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

Authors: David W. Pravica; Njinasoa Randriampiry; Michael J. Spurr
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The family ofnth 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 0≤k<n, 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.


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## Body

## 1. Introduction

The Legendre polynomialsPn(ω) of degree n can be obtained by starting with P0(ω)=1, P1(ω)=ω and then relying on the recursive relation
(1)(n+1)Pn+1(ω)=(2n+1)ωPn(ω)-nPn-1(ω)
for n≥1 to obtain Legendre polynomials of higher degree. For instance, take n=1 in (1) and solve for P2(ω) to find P2(ω)=(3/2)ωP1(ω)-(1/2)P0(ω)=(3/2)ω2-(1/2). Proceeding with (1), one obtains the remaining Pn(ω).ThePn(ω) have many interesting properties. In addition to satisfying (1), the Pn(ω) satisfy Legendre’s ordinary differential equation
(2)Dω2Pn(ω)=ω2Dω2Pn(ω)+2ωDωPn(ω)-n(n+1)Pn(ω),
where Dω denotes differentiation in the variable ω; see [1]. Furthermore, the Pn(ω) restricted to the interval [-1,1] form an orthogonal complete set for the square integrable functions L2([-1,1]) in the norm ∥·∥2,[-1,1]. The Pn(ω) can also be constructed by a Gram-Schmidt orthogonalization on the polynomials {1,ω,ω2,…,ωk,…} on the interval [-1,1] scaling so that Pn(1)=1, as required by (1). This implies the vanishing of the following moments:
(3)∫-11‍ωkPn(ω)dω=0for0≤k<n
on [-1,1] for n≥1; see [2]. From [3], we also have that
(4)inπ2Pn(ω)χ~[-1,1](ω)=12π∫-∞∞‍eiωtjn(t)dt=F-1[jn(t)](ω),
where F-1 denotes the inverse Fourier transform
(5)F-1[f(t)](ω)=12π∫-∞∞‍eiωtf(t)dt,
where jn(t) denotes the nth spherical Bessel function of the first kind
(6)jn(t)≡(-t)n(1tddt)nsin⁡(t)t,
where, for χΩ(ω) denoting the characteristic function of the set Ω, we have
(7)χ~[-1,1](ω)=12[χ[-1,1](ω)+χ(-1,1)(ω)]={0,ω∉[-1,1],1,ω∈(-1,1),12,ω∈∂[-1,1].
We refer to the Pn(ω)χ~[-1,1](ω) in (4) as the truncated nth degree Legendre polynomials.In [4], the nth order q-advanced spherical Bessel functions of the first kind jn(q;t) are introduced. Paralleling (6), one has that, for q>1,  n∈N0=N∪{0},
(8)jn(q;t)≡(-t)n(1tddt)nSin⁡q(t)t=(-t)n(1tddt)nj0(q;t),
where Sin⁡q(t) is the q-advanced sine function
(9)Sin⁡q(t)≡sign⁡(t)∑M=-∞∞‍(-1)Me-qM|t|q-M(M-1)∑M=-∞∞‍(-1)Mq-M2.
Since Sin⁡q(t) is defined to be odd, Sin⁡q(t)/t is then even, and (8) reveals jn(q;t) to be even in t when the order n is even and odd when the order n is odd. Many further interesting properties of jn(q;t) and Sin⁡q(t) 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 jn(q;t). First, the jn(q;t) belong to the class of Schwartz functions S(R) and they are solutions to the multiplicatively advanced differential equation (MADE)
(10)jn′′(q;t)+2tjn′(q;t)-n(n+1)t2jn(q;t)=-qn+3jn(q;qt),
as is proven in [4]. Note that (10) is a MADE from the fact that the argument qt in the right-hand side of (10) is a multiple of t by q>1. The inverse Fourier transforms of F-1[jn(q;t)](ω) are developed in [4] and given there as
(11)F-1[j0(q;t)](ω)=-2(μq2)3f0,q(0)2πAω(ωθ(q2;ω2)),(12)F-1[jn+1(q;t)](ω)=in+1×Dωn{ω(Aωω)nF-1[j0(q;t)](ω)},
where the integral operator Aω appearing in (11) and (12) acts on S(R) and is defined by
(13)Aωf(ω)≡∫-∞ω‍f(v)dv.
In (11), one has that θ is the Jacobi theta function
(14)θ(Q;W)≡μQ∏n=0∞‍(1+WQn)(1+1WQn+1)=∑n=-∞∞‍WnQn(n-1)/2
for Q>1, where
(15)μQ≡∏n=0∞‍(1-1Qn+1)>0,(16)f0,q(0)≡∑M=-∞∞(-1)Mq-M2=θ(q2;-1q)=μq2∏n≥0‍(1-1q2n+1)2>0
for q>1. From [4], one has the q-Wallis limit which relates μq2 from (15) to f0,q(0) from (16) asymptotically as q→1+:
(17)lim⁡q→1+ln⁡(q)(μq2)3f0,q(0)=π2.Since most of the functions studied here will exhibit wavelet properties, we mention that functionϕ is considered to be a wavelet if
(18)ϕ∈L1(R)∩L2(R)∩L∞(R),∫-∞∞‍ϕ(t)dt=0,(equivalentlyF[ϕ(t)](0)=0),∫-∞∞‍|F[ϕ(t)](ω)|2|ω|dω<∞.
See [7] for further background on wavelets.Solving (4) for Pn(ω)χ~[-1,1](ω) yields
(19)Pn(ω)χ~[-1,1](ω)=(-i)n2πF-1[jn(t)](ω).
In analogy to (19), we make the following definition.Definition 1.
Forn∈N0 and q>1, the nth order q-Legendre polynomials P~n(q;ω) are given by
(20)P~n(q;ω)≡(-i)n2πF-1[jn(q;t)](ω).See Figure1 for graphical representations of P~n(q;ω). A main purpose of this paper is to study the functions P~n(q;ω)∈S(R). These q-Legendre polynomials are Schwartz approximations to the truncated Legendre polynomials Pn(ω)χ~[-1,1](ω)∉S(R), as the next theorem shows.(a)P~0(q;ω) is shown in solid for q=1.20 and is compared with P0(ω)χ~[-1,1](ω) which is dashed. (b) P~1(q;ω) is shown in solid for q=1.20 and is compared with P1(ω)χ~[-1,1](ω) which is dashed. (c) P~2(q;ω) is shown in solid for q=1.20 and is compared with P2(ω)χ~[-1,1](ω) which is dashed.
(a)
(b)
(c)Theorem 2.
Theq-Legendre polynomials are Schwartz functions and are expressible in terms of the Jacobi theta function as follows:
(21)P~0(q;ω)=-2(μq2)3πf0,q(0)Aω(ωθ(q2;ω2)),(22)P~n+1(q;ω)=-2(μq2)3πf0,q(0)×Dωn{(ωAω)n+1(ωθ(q2;ω2))}forn≥0,
where Aω is as in (13). Furthermore, for each 1≤p<∞, one has convergence in Lp(R) norm
(23)0=lim⁡q→1+∥P~n(q;ω)-Pn(ω)χ~[-1,1](ω)∥p.
In addition, P~n(q;ω) converges pointwise to Pn(ω)χ~[-1,1](ω) on R. For n≥1, the P~n(q;ω) are wavelets. Finally, P~n(q;ω) is even in ω for n even and odd for n 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 n+1 case of (20) and then one substitutes (11) into the result to give
(24)P~n+1(q;ω)=(-i)n+12πF-1[jn+1(q;t)](ω)=(-i)n+12πin+1×Dωn{ω(Aωω)nF-1[j0(q;t)](ω)}(25)=2π-2(μq2)3f0,q(0)2π×Dωn{ω(Aωω)nAω(ωθ(q2;ω2))}.
Examining (20), one has that for n∈N0 the P~n(q;ω) are Schwartz from the fact that the F-1[jn(q;t)](ω) are Schwartz, which in turn follows from the fact that the jn(q;t) are Schwartz and that F-1 preserves the Schwartz property. Similarly, for n∈N, the fact that the P~n(q;ω) are wavelets follows from the fact that the F-1[jn(q;t)](ω) are wavelets, because the order of vanishing at t=0 of F[F-1[jn(q;t)]]=jn(q;t) is n-1 as is observed from (8) using Taylor’s remainder theorem. See Theorem 8 for further discussion. The Lp 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 jn(q;t) as even in t when n is even and odd when n is odd, and since F-1 preserves evenness or oddness of a function, one sees from (20) that P~n(q;ω) is even in ω when n is even and odd when n is odd.Remark 3.
Equation (22) in the n+1=0 case reduces to (21) if one interprets Dω-1=Aω. 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 θ(q2;ω2):
(26)1θ(q2;ω2)≤[q1/4|ω|e(ln⁡(|ω|))2/ln⁡(q){πln⁡(q)-1}]-1(27)=ln⁡(q)[1ln⁡(q)q-1/4|ω|-1e-(ln⁡(|ω|))2/ln⁡(q){π-ln⁡(q)}]
for 1<q<eπ. This bound will be especially useful in analyzing the decay rate of the functions of interest for ω in the tails |ω|≥1+ε.Second, there is also aq-advanced cosine function
(28)Cos⁡q(t)≡∑M=-∞∞‍(-1)Me-qM|t|q-M2∑M=-∞∞‍(-1)Mq-M2.
From [4, 5], we have the Fourier transforms
(29)F[Cos⁡q(t)](ω)=2π(μq2)3f0,q(0)1θ(q2;ω2),F[Sin⁡q(t)](ω)=-i2π(μq2)3f0,q(0)ωθ(q2;ω2).F[Cos⁡q(t)](ω) will be utilized to obtain Proposition 20, which in turn helps in yielding the uniform convergence results.


