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

and Applied Analysis 3

In [4], the th order -advanced spherical Bessel functions of the first kind   (; ) are introduced.Paralleling ( 6), one has that, for  > 1,  ∈ N 0 = N ∪ {0},   (; ) ≡ (−) where  Sin() is the -advanced sine function Since  Sin() is defined to be odd,  Sin()/ 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  Sin() and other related functions are developed in [4][5][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 S(R) and they are solutions to the multiplicatively advanced differential equation (MADE) (; ) + 2     (; ) −  ( + 1)  2   (; ) = − +3   (; ) , (10) 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  > 1.The inverse Fourier transforms of F −1 [  (; )]() are developed in [4] and given there as where the integral operator A  appearing in (11) and (12) acts on S(R) and is defined by In (11), one has that  is the Jacobi theta function (−1)/2 (14) for  > 1, where for  > 1.From [4], one has the -Wallis limit which relates   2 from (15) to  0, (0) from ( 16) asymptotically as  → 1 + : lim 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   () χ[−1,1] () yields In analogy to (19), we make the following definition.
See Figure 1 for graphical representations of P (; ).A main purpose of this paper is to study the functions P (; ) ∈ S(R).These -Legendre polynomials are Schwartz approximations to the truncated Legendre polynomials   () χ[−1,1] () ∉ S(R), as the next theorem shows.Theorem 2. The -Legendre polynomials are Schwartz functions and are expressible in terms of the Jacobi theta function as follows: where A  is as in (13).Furthermore, for each 1 ≤  < ∞, one has convergence in L  (R) norm In addition, P (; ) converges pointwise to   () χ[−1,1] () on R. For  ≥ 1, the P (; ) are wavelets.Finally, P (; ) 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  + 1 case of (20) and then one substitutes (11) into the result to give Examining (20), one has that for  ∈ N 0 the P (; ) are Schwartz from the fact that the F −1 [  (; )]() are Schwartz, which in turn follows from the fact that the   (; ) are Schwartz and that F −1 preserves the Schwartz property.Similarly, for  ∈ N, the fact that the P (; ) are wavelets follows from the fact that the F −1 [  (; )]() are wavelets, because the order of vanishing at  = 0 of F[F −1 [  (; )]] =   (; ) is  − 1 as is observed from (8) using Taylor's remainder theorem.See Theorem 8 for further discussion.The L  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 F −1 preserves evenness or oddness of a function, one sees from (20) that P (; ) is even in  when  is even and odd when  is odd.
To conclude this section, we mention some useful results here.First, from [4,6], the following bound holds on the reciprocal of ( 2 ;  2 ): for 1 <  <   .This bound will be especially useful in analyzing the decay rate of the functions of interest for  in the tails || ≥ 1 + .Second, there is also a -advanced cosine function From [4,5], we have the Fourier transforms ( 2 ;  2 ) . (29) F[  Cos()]() will be utilized to obtain Proposition 20, which in turn helps in yielding the uniform convergence results.

Recursive Relations for the P𝑛 (𝑞; 𝜔)
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 is the starting point in proving the following recursion relations.(34) Proof.The  = 0 case is handled directly.Namely, from ( 8), one has To obtain (34), one takes the inverse Fourier transform of (33), relying on the fact that One next utilizes the fact that to reexpress F −1 [ −1 (; )]() in (45), obtaining Multiplying (47) through by (−) −1 √2/ gives Relying on (20) from Definition 1 gives Relying on the facts that one applies the inverse Fourier transform F −1 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 Letting  = / with   = (1/)  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.

Vanishing of Moments for the P𝑛 (𝑞; 𝜔)
Let  ≥ 1.In light of ( 19), (3) can be rewritten as which tells us that the 0th through ( − 1)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  ≥ 1.The 0th through ( − 1)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 F[] vanishing at 0. From (20), one has immediately that, for all  ∈ N, where the last equality follows from (8).Now, the (−)  factor and the outer 1/ factor in (69) guarantee that the first  − 1 derivatives of F[ P (; )]() vanish at  = 0, after noting that the derivatives of ((1/)(/))   0 (; ) are bounded for all  ∈ N 0 .This gives (68).In contrast, it is shown in [4] that the th derivative of F[ P (; )]() does not vanish at  = 0.

A Reciprocal Symmetry for P0 (𝑞; 𝜔)
There is an interesting reciprocal symmetry satisfied by P0 (; ), and this will help produce a pointwise convergence result in Section 9.

Nearly Orthonormal Frames from the P𝑛 (𝑞; 𝜔)
As in [13], a countable set of functions The frame condition (81) is equivalent to We construct a frame from the P (; ) in the following manner.For each  ∈ N 0 = N ∪ {0} and  ∈ Z, let where Thus, for each 1 >  > 0, there is a  (,,) > 1 such that for all  with 1 <  ≤  (,,) Similarly, one bounds (92) from the above by bounding the last two terms in (92) via (94) as follows: where    is the Kronecker delta function.We conclude that {w , } is a nearly orthonormal frame.

