272) as the MADE

which simplifies to the following equation:

We exhibit a function with a solution of (272) (and with a solution of (276) as well) satisfying the condition that . Set

To verify (272), one proceeds as follows:

We have relied on the facts that

where is given by (9) and is given by (28). These facts are proven in [5].
Note that since is Schwartz, one has . And thus the solution to the MADE (268) given by

exhibits wavelet properties, since its 0th moment vanishes. A change of variables was made to obtain the last two equalities and express the solution in terms of the 0th order -advanced spherical Bessel function.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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