## 2. Main Results

Specific properties of theq-Legendre polynomials P~n(q;ω) are established. First, we show that the P~n(q;ω) satisfy an analogue of the recursion relation (1), namely, (34) below. Next, P~n(q;ω) also satisfy a multiplicatively advanced analogue of the Legendre ordinary differential equation (2), namely, the q-Legendre multiplicatively advanced differential equation (MADE) given by (56) below. Moment vanishing properties of the P~n(q;ω) exactly analogous to those of the truncated Pn(ω) in (3) are shown in (68). We obtain a reciprocal symmetry for P~0(q;ω) in (71). The P~n(q;ω) are used to generate a nearly orthonormal frame for L2(R) in Section 7. Alternative expressions for the P~n(q;ω) are obtained in Section 8. In Section 9, we obtain uniform convergence of the P~n(q;ω) to Pn(ω)χ~[-1,1](ω) as q→1+ on all closed sets of R not containing ±1. This result combined with the reciprocal symmetry property then gives pointwise convergence of P~n(q;ω) to Pn(ω)χ~[-1,1](ω) on R. In Section 10, Lp convergence of these functions is demonstrated. Approximations to the P~n(q;ω) are provided in Section 11. Finally, as encountered in the process of showing the q-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 theP~n(q;ω)

In this section, we obtain aq-version of the recursion formula (1) for q-Legendre polynomials, namely, (34) below. This follows from a recursion relation on the jn(q;t) given by (33). We begin with a lemma describing the derivative of jn(q;t).Lemma 4.
Forn∈N0 and q≥1, one has
(30)Dtjn(q;t)=ntjn(q;t)-jn+1(q;t).Proof.
First, recall thatj0(q;t)=Sin⁡q(t)/t and take j0(1;t)=j0(t)=sin(t)/t. Then,
(31)jn(q;t)=(-t)n(1tDt)nj0(q;t)
is well defined. Differentiation now yields
(32)Dtjn(q;t)=-n(-t)n-1(1tDt)nj0(q;t)+(-t)nDt(1tDt)nj0(q;t)=nt(-t)n(1tDt)nj0(q;t)-(-t)n+1(1tDt)n+1j0(q;t)=ntjn(q;t)-jn+1(q;t),
giving the lemma.Lemma4 is the starting point in proving the following recursion relations.Theorem 5.
Forn∈N0,
(33)(2n+1)Dtjn(q;t)=nqn+2jn-1(q;qt)-(n+1)jn+1(q;t).
Or, equivalently,
(34)(n+1)P~n+1(q;ω)=(2n+1)ωP~n(q;ω)-nqn+1P~n-1(q;ωq).Proof.
Then=0 case is handled directly. Namely, from (8), one has
(35)Dtj0(q;t)=-(-t)(1tDt)j0(q;t)=-j1(q;t),
giving (33) for n=0. Taking inverse Fourier transforms of both sides of (35) and multiplying the resulting equation by i2/π yield
(36)ωP~0(q;ω)=P~1(q;ω),
which is (34) for n=0. Thus, we assume n≥1 from this point on and begin first by showing (33). Setting the index n equal to n-1 in (30) and solving the result for jn(q;t) yield
(37)jn(q;t)=n-1tjn-1(q;t)-Dtjn-1(q;t).
From (10), with the index set to be n-1, one obtains
(38)Dt2jn-1(q;t)=n(n-1)t2jn-1(q;t)-2tDtjn-1(q;t)-qn+2jn-1(q;qt).
Differentiating (37) yields
(39)Dtjn(q;t)=-(n-1)t2jn-1(q;t)+(n-1)tDtjn-1(q;t)-Dt2jn-1(q;t).
Replacing the second derivative term in (39) with that in (38) yields
(40)Dtjn(q;t)=-(n-1)t2jn-1(q;t)+(n-1)tDtjn-1(q;t)-n(n-1)t2jn-1(q;t)+2tDtjn-1(q;t)+qn+2jn-1(q;qt)(41)=(n+1)t[Dtjn-1(q;t)-(n-1)tjn-1(q;t)]+qn+2jn-1(q;qt)(42)=(n+1)t[-jn(q;t)]+qn+2jn-1(q;qt),
where (37) was used to replace the bracketed expression in (41) and obtain (42). Continuing yields
(43)Dtjn(q;t)=-(n+1)n[ntjn(q;t)]+qn+2jn-1(q;qt)(44)=-(n+1)n[Dtjn(q;t)+jn+1(q;t)]+qn+2jn-1(q;qt),
where (30) was used to replace the bracketed expression in (43) and obtain (44). Multiplying (43) and (44) through by n and combining terms gives (33). Note the multiplicative advance qt in the argument of the jn-1 term in (33) and (44).
To obtain (34), one takes the inverse Fourier transform of (33), relying on the fact that F-1[Dtf(t)](ω)=(-iω)F-1[f(t)](ω) to obtain
(45)(2n+1)(-iω)F-1[jn(q;t)](ω)=nqn+2F-1[jn-1(q;qt)](ω)-(n+1)F-1[jn+1(q;t)](ω).
One next utilizes the fact that
(46)F-1[f(qt)](ω)=12π∫-∞∞‍eiωtf(qt)dt=12π∫-∞∞‍ei(ω/q)qtf(qt)d(qt)q=1qF-1[f(t)](ωq)
to reexpress F-1[jn-1(q;qt)](ω) in (45), obtaining
(47)(2n+1)(-iω)F-1[jn(q;t)](ω)=nqn+2q-1F-1[jn-1(q;t)](ωq)-(n+1)F-1[jn+1(q;t)](ω).
Multiplying (47) through by (-i)n-12/π gives
(48)(2n+1)ω(-i)n2πF-1[jn(q;t)](ω)=nqn+1(-i)n-12πF-1[jn-1(q;t)](ωq)+(n+1)(-i)n+12πF-1[jn+1(q;t)](ω).
Relying on (20) from Definition 1 gives
(49)(2n+1)ωP~n(q;ω)=nqn+1P~n-1(q;ωq)+(n+1)P~n+1(q;ω).
Solving for (n+1)P~n+1(q;ω) gives (34) and finishes the proof. Note the multiplicative delay ω/q in the argument of the P~n-1 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 q-Legendre polynomials. Observe that P~0(q;ω) is given directly by (21). Next, from (22), with n set to 0, we obtain
(50)P~1(q;ω)=ωP~0(q;ω).
From (34), with n=1, one obtains
(51)P~2(q;ω)=32ωP~1(q;ω)-12q2P~0(q;ωq)=32ω2P~0(q;ω)-12q2P~0(q;ωq).
From (34), with n=2, one obtains
(52)P~3(q;ω)=53ωP~2(q;ω)-23q3P~1(q;ωq)=53ω{32ω2P~0(q;ω)-12q2P~0(q;ωq)}-23q3{ωqP~0(q;ω/q)}(53)=52ω3P~0(q;ω)-32q2ωP~0(q;ωq).
From (34), with n=3, one obtains
(54)P~4(q;ω)=74ωP~3(q;ω)-34q4P~2(q;ωq)=74ω{52ω3P~0(q;ω)-32q2ωP~0(q;ωq)}-34q4{32(ωq)2P~0(q;ωq)-12q2P~0(q;ωq2)}=358ω4P~0(q;ω)-218q2ω2P~0(q;ωq)-98q2ω2P~0(q;ωq)+38q6P~0(q;ωq2)(55)=358ω4P~0(q;ω)-308q2ω2P~0(q;ωq)+38q6P~0(q;ωq2).
Proceeding on, one obtains the general nth order q-Legendre polynomial by multiplying each term of the nth degree Legendre polynomial by a power of q and by a multiplicative delay of P~0(q;ω) by a power of q and then summing. The expression extending (50)–(55) to general n will be given in Theorem 11 and Corollary 13 in Section 8 below.


## 4. MADEs for theP~n(q;ω)

In this section theq-Legendre polynomials are shown to satisfy a q-version of Legendre’s ODE (2), namely, the multiplicatively advanced differential equation (MADE) given by (56) below.Theorem 6.
Theq-Legendre polynomial P~n(q;ω)=(-i)n2/πF-1[jn(q;t)](ω) satisfies the multiplicatively advanced differential equation
(56)[Dω2P~n](q;ω)=q-n(qω)2[Dω2P~n](q;qω)+2q-n(qω)[DωP~n](q;qω)-q-nn(n+1)P~n(q;qω).
This reduces to the Legendre differential equation (2) as q→1+ with P~n(1;ω)≡Pn(ω)χ~[-1,1](ω) for |ω|<1.Proof.
Multiplying (10) by t2 yields
(57)t2jn′′(q;t)+2tjn′(q;t)-n(n+1)jn(q;t)=-qn+3t2jn(q;qt).
Relying on the facts that
(58)F-1[tf(t)](ω)=(-iDω)F-1[f(t)](ω),F-1[Dtf(t)](ω)=(-iω)F-1[f(t)](ω),
one applies the inverse Fourier transform F-1 to (57) to obtain
(59)(-iDω)2(-iω)2F-1[jn(q;t)](ω)+2(-iDω)(-iω)F-1[jn(q;t)](ω)-n(n+1)F-1[jn(q;t)](ω)=-qn+3(-iDω)2F-1[jn(q;qt)](ω).
Simplifying the left-hand side of (59) and relying on (46) to simplify the right-hand side of (59) yields
(60)Dω2ω2F-1[jn(q;t)](ω)-2DωωF-1[jn(q;t)](ω)-n(n+1)F-1[jn(q;t)](ω)(61)=qn+3Dω2{1qF-1[jn(q;t)](ωq)}.
Using the derivation property of Dω on the left-hand side of (61) gives
(62)2F-1[jn(q;t)](ω)+4ωDωF-1[jn(q;t)](ω)+ω2Dω2F-1[jn(q;t)](ω)-2F-1[jn(q;t)](ω)-2ωDωF-1[jn(q;t)](ω)-n(n+1)F-1[jn(q;t)](ω)(63)=qn+2Dω2{F-1[jn(q;t)](ωq)}.
Simplifying (62)-(63) and multiplying through by (-i)n2/π yield
(64)ω2Dω2P~n(q;ω)+2ωDωP~n(q;ω)-n(n+1)P~n(q;ω)=qn+2Dω2{P~n(q;ωq)}.
Letting u=ω/q with Dω=(1/q)Du yields
(65)(qu)2{1qDu}2{P~n(q;qu)}+2(qu){1qDu}{P~n(q;qu)}-n(n+1)P~n(q;qu)=qn+2{1qDu}2P~n(q;u).
Thus, we solve for the right-hand side of (65) scaled by q-n to obtain
(66)Du2P~n(q;u)=q-n(qu)2{1qDu}2{P~n(q;qu)}+2q-n(qu){1qDu}{P~n(q;qu)}-q-nn(n+1)P~n(q;qu),
which simplifies to (56) after a final substitution u=ω. 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 |ω| and thus expressed in terms of the unscaled variable. This will be further addressed in Section 12.