Alternative Expressions for P𝑛 (𝑞; 𝜔)
The goal of this section is to provide alternative expressions for P (; ) 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  ≥ 1, 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  ≥ 0, the th order -Legendre polynomial is given by where  ,−2 is the coefficient of  −2 in the th degree Legendre polynomial   (), as given by (104) and (105).
Proof.Note that (109) is true in the  = 0 case as it is a tautology, and it has been shown to hold for 1 ≤  ≤ 4 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 P (; ) by ( 109) is that there are no nested integrals in (109), whereas the previous expressions (22) for higher order P (; ) involve nested integrals.Thus, we have gained computational efficiency.
Next, we obtain uniform convergence on closed sets not containing ±1.
Proof.Subtracting (180) from ( 117), one has The change of variables  = 1/ is made on the first integral in (182), and the algebraic identity  2 ( 2 ;  −2 ) = ( 2 ;  2 ) is used to obtain Now ( 183) is used to reexpress (182) as from which one sees that P0 (; 0) > 1 for all  > 1. Deploying the bound (27) within the integral in (185) gives where It follows from the -Wallis formula (17) that We now show that P0 (; 0) − 1 can be made arbitrarily small for all  > 1 sufficiently close to 1 + .In the light of (188), this is accomplished by first showing that the corresponding statement holds for the bracketed expression in (186).

Convergence in L 𝑝 (R)
We turn next to convergence in L  (R).Proof.We first handle the case  = 1, which, by boundedness of the functions under study, turns out to be sufficient to handle the remaining cases 1 <  < ∞.

Approximating P𝑛 (𝑞; 𝜔)
The goal of this section is to provide and analyze the two approximations for P (; ) given by ( 228) and (229) below.These two expressions are related to the exact expression (109) for P (; ), but they are computationally simpler approximations of P (; ).We will also show that as  → 1 + the difference of P (; ) with each of the approximations (228) and ( 229) is converging uniformly to 0 on compact sets of R.
The first expression, (228), is obtained by removing all delays in (109): The second expression, (229), is obtained by setting  = 1 in the summation for (228): P (; ) ≈ P0 (; )   () = P0 (; ) The convergence is as follows.Given any compact set  ⊂ R, there is an  0 with  ⊂ [− 0 ,  0 ] ⊂ R. Given such  and corresponding  0 and given  > 0, there is a q > 1 such that for all 1 <  < q one has where the maximum of V/( 2 ; V 2 ) occurring at 1 is shown in [4].From (26), we bound 1/( 2 ; 1 2 ) in (232) from the above to give 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  7 such that, for all 1 <  <

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 0 value at  = 0, because its inverse Fourier transform will be a wavelet solution to a corresponding MADE.(257) )  = 0, (equivalentlyF [ ()] (0) = 0) , ∫ ∞ −∞     F [ ()] ()     2 ||  < ∞.

1 ∑
refer to (249) as a MADE if it formally converges to (253) where deg () ≤ deg () as  → 0 (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 F −1 [] = f.Then, from (46) and (58), one obtains  (−  , −) f ()  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 â(, ) and b(, ) in the two variables  and  withâ (,   ) f () = [ b (,   ) f] (   ) .(255)Now, suppose that as  → 0 in the operators of (255) one obtains â (, ) → deg Â and deg B are not both −∞.(i.e., âN  ̸ = 0 or bN  ̸ = 0).Then, similar to the process for (253), one has (255) that formally converges to the equation âN â [ Nâ  f] () + Nâ −=0 â [   f] () = bM b [ Mb  f] ( [4][5][6][8][9][10][11][12]plicatively 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 P (; ) exactly analogous to those of the truncated   () in (3) are shown in (68).We obtain a reciprocal symmetry for P0 (; ) in (71).The P (; ) are used to generate a nearly orthonormal frame for L 2 (R) in Section 7. Alternative expressions for the P (; ) are obtained in Section 8.In Section 9, we obtain uniform convergence of the P (; ) to   () χ[−1,1] () as  → 1 + on all closed sets of R not containing ±1.This result combined with the reciprocal symmetry property then gives pointwise convergence of P (; ) to   () χ[−1,1] () on R. In Section 10, L  convergence of these functions is demonstrated.Approximations to the P (; ) 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][5][6][8][9][10][11][12].
Specific properties of the -Legendre polynomials P (; ) are established.First, we show that the P (; ) satisfy an analogue of the recursion relation (1), namely, (34) below.Next,
Proposition 14.The propositions in this subsection provide the estimates on which Proposition 14 is based.