## 5. Vanishing of Moments for theP~n(q;ω)

Letn≥1. In light of (19), (3) can be rewritten as
(67)0=∫-∞∞‍ωkχ~[-1,1](ω)Pn(ω)dω=∫-∞∞‍ωk(-i)n2πF-1[jn(t)](ω)dωfor0≤k<n,
which tells us that the 0th through (n-1)th moments of the truncated nth 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.
Letn≥1. The 0th through (n-1)th moments of the nth order q-Legendre polynomial vanish. Consider
(68)0=∫-∞∞‍ωkP~n(q;ω)dω=∫-∞∞‍ωk(-i)n2πF-1[jn(q;t)](ω)dωfor0≤k<n.The proof is outlined here. Recall that thekth moment of f vanishing is equivalent to the kth derivative of F[f] vanishing at 0. From (20), one has immediately that, for all n∈N,
(69)F[P~n(q;ω)](t)=(-i)n2πF[F-1[jn(q;t)](ω)](t)=(-i)n2πjn(q;t)=(-i)n2π(-t)n(1tddt)nj0(q;t),
where the last equality follows from (8). Now, the (-t)n factor and the outer 1/t factor in (69) guarantee that the first n-1 derivatives of F[P~n(q;ω)](t) vanish at t=0, after noting that the derivatives of ((1/t)(d/dt))nj0(q;t) are bounded for all n∈N0. This gives (68). In contrast, it is shown in [4] that the nth derivative of F[P~n(q;ω)](t) does not vanish at t=0.


## 6. A Reciprocal Symmetry forP~0(q;ω)

There is an interesting reciprocal symmetry satisfied byP~0(q;ω), and this will help produce a pointwise convergence result in Section 9.Theorem 9.
For allω∈R. one has
(70)P~0(q;0)-P~0(q;ω)=P~0(q;qω),
or equivalently
(71)P~0(q;0)-P~0(q;qω)=P~0(q;qω).Proof.
From (21), we have, for ω<0,
(72)P~0(q;0)-P~0(q;ω)=-2(μq2)3πf0,q(0)∫-∞0‍vθ(q2;v2)dv--2(μq2)3πf0,q(0)∫-∞ω‍vθ(q2;v2)dv(73)=-2(μq2)3πf0,q(0)∫ω0‍vθ(q2;v2)dv(74)=-2(μq2)3πf0,q(0)∫1/ω-∞‍u-1θ(q2;u-2)-duu2(75)=-2(μq2)3πf0,q(0)∫-∞1/ω‍u-1θ(q2;u-2)duu2(76)=-2(μq2)3πf0,q(0)∫-∞1/ω‍u-1θ(q2;u2)du(77)=-2(μq2)3πf0,q(0)∫-∞1/ω‍q2uq2u2θ(q2;u2)du(78)=-2(μq2)3πf0,q(0)∫-∞1/ω‍quθ(q2;(qu)2)d(qu)(79)=-2(μq2)3πf0,q(0)∫-∞q/ω‍vθ(q2;v2)dv(80)=P~0(q;qω).
Here, the change of variables u=1/v was used to obtain (74), the identity u2θ(q2;u-2)=θ(q2;u2) was used to obtain (76), the integrand in (76) was multiplied by q2u2/(q2u2) to obtain (77), the identity q2u2θ(q2;u2)=θ(q2;q2u2) was used to obtain (78), the change of variables v=qu was used to obtain (79), and (21) was used to obtain (80) above. Thus, (70) holds for ω<0. By replacing ω by qω in the expression (70), one obtains (71) for ω<0. Note that, by evenness of P~0(q;ω), one has that both the identities (70) and (71) hold for all ω∈R.


## 7. Nearly Orthonormal Frames from theP~n(q;ω)

As in [13], a countable set of functions {gk(t)∣k∈N}⊂L2(R) is a frame for L2(R) if there are constants 0<A≤B<∞ with
(81)A∥f∥22≤∑k∈N‍|〈f(t),gk(t)〉|2≤B∥f∥22∀f∈L2(R).
The frame condition (81) is equivalent to
(82)A≤∑k∈N‍|〈f(t)∥f∥2,gk(t)〉|2≤B∀f∈L2(R)∖0.
We construct a frame from the P~n(q;ω) in the following manner. For each n∈N0=N∪{0} and k∈Z, let
(83)wn,k(ω)≡CnPn(ω-2k)χ~[-1,1](ω-2k),(84)w^n,k,q(ω)≡CnP~n(q;ω-2k),
where Cn=1/(n+1/2) normalizes wn,k to give ∥wn,k∥2=1, see [1].From (23), it follows that
(85)lim⁡q→1+∥w^n,k,q(ω)-wn,k(ω)∥2=0.
Thus, for each 1>ε>0, there is a q(n,k,ε)>1 such that for all q with 1<q≤q(n,k,ε)(86)∥w^n,k,q(ω)-wn,k(ω)∥2<ε18·2n·2|k|.
For conciseness, by suppressing ε and q(n,k,ε), set
(87)w~n,k(ω)≡w^n,k,q(n,k,ε)(ω).Theorem 10.
Givenε∈(0,1), the associated set of functions {w~n,k(ω)∣n∈N0,k∈Z}, with w~n,k(ω), as in (87), is a nearly orthonormal frame for L2(R) in the sense that
(88)|〈w~N,K,w~n,k〉-δNn·δKk|<ε,
where the Kronecker delta function δkj satisfies δkj=0 for k≠j and δjj=1.Proof.
We start by noting that(89)∑n∈N0∑k∈Z‍|〈f(t)∥f∥2,wn,k(t)〉|2=1∀f∈L2(R)∖0.
This follows since the wn,0 are orthonormal and complete on [-1,1], as in [2]. Thus the translates of wn,0 by multiples 2, namely the wn,k, are orthonormal and complete in L2(R). Let 1>ε>0 be given. We bootstrap on the fact that {wn,k} is an orthonormal frame to show that smooth approximations {w~n,k} given by (87) are also a frame for L2(R). For all functions f∈L2(R)∖{0}, one computes that
(90)∑n∈N0‍∑k∈Z‍|〈f(t)∥f∥2,w~n,k(t)〉|2=∑n∈N0‍∑k∈Z|〈f(t)∥f∥2,wn,k(t)〉+〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉|2(91)=∑n∈N0‍∑k∈Z{|〈f(t)∥f∥2,wn,k(t)〉|2+2R{〈f(t)∥f∥2,wn,k(t)〉〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}×〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}+|〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉|2}(92)=1+∑n∈N0‍∑k∈Z‍2R{〈f(t)∥f∥2,wn,k(t)〉〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}×〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}+∑n∈N0‍∑k∈Z‍|〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉|2.
Now, one uses Cauchy-Schwarz and (86) to obtain the bound
(93)|2R{〈f(t)∥f∥2,wn,k(t)〉〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}|≤2|〈f(t)∥f∥2,wn,k(t)〉〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉|(94)≤2∥f(t)∥f∥2∥2∥wn,k(t)∥2∥f(t)∥f∥2∥2∥w~n,k(t)-wn,k(t)∥2=2·1·1·1∥w~n,k(t)-wn,k(t)∥2≤2ε18·2n·2|k|.
Thus, one can bound (90) from below by discarding the last term of (92) and relying on (94) to obtain (96) as follows:
(95)∑n∈N0‍∑k∈Z‍|〈f(t)∥f∥2,w~n,k(t)〉|2≥1+∑n∈N0‍∑k∈Z‍2R{〈f(t)∥f∥2,wn,k(t)〉〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}×〈f(t)∥f∥2,w~n,k(t)-wn,k(t)〉¯}(96)≥1-∑n∈N0‍∑k∈Z‍2ε18·2n·2|k|(97)=1-2ε18∑n=0∞‍12n{∑k=0∞‍12|k|+∑k=-∞-1‍12|k|}=1-2ε182{2+1}=1-12ε18(98)≥1-ε.
Similarly, one bounds (92) from the above by bounding the last two terms in (92) via (94) as follows:
(99)∑n∈N0‍∑k∈Z‍|〈f(t)∥f∥2,w~n,k(t)〉|2≤1+∑n∈N0‍∑k∈Z‍2ε18·2n·2|k|+∑n∈N0‍∑k∈Z‍(ε18·2n·2|k|)2≤1+∑n∈N0‍∑k∈Z‍2ε18·2n·2|k|+∑n∈N0‍∑k∈Z‍1·(ε18·2n·2|k|)=1+3ε18∑n=0∞‍12n{∑k=0∞‍12|k|+∑k=-∞-1‍12|k|}=1+3ε182{2+1}=1+18ε18(100)≤1+ε.
Thus, combining (98) with (100) yields that, for all functions f∈L2(R)∖{0},
(101)1-ε≤∑n∈N0‍∑k∈Z‍|〈f(t)∥f∥2,w~n,k(t)〉|2≤1+ε.
Hence, given ε<1, the associated set {w~n,k} is a frame for L2(R).
Next, we show near orthonormality of the frame{w~n,k}. Observe that, given ε<1, one has
(102)|〈w~N,K,w~n,k〉-〈wN,K,wn,k〉|=|〈w~N,K-wN,K,w~n,k〉+〈wN,K,w~n,k-wn,k〉|≤|〈w~N,K-wN,K,w~n,k〉|+|〈wN,K,w~n,k-wn,k〉|≤∥w~N,K-wN,K∥2∥w~n,k∥2+∥wN,K∥2∥w~n,k-wn,k∥2=∥w~N,K-wN,K∥2∥wn,k+w~n,k-wn,k∥2+1·∥w~n,k-wn,k∥2≤∥w~N,K-wN,K∥2{∥wn,k∥2+∥w~n,k-wn,k∥2}+∥w~n,k-wn,k∥2≤ε18·2N·2|K|{1+ε18·2n·2|k|}+ε18·2n·2|k|≤ε18{1+118}+ε18<ε.
Thus,
(103)|〈w~N,K,w~n,k〉-〈wN,K,wn,k〉|=|〈w~N,K,w~n,k〉-δNn·δKk|<ε,
where δNn is the Kronecker delta function. We conclude that {w~n,k} is a nearly orthonormal frame.


## 8. Alternative Expressions forP~n(q;ω)

The goal of this section is to provide alternative expressions forP~n(q;ω) 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 nth degree Legendre polynomial as
(104)Pn(ω)=∑k=0⌊n/2⌋‍Cn,n-2kωn-2k,
where
(105)Cn,n-2k=(-1)k(2n-2k)!2nk!(n-k)!(n-2k)!
and ⌊x⌋ denotes the greatest integer function. For n≥1, the recursion relation (1) in this notation takes the form
(106)∑k=0⌊(n+1)/2⌋‍Cn+1,n+1-2kωn+1-2k(107)=∑k=0⌊n/2⌋‍(2n+1)(n+1)Cn,n-2kωn+1-2k-∑k=0⌊(n-1)/2⌋‍n(n+1)Cn-1,n-1-2kωn-1-2k(108)=∑k=0⌊n/2⌋‍(2n+1)(n+1)Cn,n-2kωn+1-2k-∑k=1⌊(n-1)/2⌋+1‍n(n+1)Cn-1,n+1-2kωn+1-2k
after reindexing k in the rightmost summation in (107) to obtain (108). This implies a recursion relation on the coefficients of like powers of ω obtained in setting (106) equal to (108).We are now prepared to state the next theorem generalizing (50)–(55).Theorem 11.
Forn≥0, the nth order q-Legendre polynomial is given by
(109)P~n(q;ω)=∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)P~0(q;ωqk),
where Cn,n-2k is the coefficient of ωn-2k in the nth degree Legendre polynomial Pn(ω), as given by (104) and (105).Proof.
Note that (109) is true in the n=0 case as it is a tautology, and it has been shown to hold for 1≤n≤4 via (50)–(55). Assume that (109) has been established up through order n. Then, the recursion relation (34) expressed in terms of (109) gives that
(110)P~n+1(q;ω)=2n+1n+1ωP~n(q;ω)-nn+1qn+1P~n-1(q;ωq)=∑k=0⌊n/2⌋‍2n+1n+1Cn,n-2kωn+1-2kqk(k+1)P~0(q;ωqk)-∑k=0⌊(n-1)/2⌋‍{nn+1Cn-1,n-1-2k(ωq)n-1-2k×qn+1+k(k+1)P~0(q;ωqk+1)}=∑k=0⌊n/2⌋‍2n+1n+1Cn,n-2kωn+1-2kqk(k+1)P~0(q;ωqk)(111)-∑k=0⌊(n-1)/2⌋‍{nn+1Cn-1,n-1-2kωn-1-2k×q(k+2)(k+1)P~0(q;ωqk+1)}(112)=∑k=0⌊n/2⌋‍2n+1n+1Cn,n-2kωn+1-2kqk(k+1)P~0(q;ωqk)(113)-∑k=1⌊(n-1)/2⌋+1‍{nn+1Cn-1,n+1-2kωn+1-2k×qk(k+1)P~0(q;ωqk)}(114)=∑k=0⌊(n+1)/2⌋‍Cn+1,n+1-2kωn+1-2kqk(k+1)P~0(q;ωqk),
where consolidating powers of q 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 representingP~n(q;ω) by (109) is that there are no nested integrals in (109), whereas the previous expressions (22) for higher order P~n(q;ω) involve nested integrals. Thus, we have gained computational efficiency.Replacing eachP~0(q;ω/qk) in (109) by the corresponding integral expression in (21) yields the following.Corollary 13.
Forn≥0, the nth order q-Legendre polynomial is given by
(115)P~n(q;ω)=-2(μq2)3πf0,q(0)∑k=0⌊n/2⌋‍{×∫-∞ω/qk‍vθ(q2;v2)dvCn,n-2kωn-2kqk(k+1)×∫-∞ω/qk‍vθ(q2;v2)dv}.


## 9. Convergence Results forP~n(q;ω)

On closed sets not containing±1, one has uniform convergence of P~n(q;ω) to Pn(ω)χ~[-1,1](ω) as q→1+. We will see that, via expression (109), one can obtain this result by relying on the fact that P~0(q;ω) converges uniformly to P0(ω)χ~[-1,1](ω) away from ±1. These results, coupled with an application of the reciprocal symmetry (71), will let us obtain pointwise convergence on all of R.


### 9.1. Uniform Convergence Away from±1

First, we obtain a uniform convergence forP~0(q;ω) on subsets of the form [-1+ε,1-ε].Proposition 14.
Given0<ε<1, one has that P~0(q;ω) converges uniformly to P0(ω)χ~[-1,1](ω)=1 on the interval [-1+ε,1-ε] as q→1+.Proof.
From (21), it follows that
(116)ddωP~0(q;ω)=-2(μq2)3πf0,q(0)ωθ(q2;ω2),
from which we see that P~0(q;ω) is increasing on (-∞,0) and decreasing on (0,∞) with a maximum at ω=0 of
(117)P~0(q;0)=-2(μq2)3πf0,q(0)∫-∞0‍ωθ(q2;ω2)dω=2(μq2)3πf0,q(0)∫0∞‍ωθ(q2;ω2)dω.
From (21), along with the oddness of ω/θ(q2;ω2), we have that P~0(q;ω) is even and lim⁡ω→±∞P~0(q;ω)=0. Thus,
(118)0<P~0(q;ω)≤P~0(q;0)
for all ω∈R. Furthermore, on the interval [-1+ε,0], the function |P~0(q;ω)-P0(ω)χ~[-1,1](ω)|=|P~0(q;ω)-1| assumes its maximum value at an endpoint ω=-1+ε or ω=0.
From Proposition19, we have that, given ε1>0, there is a Q2>1 such that for all 1<q<Q2(119)|P~0(q;-1+ε)-1|<ε1.
From Proposition 20, we have that, given ε1>0, there is a Q1>1 with
(120)|P~0(q;0)-1|=P~0(q;0)-1<ε1
for all 1<q<Q1. From the increasing property of P~0(q;ω) on [-1+ε,0], we have
(121)|P~0(q;ω)-P0(ω)χ~[-1,1](ω)|=|P~0(q;ω)-1|≤max⁡{|P~0(q;-1+ε)-1|,|P~0(q;0)-1|}<ε1,
for all q with 1<q<min⁡{Q1,Q2}. By evenness, (121) also gives uniform convergence on the interval [-1+ε,1-ε], and the proposition is proven.Next, we obtain uniform convergence on closed sets not containing±1.Theorem 15.
Givenε with 1>ε>0, let Sε≡(-∞,-1-ε]∪[-1+ε,1-ε]∪[1+ε,∞). Then P~n(q;ω) converges uniformly to Pn(ω)χ~[-1,1](ω) on Sε as q→1+.Proof.
The proof breaks into two parts: (1) uniform convergence on the middle, that is on[-1+ε,1-ε], and (2) uniform convergence on the tails, that is on (-∞,-1-ε]∪[1+ε,∞).
We first handle the middle case. From Proposition14, P~0(q;ω) converges uniformly to 1=P0(ω)χ~[-1,1](ω) on the interval [-1+ε,1-ε] as q→1+. From (104) and (105), one has that on [-1+ε,1-ε](122)|Pn(ω)|=|∑k=0⌊n/2⌋‍Cn,n-2kωn-2k|≤∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k<∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2k≡An.
Thus, given ε1 with 1>ε1>0, there is q1>1 such that
(123)|P~0(q;ω)-1|<ε12An
for all q with 1<q<q1 and for all ω with |ω|≤1-ε. Then, for each ω∈[-1+ε,1-ε] and k≥0, one has ω/qk∈[-1+ε,1-ε], given that
(124)|P~0(q;ωqk)-1|<ε12An
for all 1<q<q1. Thus, for ω∈[-1+ε,1-ε] and 1<q<q1, by relying on (109) and (104), we have
(125)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|=|∑k=0⌊n/2⌋Cn,n-2kωn-2kqk(k+1)P~0(q;ωqk)-∑k=0⌊n/2⌋Cn,n-2kωn-2kχ~[-1,1](ω)|≤|∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)P~0(q;ωqk)-∑k=0⌊n/2⌋‍Cn,n-2kωn-2kP~0(q;ωqk)|+|∑k=0⌊n/2⌋‍Cn,n-2kωn-2kP~0(q;ωqk)-∑k=0⌊n/2⌋‍Cn,n-2kωn-2kχ~[-1,1](ω)|≤∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k(qk(k+1)-1)|P~0(q;ωqk)|+∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k|P~0(q;ωqk)-1|<∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2k(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(1+ε12An)+∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2kε12An=An(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(1+ε12An)+Anε12An(126)=(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(An+ε12)+ε12.
When
(127)(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(An+ε12)<ε12,
we see that (126) is less than ε1. However, (127) holds when
(128)(q(⌊n/2⌋)(⌊n/2⌋+1)-1)<ε12(An+ε1/2),
which is true for all q<q2 with
(129)q2=(1+ε12An+1)1/[(⌊n/2⌋)(⌊n/2⌋+1)].
Thus, given 1>ε1>0, for all q with 1<q<min⁡{q1,q2} from (126) and (127), one has
(130)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|<ε1
for all ω∈[-1+ε,1-ε]. Thus, uniform convergence is obtained on [-1+ε,1-ε], and part 1 of the proof is shown.
We proceed to part 2 of the proof, uniform convergence in the tails|ω|≥1+ε. Let ε1>0 be given. Then, for ω∈(-∞,-1-ε]∪[1+ε,∞), one has
(131)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|=|∑k=0⌊n/2⌋‍Cn,n-2kqk(k+1)ωn-2kP~0(q;ωqk)-0|≤∑k=0⌊n/2⌋‍|Cn,n-2k|qk(k+1)|ωn-2kP~0(q;ωqk)|≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(132)·∑k=0⌊n/2⌋‍{|Cn,n-2k|qk(k+1)q[(n-2k)2+2(n-2k)(2k+1)]/4×e-[ln⁡(|ω|)-(n-2k+2k+1)ln⁡(q)/2]2/ln⁡(q)}(133)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(134)·qn(n+2)/4(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω|)-(n+1)ln⁡(q)/2]2/ln⁡(q)(135)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(136)·qn(n+2)/4(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q),
where (132) follows from (143) of Proposition 17, with p=n-2k, and holds for 1<q<q3 and |ω|≥|ω0| with
(137)1<q3(n-2k+2k+1)/2=q3(n+1)/2<|ω0|<1+ε
as obtained in (154).
From (17), one has
(138)lim⁡q→1+[2(μq2)3ln⁡(q)πf0,q(0)]=1,lim⁡q→1+{π-ln⁡(q)}-1π2qn(n+2)/4=12,
from which it follows via (136) that there is a q4 with 1<q4<q3 such that for all 1<q<q4(139)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|≤1·(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q)
for all |ω|≥|ω0|. Since
(140)lim⁡q→1+e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q)=0,
there is a q5 with 1<q5<q4 such that
(141)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|<ε1
for all 1<q<q5 and all |ω|≥|ω0|. Thus, since 1+ε>|ω0|, 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 Theorem15.Proposition 17.
Givenp,k∈N0 and given ε>0, then for each ω0 with 1<ω0<1+ε there exists q3>1 such that
(142)|ωpP~0(q;ωqk)|(143)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q)(144)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω0|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q)
for all 1<q<q3 and all |ω|≥ω0.Proof.
From (21), one has
(145)|ωpP~0(q;ωqk)|=|ωp2(μq2)3πf0,q(0)∫-∞ω/qk‍vθ(q2;v2)dv|(146)=|ω|p2(μq2)3πf0,q(0)|∫-∞-|ω|/qk‍vθ(q2;v2)dv|=|ω|p2(μq2)3πf0,q(0)∫|ω|/qk∞‍vθ(q2;v2)dv(147)≤|ω|p2(μq2)3πf0,q(0)q-1/4{πln⁡(q)-1}-1×∫|ω|/qk∞‍e-[ln⁡(v)]2/ln⁡(q)dv(148)≤|ω|p2(μq2)3πf0,q(0)q-1/4{πln⁡(q)-1}-1×π2ln⁡(q)e[ln⁡(q)]/4e-[ln⁡(|ω|/qk)-ln⁡(q)/2]2/ln⁡(q)(149)=epln⁡(|ω|)[2(μq2)3ln⁡(q)πf0,q(0)]{ln⁡(q)}-1×{πln⁡(q)-1}-1π2e-[ln⁡(|ω|)-(k+1/2)ln⁡(q)]2/ln⁡(q)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2e-[ln⁡2(|ω|)-(p+2k+1)ln⁡(|ω|)ln⁡(q)+(k+1/2)2ln⁡2(q)]/ln⁡(q)(150)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q).
Here, (146) follows from oddness of the integrand v/θ(q2;v2). Then, (147) follows from the bound (27). Also, (148) follows from the bound
(151)∫C∞‍e-A[ln⁡(x)+B]2dx≤π2e1/(4A)-BAe-A[ln⁡(C)+B-1/(2A)]2
for C>e-B+1/(2A), from [4, 14], with A=1/ln⁡(q), B=0, and C=|ω|/qk. Finally, (150) follows by a completion of squares. Note that the requirement that C>e-B+1/(2A) in (151) becomes |ω|/qk>eln⁡(q)/2 or |ω|>qk+1/2 in (148). However, in order for (150) to be decreasing in |ω|, we require the slightly stronger condition that
(152)ln⁡(|ω|)-(p+2k+1)ln⁡(q)2>0,
or equivalently
(153)|ω|>q(p+2k+1)/2.
Now, given ε>0, and p,k∈N0 as in the hypotheses of the proposition, first choose ω0 with 1<ω0<1+ε and then choose q3 with 1<q3<|ω0|2/[p+2k+1] so that
(154)1<q3(p+2k+1)/2<|ω0|<1+ε.
Then, for 1<q<q3 and |ω|≥|ω0|, one has that (153) holds, from which (150) is decreasing in |ω|. This then gives (143) and (144) and completes the proof of the proposition.


### 9.2. Pointwise Convergence onR

It is now possible to give a pointwise convergence result.Theorem 18.
P
~
n
(
q
;
ω
) converges pointwise to Pn(ω)χ~[-1,1](ω) as q→1+.Proof.
By virtue of Theorem15, one automatically has pointwise convergence for ω≠±1. Thus, we concentrate on ω=±1 to obtain the result. Observe first that for any p∈R one has
(155)lim⁡q→1+[P~0(q;qp)-P~0(q;1)]=0.
This follows from the estimate
(156)|P~0(q;qp)-P~0(q;1)|=|-2π(μq2)3f0,q(0)∫-∞qp‍ωθ(q2;ω2)dω--2π(μq2)3f0,q(0)∫-∞1‍ωθ(q2;ω2)dω|=2π(μq2)3f0,q(0)|∫1qp‍ωθ(q2;ω2)dω|(157)≤2πln⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡(q)}×[1ln⁡(q)|∫1qp‍1e(ln⁡(|ω|))2/ln⁡(q)dω|](158)≤2πln⁡⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡⁡(q)}×[1ln⁡⁡(q)|qp-1|],
where (27) was used to obtain (157) and where the integrand in (157) being at most 1 was used to obtain (158). Now, by L’Hopital’s rule, one has, for each p∈R, that
(159)lim⁡q→1+qp-1ln⁡(q)=lim⁡q→1+pqp-1(1/2)(ln⁡(q))-1/2(1/q)=lim⁡q→1+2pqpln⁡(q)=0.
By the q-Wallis limit (17), one also has
(160)lim⁡q→1+2πln⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡(q)}=1π.
Applying (159) and (160) to (158) gives (155).
Next, observe from the reciprocal identity (71) with ω set equal to 1, that
(161)P~0(q;0)-P~0(q;q)=P~0(q;q),
from which one obtains
(162)12P~0(q;0)=P~0(q;q).
Proposition 20 gives that lim⁡q→1+P~0(q;0)=1. Applying this to (162) gives
(163)lim⁡q→1+P~0(q;q)=12lim⁡q→1+P~0(q;0)=12.
Setting p=1/2 in (155) gives
(164)lim⁡q→1+P~0(q;1)=lim⁡q→1+P~0(q;q)=12.
Letting p be arbitrary in (155) gives
(165)lim⁡q→1+P~0(q;qp)=lim⁡q→1+P~0(q;1)=12
for all p∈R.
Next, from (109), we have
(166)P~n(q;1)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2kqk(k+1)P~0(q;1qk).
From (165), one has that lim⁡q→1+P~0(q;1/qk)=1/2. Taking limits of (166) with this in mind yields
(167)lim⁡q→1+P~n(q;1)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2klim⁡q→1+[qk(k+1)P~0(q;1qk)](168)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2k12(169)=Pn(1)12=Pn(1)χ~[-1,1](1)=12,
where (168) follows from (165) and (169) follows from (104). By evenness/oddness, respectively, one has
(170)lim⁡q→1+P~n(q;-1)=lim⁡q→1+[(-1)nP~n(q;1)]=(-1)nPn(1)χ~[-1,1](1)=Pn(-1)χ~[-1,1](-1)=(-1)n12.
Equations (169) and (170) give pointwise convergence at ±1 and the theorem is proven.


### 9.3. Estimates Giving Proposition14

The propositions in this subsection provide the estimates on which Proposition14 is based.Proposition 19.
For0<ε<1, given ε1>0, there is a Q2>1 such that
(171)|P~0(q;-1+ε)-1|<ε1
for all 1<q<Q2.Proof.
(172)
|
P
~
0
(
q
;
-
1
+
ε
)
-
1
|
≤
|
P
~
0
(
q
;
-
1
+
ε
)
-
P
~
0
(
q
;
0
)
|
+
|
P
~
0
(
q
;
0
)
-
1
|
.
By virtue of Proposition 20 below, which bounds |P~0(q;0)-1|, we only need bound the first term, |P~0(q;-1+ε)-P~0(q;0)|, in the right-hand side of inequality (172).
Evenness ofP~0(q;ω) and the setting of ω=-1+ε in the reciprocal identity (70) together give
(173)0<P~0(q;0)-P~0(q;-1+ε)=P~0(q;q(-1+ε))=P~0(q;q(1-ε)),
for all q>1. Now, since 1+ε<1/(1-ε)<q/(1-ε) and P~0(q;ω) is decreasing for ω>0, we have that P~0(q;1+ε)>P~0(q;1/(1-ε))>P~0(q;q/(1-ε)), and this applied to (173) yields
(174)0<P~0(q;0)-P~0(q;-1+ε)=P~0(q;q(1-ε))<P~0(q;1+ε).
Next, pick ω0 with 1<ω0<1+ε, and apply Proposition 17 with p=k=0 and ω=1+ε to obtain
(175)P~0(q;1+ε)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2e-[ln⁡(|ω0|)-ln⁡(q)/2]2/ln⁡(q).
By (17), one has
(176)lim⁡q→1+[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2=12,
and one sees directly that
(177)lim⁡q→1+e-[ln⁡(|ω0|)-ln⁡(q)/2]2/ln⁡(q)=0.
Thus, the right-hand side of (175) approaches 0 as q→1+. So, from (175) and (174), given ε1>0, there is a Q2>1 such that, for all 1<q<Q2,
(178)0<P~0(q;0)-P~0(q;-1+ε)<P~0(q;1+ε)<ε12.
This gives the proposition.Next, by (29), we have
(179)Cos⁡q⁡(t)=F-1(F(Cos⁡q⁡(t)))=12π∫-∞∞‍eiωt2π(μq2)3f0,q(0)1θ(q2;ω2)dω=1π(μq2)3f0,q(0)∫-∞∞‍eiωt1θ(q2;ω2)dω,
from which one obtains that
(180)1=Cos⁡q⁡(0)=1π(μq2)3f0,q(0)∫-∞∞‍1θ(q2;ω2)dω=2π(μq2)3f0,q(0)∫0∞‍1θ(q2;ω2)dω.Proposition 20.
For eachε>0, there exists a Q>1 such that for all 1<q<Q(181)0<P~0(q;0)-1<ε.Proof.
Subtracting (180) from (117), one has
(182)P~0(q;0)-1=2(μq2)3πf0,q(0)∫0∞‍ω-1θ(q2;ω2)dω=2(μq2)3πf0,q(0)×[∫01‍ω-1θ(q2;ω2)dω+∫1∞‍ω-1θ(q2;ω2)dω].
The change of variables ω=1/u is made on the first integral in (182), and the algebraic identity u2θ(q2;u-2)=θ(q2;u2) is used to obtain
(183)∫01ω-1θ(q2;ω2)dω=∫∞1‍u-1-1θ(q2;u-2)(-du)u2=∫1∞‍u-1-1θ(q2;u2)du.
Now (183) is used to reexpress (182) as
(184)P~0(q;0)-1=2(μq2)3πf0,q(0)[∫1∞‍ω-1-1θ(q2;ω2)dω+∫1∞‍ω-1θ(q2;ω2)dω](185)=2(μq2)3πf0,q(0)∫1∞‍ω-1-2+ωθ(q2;ω2)dω=2(μq2)3πf0,q(0)∫1∞‍(ω-1)2ωθ(q2;ω2)dω>0,
from which one sees that P~0(q;0)>1 for all q>1. Deploying the bound (27) within the integral in (185) gives
(186)0<P~0(q;0)-1≤F(q)[1ln⁡(q)∫1∞‍(ω-1)2ω-2e(ln⁡ω)2/ln⁡(q)dω]=F(q)[1ln⁡(q)∫1∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω],
where
(187)F(q)=1q1/42π[ln⁡(q)(μq2)3f0,q(0)]1π-ln⁡(q).
It follows from the q-Wallis formula (17) that
(188)lim⁡q→1+F(q)=1π.
We now show thatP~0(q;0)-1 can be made arbitrarily small for all q>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).
Letα>0 be arbitrary, with α being specified later. The integral in ω over the interval [1,∞) in (186) is now subdivided into two integrals, the first over [1,eαln⁡⁡(q)] and the second over [eαln⁡(q),∞). First, on [1,eαln⁡⁡(q)], one computes
(189)1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω.
Now, the function (1-ω-1)2 is increasing on [1,eαln⁡(q)], and it assumes its maximum value of (1-e-αln⁡⁡(q))2 at the right endpoint eαln⁡(q). Thus, we bound the integral in (189) by the length of the interval times the bound (1-e-αln⁡⁡(q))2 on the numerator. This gives
(190)1ln⁡⁡(q)∫1eαln⁡⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω≤1ln⁡(q)∫1eαln⁡(q)‍(1-e-αln⁡⁡(q))2dω(191)≤1ln⁡⁡(q)(eαln⁡⁡(q)-1)(1-e-αln⁡⁡(q))2.
An application of L’Hopital’s rule gives that
(192)lim⁡q→1+(eαln⁡⁡(q)-1)ln⁡(q)=α,
which implies that in (191) we have
(193)lim⁡q→1+[(eαln⁡⁡(q)-1)ln⁡(q)(1-e-αln⁡⁡(q))2]=α·02=0.
Combining (193) with (191) gives that
(194)lim⁡q→1+[1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω]=0.
We next estimate the portion of the integral in (186) over the interval [eαln⁡(q),∞). First, since 1-ω-1≤1 on this interval, one has
(195)1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω≤1ln⁡(q)∫eαln⁡(q)∞‍1e(ln⁡ω)2/ln⁡(q)dω.
We next bound the right-hand side of (195), using the following estimate from [4, 14]:
(196)∫C∞‍e-A[ln⁡ω+B]2dω≤π2e1/(4A)-BAe-A[ln⁡⁡(C)+B-1/(2A)]2,
which holds for C>e-B+1/(2A). Setting A=1/ln⁡(q), B=0, and C=eαln⁡(q) as in (195) yields that
(197)1ln⁡(q)∫eαln⁡(q)∞‍1e(ln⁡ω)2/ln⁡(q)dω≤1ln⁡(q)π2eln⁡(q)/41/ln⁡(q)e-[αln⁡(q)-ln⁡(q)/2]2/ln⁡(q)=π2e-α2[eαln⁡(q)]
for eαln⁡(q)>eln⁡(q)/2, or equivalently for 1<q<e4α2. Since the final expression in (197) approaches πe-α2/2 as q→1+, by choosing α sufficiently large one can make (197) and hence (195), arbitrarily small for q and sufficiently close to 1.
Now, letε>0 be given. By (186), we have for all α>0(198)0<P~0(q;0)-1≤F(q)[1ln⁡(q)∫1eαln⁡(q)(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω](199)+F(q)[1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω].
One has from (188) and (197) that
(200)lim⁡q→1+{F(q)π2e-α2[eαln⁡(q)]}=12e-α2<e-α2.
Fix α>0 such that e-α2<ε/2. By (197) and (200), there exists q1=q1(α,ε) such that for all 1<q<q1 one has
(201)0<F(q)[1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω](202)≤F(q)π2e-α2[eαln⁡⁡(q)]<e-α2<ε2.
For this value of α, by virtue of (194), one has
(203)lim⁡q→1+F(q)[1ln⁡⁡(q)∫1eαln⁡⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω]=0,
which in turn says that there is a q2=q2(α,ε) such that for all 1<q<q2 one has
(204)0<F(q)[1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω]<ε2.
Thus, for Q(α,ε)≡min⁡{q1,q2} and for all 1<q<Q(α,ε), applying (204) and (202) to (198) and (199) yields
(205)0<P~0(q;0)-1<ε2+ε2<ε.
This gives the proposition.


## 9.1. Uniform Convergence Away from±1

First, we obtain a uniform convergence forP~0(q;ω) on subsets of the form [-1+ε,1-ε].Proposition 14.
Given0<ε<1, one has that P~0(q;ω) converges uniformly to P0(ω)χ~[-1,1](ω)=1 on the interval [-1+ε,1-ε] as q→1+.Proof.
From (21), it follows that
(116)ddωP~0(q;ω)=-2(μq2)3πf0,q(0)ωθ(q2;ω2),
from which we see that P~0(q;ω) is increasing on (-∞,0) and decreasing on (0,∞) with a maximum at ω=0 of
(117)P~0(q;0)=-2(μq2)3πf0,q(0)∫-∞0‍ωθ(q2;ω2)dω=2(μq2)3πf0,q(0)∫0∞‍ωθ(q2;ω2)dω.
From (21), along with the oddness of ω/θ(q2;ω2), we have that P~0(q;ω) is even and lim⁡ω→±∞P~0(q;ω)=0. Thus,
(118)0<P~0(q;ω)≤P~0(q;0)
for all ω∈R. Furthermore, on the interval [-1+ε,0], the function |P~0(q;ω)-P0(ω)χ~[-1,1](ω)|=|P~0(q;ω)-1| assumes its maximum value at an endpoint ω=-1+ε or ω=0.
From Proposition19, we have that, given ε1>0, there is a Q2>1 such that for all 1<q<Q2(119)|P~0(q;-1+ε)-1|<ε1.
From Proposition 20, we have that, given ε1>0, there is a Q1>1 with
(120)|P~0(q;0)-1|=P~0(q;0)-1<ε1
for all 1<q<Q1. From the increasing property of P~0(q;ω) on [-1+ε,0], we have
(121)|P~0(q;ω)-P0(ω)χ~[-1,1](ω)|=|P~0(q;ω)-1|≤max⁡{|P~0(q;-1+ε)-1|,|P~0(q;0)-1|}<ε1,
for all q with 1<q<min⁡{Q1,Q2}. By evenness, (121) also gives uniform convergence on the interval [-1+ε,1-ε], and the proposition is proven.Next, we obtain uniform convergence on closed sets not containing±1.Theorem 15.
Givenε with 1>ε>0, let Sε≡(-∞,-1-ε]∪[-1+ε,1-ε]∪[1+ε,∞). Then P~n(q;ω) converges uniformly to Pn(ω)χ~[-1,1](ω) on Sε as q→1+.Proof.
The proof breaks into two parts: (1) uniform convergence on the middle, that is on[-1+ε,1-ε], and (2) uniform convergence on the tails, that is on (-∞,-1-ε]∪[1+ε,∞).
We first handle the middle case. From Proposition14, P~0(q;ω) converges uniformly to 1=P0(ω)χ~[-1,1](ω) on the interval [-1+ε,1-ε] as q→1+. From (104) and (105), one has that on [-1+ε,1-ε](122)|Pn(ω)|=|∑k=0⌊n/2⌋‍Cn,n-2kωn-2k|≤∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k<∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2k≡An.
Thus, given ε1 with 1>ε1>0, there is q1>1 such that
(123)|P~0(q;ω)-1|<ε12An
for all q with 1<q<q1 and for all ω with |ω|≤1-ε. Then, for each ω∈[-1+ε,1-ε] and k≥0, one has ω/qk∈[-1+ε,1-ε], given that
(124)|P~0(q;ωqk)-1|<ε12An
for all 1<q<q1. Thus, for ω∈[-1+ε,1-ε] and 1<q<q1, by relying on (109) and (104), we have
(125)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|=|∑k=0⌊n/2⌋Cn,n-2kωn-2kqk(k+1)P~0(q;ωqk)-∑k=0⌊n/2⌋Cn,n-2kωn-2kχ~[-1,1](ω)|≤|∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)P~0(q;ωqk)-∑k=0⌊n/2⌋‍Cn,n-2kωn-2kP~0(q;ωqk)|+|∑k=0⌊n/2⌋‍Cn,n-2kωn-2kP~0(q;ωqk)-∑k=0⌊n/2⌋‍Cn,n-2kωn-2kχ~[-1,1](ω)|≤∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k(qk(k+1)-1)|P~0(q;ωqk)|+∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2k|P~0(q;ωqk)-1|<∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2k(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(1+ε12An)+∑k=0⌊n/2⌋‍|Cn,n-2k|(1)n-2kε12An=An(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(1+ε12An)+Anε12An(126)=(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(An+ε12)+ε12.
When
(127)(q(⌊n/2⌋)(⌊n/2⌋+1)-1)(An+ε12)<ε12,
we see that (126) is less than ε1. However, (127) holds when
(128)(q(⌊n/2⌋)(⌊n/2⌋+1)-1)<ε12(An+ε1/2),
which is true for all q<q2 with
(129)q2=(1+ε12An+1)1/[(⌊n/2⌋)(⌊n/2⌋+1)].
Thus, given 1>ε1>0, for all q with 1<q<min⁡{q1,q2} from (126) and (127), one has
(130)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|<ε1
for all ω∈[-1+ε,1-ε]. Thus, uniform convergence is obtained on [-1+ε,1-ε], and part 1 of the proof is shown.
We proceed to part 2 of the proof, uniform convergence in the tails|ω|≥1+ε. Let ε1>0 be given. Then, for ω∈(-∞,-1-ε]∪[1+ε,∞), one has
(131)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|=|∑k=0⌊n/2⌋‍Cn,n-2kqk(k+1)ωn-2kP~0(q;ωqk)-0|≤∑k=0⌊n/2⌋‍|Cn,n-2k|qk(k+1)|ωn-2kP~0(q;ωqk)|≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(132)·∑k=0⌊n/2⌋‍{|Cn,n-2k|qk(k+1)q[(n-2k)2+2(n-2k)(2k+1)]/4×e-[ln⁡(|ω|)-(n-2k+2k+1)ln⁡(q)/2]2/ln⁡(q)}(133)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(134)·qn(n+2)/4(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω|)-(n+1)ln⁡(q)/2]2/ln⁡(q)(135)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2(136)·qn(n+2)/4(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q),
where (132) follows from (143) of Proposition 17, with p=n-2k, and holds for 1<q<q3 and |ω|≥|ω0| with
(137)1<q3(n-2k+2k+1)/2=q3(n+1)/2<|ω0|<1+ε
as obtained in (154).
From (17), one has
(138)lim⁡q→1+[2(μq2)3ln⁡(q)πf0,q(0)]=1,lim⁡q→1+{π-ln⁡(q)}-1π2qn(n+2)/4=12,
from which it follows via (136) that there is a q4 with 1<q4<q3 such that for all 1<q<q4(139)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|≤1·(∑k=0⌊n/2⌋‍|Cn,n-2k|)×e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q)
for all |ω|≥|ω0|. Since
(140)lim⁡q→1+e-[ln⁡(|ω0|)-(n+1)ln⁡(q)/2]2/ln⁡(q)=0,
there is a q5 with 1<q5<q4 such that
(141)|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|<ε1
for all 1<q<q5 and all |ω|≥|ω0|. Thus, since 1+ε>|ω0|, 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 Theorem15.Proposition 17.
Givenp,k∈N0 and given ε>0, then for each ω0 with 1<ω0<1+ε there exists q3>1 such that
(142)|ωpP~0(q;ωqk)|(143)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q)(144)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω0|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q)
for all 1<q<q3 and all |ω|≥ω0.Proof.
From (21), one has
(145)|ωpP~0(q;ωqk)|=|ωp2(μq2)3πf0,q(0)∫-∞ω/qk‍vθ(q2;v2)dv|(146)=|ω|p2(μq2)3πf0,q(0)|∫-∞-|ω|/qk‍vθ(q2;v2)dv|=|ω|p2(μq2)3πf0,q(0)∫|ω|/qk∞‍vθ(q2;v2)dv(147)≤|ω|p2(μq2)3πf0,q(0)q-1/4{πln⁡(q)-1}-1×∫|ω|/qk∞‍e-[ln⁡(v)]2/ln⁡(q)dv(148)≤|ω|p2(μq2)3πf0,q(0)q-1/4{πln⁡(q)-1}-1×π2ln⁡(q)e[ln⁡(q)]/4e-[ln⁡(|ω|/qk)-ln⁡(q)/2]2/ln⁡(q)(149)=epln⁡(|ω|)[2(μq2)3ln⁡(q)πf0,q(0)]{ln⁡(q)}-1×{πln⁡(q)-1}-1π2e-[ln⁡(|ω|)-(k+1/2)ln⁡(q)]2/ln⁡(q)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2e-[ln⁡2(|ω|)-(p+2k+1)ln⁡(|ω|)ln⁡(q)+(k+1/2)2ln⁡2(q)]/ln⁡(q)(150)=[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2q[p2+2p(2k+1)]/4e-[ln⁡(|ω|)-(p+2k+1)ln⁡(q)/2]2/ln⁡(q).
Here, (146) follows from oddness of the integrand v/θ(q2;v2). Then, (147) follows from the bound (27). Also, (148) follows from the bound
(151)∫C∞‍e-A[ln⁡(x)+B]2dx≤π2e1/(4A)-BAe-A[ln⁡(C)+B-1/(2A)]2
for C>e-B+1/(2A), from [4, 14], with A=1/ln⁡(q), B=0, and C=|ω|/qk. Finally, (150) follows by a completion of squares. Note that the requirement that C>e-B+1/(2A) in (151) becomes |ω|/qk>eln⁡(q)/2 or |ω|>qk+1/2 in (148). However, in order for (150) to be decreasing in |ω|, we require the slightly stronger condition that
(152)ln⁡(|ω|)-(p+2k+1)ln⁡(q)2>0,
or equivalently
(153)|ω|>q(p+2k+1)/2.
Now, given ε>0, and p,k∈N0 as in the hypotheses of the proposition, first choose ω0 with 1<ω0<1+ε and then choose q3 with 1<q3<|ω0|2/[p+2k+1] so that
(154)1<q3(p+2k+1)/2<|ω0|<1+ε.
Then, for 1<q<q3 and |ω|≥|ω0|, one has that (153) holds, from which (150) is decreasing in |ω|. This then gives (143) and (144) and completes the proof of the proposition.


## 9.2. Pointwise Convergence onR

It is now possible to give a pointwise convergence result.Theorem 18.
P
~
n
(
q
;
ω
) converges pointwise to Pn(ω)χ~[-1,1](ω) as q→1+.Proof.
By virtue of Theorem15, one automatically has pointwise convergence for ω≠±1. Thus, we concentrate on ω=±1 to obtain the result. Observe first that for any p∈R one has
(155)lim⁡q→1+[P~0(q;qp)-P~0(q;1)]=0.
This follows from the estimate
(156)|P~0(q;qp)-P~0(q;1)|=|-2π(μq2)3f0,q(0)∫-∞qp‍ωθ(q2;ω2)dω--2π(μq2)3f0,q(0)∫-∞1‍ωθ(q2;ω2)dω|=2π(μq2)3f0,q(0)|∫1qp‍ωθ(q2;ω2)dω|(157)≤2πln⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡(q)}×[1ln⁡(q)|∫1qp‍1e(ln⁡(|ω|))2/ln⁡(q)dω|](158)≤2πln⁡⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡⁡(q)}×[1ln⁡⁡(q)|qp-1|],
where (27) was used to obtain (157) and where the integrand in (157) being at most 1 was used to obtain (158). Now, by L’Hopital’s rule, one has, for each p∈R, that
(159)lim⁡q→1+qp-1ln⁡(q)=lim⁡q→1+pqp-1(1/2)(ln⁡(q))-1/2(1/q)=lim⁡q→1+2pqpln⁡(q)=0.
By the q-Wallis limit (17), one also has
(160)lim⁡q→1+2πln⁡(q)(μq2)3f0,q(0)q-1/4{π-ln⁡(q)}=1π.
Applying (159) and (160) to (158) gives (155).
Next, observe from the reciprocal identity (71) with ω set equal to 1, that
(161)P~0(q;0)-P~0(q;q)=P~0(q;q),
from which one obtains
(162)12P~0(q;0)=P~0(q;q).
Proposition 20 gives that lim⁡q→1+P~0(q;0)=1. Applying this to (162) gives
(163)lim⁡q→1+P~0(q;q)=12lim⁡q→1+P~0(q;0)=12.
Setting p=1/2 in (155) gives
(164)lim⁡q→1+P~0(q;1)=lim⁡q→1+P~0(q;q)=12.
Letting p be arbitrary in (155) gives
(165)lim⁡q→1+P~0(q;qp)=lim⁡q→1+P~0(q;1)=12
for all p∈R.
Next, from (109), we have
(166)P~n(q;1)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2kqk(k+1)P~0(q;1qk).
From (165), one has that lim⁡q→1+P~0(q;1/qk)=1/2. Taking limits of (166) with this in mind yields
(167)lim⁡q→1+P~n(q;1)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2klim⁡q→1+[qk(k+1)P~0(q;1qk)](168)=∑k=0⌊n/2⌋‍Cn,n-2k1n-2k12(169)=Pn(1)12=Pn(1)χ~[-1,1](1)=12,
where (168) follows from (165) and (169) follows from (104). By evenness/oddness, respectively, one has
(170)lim⁡q→1+P~n(q;-1)=lim⁡q→1+[(-1)nP~n(q;1)]=(-1)nPn(1)χ~[-1,1](1)=Pn(-1)χ~[-1,1](-1)=(-1)n12.
Equations (169) and (170) give pointwise convergence at ±1 and the theorem is proven.


## 9.3. Estimates Giving Proposition14

The propositions in this subsection provide the estimates on which Proposition14 is based.Proposition 19.
For0<ε<1, given ε1>0, there is a Q2>1 such that
(171)|P~0(q;-1+ε)-1|<ε1
for all 1<q<Q2.Proof.
(172)
|
P
~
0
(
q
;
-
1
+
ε
)
-
1
|
≤
|
P
~
0
(
q
;
-
1
+
ε
)
-
P
~
0
(
q
;
0
)
|
+
|
P
~
0
(
q
;
0
)
-
1
|
.
By virtue of Proposition 20 below, which bounds |P~0(q;0)-1|, we only need bound the first term, |P~0(q;-1+ε)-P~0(q;0)|, in the right-hand side of inequality (172).
Evenness ofP~0(q;ω) and the setting of ω=-1+ε in the reciprocal identity (70) together give
(173)0<P~0(q;0)-P~0(q;-1+ε)=P~0(q;q(-1+ε))=P~0(q;q(1-ε)),
for all q>1. Now, since 1+ε<1/(1-ε)<q/(1-ε) and P~0(q;ω) is decreasing for ω>0, we have that P~0(q;1+ε)>P~0(q;1/(1-ε))>P~0(q;q/(1-ε)), and this applied to (173) yields
(174)0<P~0(q;0)-P~0(q;-1+ε)=P~0(q;q(1-ε))<P~0(q;1+ε).
Next, pick ω0 with 1<ω0<1+ε, and apply Proposition 17 with p=k=0 and ω=1+ε to obtain
(175)P~0(q;1+ε)≤[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2e-[ln⁡(|ω0|)-ln⁡(q)/2]2/ln⁡(q).
By (17), one has
(176)lim⁡q→1+[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1π2=12,
and one sees directly that
(177)lim⁡q→1+e-[ln⁡(|ω0|)-ln⁡(q)/2]2/ln⁡(q)=0.
Thus, the right-hand side of (175) approaches 0 as q→1+. So, from (175) and (174), given ε1>0, there is a Q2>1 such that, for all 1<q<Q2,
(178)0<P~0(q;0)-P~0(q;-1+ε)<P~0(q;1+ε)<ε12.
This gives the proposition.Next, by (29), we have
(179)Cos⁡q⁡(t)=F-1(F(Cos⁡q⁡(t)))=12π∫-∞∞‍eiωt2π(μq2)3f0,q(0)1θ(q2;ω2)dω=1π(μq2)3f0,q(0)∫-∞∞‍eiωt1θ(q2;ω2)dω,
from which one obtains that
(180)1=Cos⁡q⁡(0)=1π(μq2)3f0,q(0)∫-∞∞‍1θ(q2;ω2)dω=2π(μq2)3f0,q(0)∫0∞‍1θ(q2;ω2)dω.Proposition 20.
For eachε>0, there exists a Q>1 such that for all 1<q<Q(181)0<P~0(q;0)-1<ε.Proof.
Subtracting (180) from (117), one has
(182)P~0(q;0)-1=2(μq2)3πf0,q(0)∫0∞‍ω-1θ(q2;ω2)dω=2(μq2)3πf0,q(0)×[∫01‍ω-1θ(q2;ω2)dω+∫1∞‍ω-1θ(q2;ω2)dω].
The change of variables ω=1/u is made on the first integral in (182), and the algebraic identity u2θ(q2;u-2)=θ(q2;u2) is used to obtain
(183)∫01ω-1θ(q2;ω2)dω=∫∞1‍u-1-1θ(q2;u-2)(-du)u2=∫1∞‍u-1-1θ(q2;u2)du.
Now (183) is used to reexpress (182) as
(184)P~0(q;0)-1=2(μq2)3πf0,q(0)[∫1∞‍ω-1-1θ(q2;ω2)dω+∫1∞‍ω-1θ(q2;ω2)dω](185)=2(μq2)3πf0,q(0)∫1∞‍ω-1-2+ωθ(q2;ω2)dω=2(μq2)3πf0,q(0)∫1∞‍(ω-1)2ωθ(q2;ω2)dω>0,
from which one sees that P~0(q;0)>1 for all q>1. Deploying the bound (27) within the integral in (185) gives
(186)0<P~0(q;0)-1≤F(q)[1ln⁡(q)∫1∞‍(ω-1)2ω-2e(ln⁡ω)2/ln⁡(q)dω]=F(q)[1ln⁡(q)∫1∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω],
where
(187)F(q)=1q1/42π[ln⁡(q)(μq2)3f0,q(0)]1π-ln⁡(q).
It follows from the q-Wallis formula (17) that
(188)lim⁡q→1+F(q)=1π.
We now show thatP~0(q;0)-1 can be made arbitrarily small for all q>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).
Letα>0 be arbitrary, with α being specified later. The integral in ω over the interval [1,∞) in (186) is now subdivided into two integrals, the first over [1,eαln⁡⁡(q)] and the second over [eαln⁡(q),∞). First, on [1,eαln⁡⁡(q)], one computes
(189)1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω.
Now, the function (1-ω-1)2 is increasing on [1,eαln⁡(q)], and it assumes its maximum value of (1-e-αln⁡⁡(q))2 at the right endpoint eαln⁡(q). Thus, we bound the integral in (189) by the length of the interval times the bound (1-e-αln⁡⁡(q))2 on the numerator. This gives
(190)1ln⁡⁡(q)∫1eαln⁡⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω≤1ln⁡(q)∫1eαln⁡(q)‍(1-e-αln⁡⁡(q))2dω(191)≤1ln⁡⁡(q)(eαln⁡⁡(q)-1)(1-e-αln⁡⁡(q))2.
An application of L’Hopital’s rule gives that
(192)lim⁡q→1+(eαln⁡⁡(q)-1)ln⁡(q)=α,
which implies that in (191) we have
(193)lim⁡q→1+[(eαln⁡⁡(q)-1)ln⁡(q)(1-e-αln⁡⁡(q))2]=α·02=0.
Combining (193) with (191) gives that
(194)lim⁡q→1+[1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω]=0.
We next estimate the portion of the integral in (186) over the interval [eαln⁡(q),∞). First, since 1-ω-1≤1 on this interval, one has
(195)1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω≤1ln⁡(q)∫eαln⁡(q)∞‍1e(ln⁡ω)2/ln⁡(q)dω.
We next bound the right-hand side of (195), using the following estimate from [4, 14]:
(196)∫C∞‍e-A[ln⁡ω+B]2dω≤π2e1/(4A)-BAe-A[ln⁡⁡(C)+B-1/(2A)]2,
which holds for C>e-B+1/(2A). Setting A=1/ln⁡(q), B=0, and C=eαln⁡(q) as in (195) yields that
(197)1ln⁡(q)∫eαln⁡(q)∞‍1e(ln⁡ω)2/ln⁡(q)dω≤1ln⁡(q)π2eln⁡(q)/41/ln⁡(q)e-[αln⁡(q)-ln⁡(q)/2]2/ln⁡(q)=π2e-α2[eαln⁡(q)]
for eαln⁡(q)>eln⁡(q)/2, or equivalently for 1<q<e4α2. Since the final expression in (197) approaches πe-α2/2 as q→1+, by choosing α sufficiently large one can make (197) and hence (195), arbitrarily small for q and sufficiently close to 1.
Now, letε>0 be given. By (186), we have for all α>0(198)0<P~0(q;0)-1≤F(q)[1ln⁡(q)∫1eαln⁡(q)(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω](199)+F(q)[1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω].
One has from (188) and (197) that
(200)lim⁡q→1+{F(q)π2e-α2[eαln⁡(q)]}=12e-α2<e-α2.
Fix α>0 such that e-α2<ε/2. By (197) and (200), there exists q1=q1(α,ε) such that for all 1<q<q1 one has
(201)0<F(q)[1ln⁡(q)∫eαln⁡(q)∞‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω](202)≤F(q)π2e-α2[eαln⁡⁡(q)]<e-α2<ε2.
For this value of α, by virtue of (194), one has
(203)lim⁡q→1+F(q)[1ln⁡⁡(q)∫1eαln⁡⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡⁡(q)dω]=0,
which in turn says that there is a q2=q2(α,ε) such that for all 1<q<q2 one has
(204)0<F(q)[1ln⁡(q)∫1eαln⁡(q)‍(1-ω-1)2e(ln⁡ω)2/ln⁡(q)dω]<ε2.
Thus, for Q(α,ε)≡min⁡{q1,q2} and for all 1<q<Q(α,ε), applying (204) and (202) to (198) and (199) yields
(205)0<P~0(q;0)-1<ε2+ε2<ε.
This gives the proposition.


## 10. Convergence inLp(R)

We turn next to convergence inLp(R).Theorem 21.
P
~
n
(
q
;
ω
) converges to Pn(ω)χ~[-1,1](ω) in Lp(R) as q→1+ for each 1≤p<∞. That is,
(206)lim⁡q→1+∥P~n(q;ω)-Pn(ω)χ~[-1,1](ω)∥p=0.Proof.
We first handle the casep=1, which, by boundedness of the functions under study, turns out to be sufficient to handle the remaining cases 1<p<∞.
Letε1>0 be given. First note that, by (118) and (120), there is a Q1>0 such that
(207)0<P~0(q;ω)≤P~0(q;0)<1+ε1
for all 1<q<Q1. Thus, P~0(q;ω) is uniformly bounded for q. Next, note that, for ω with |ω|≤2 and 1<q<Q1, one has from (109) that
(208)|P~n(q;ω)|≤∑k=0⌊n/2⌋‍|Cn,n-2k||ω|n-2kqk(k+1)|P~0(q;ωqk)|≤∑k=0⌊n/2⌋‍|Cn,n-2k|2nq⌊n/2⌋(⌊n/2⌋+1)(1+ε1)(209)=An2nQ1⌊n/2⌋(⌊n/2⌋+1)(1+ε1)≡Bn,
where An=∑k=0⌊n/2⌋|Cn,n-2k|. Observe that, since An, 2, and Q1 are all greater than 1 in (209), one has Bn>ε1. Note also that, from (141), one has the existence of a q5>1 such that for all 1<q<q5(210)|P~n(q;ω)|<ε1
for all ω such that |ω|≥3/2. From (209) and (210) and the fact that Bn>ε1, one has that, for 1<q<q6≡min⁡{Q1,q5}, we have
(211)|P~n(q;ω)|≤Bn
for all ω∈R. Thus, P~n(q;ω) is uniformly bounded in 1<q<q6. Letting Dn=sup⁡{|Pn(ω)|:|ω|≤1}, we also have that |Pn(ω)χ~[-1,1](ω)|≤Dn. Thus, our two functions of interest are bounded on R for 1<q<q6.
Next, choose0<ε<1 with ε<ε1/[12(Bn+Dn)]. Then, by oddness/evenness, respectively, one has
(212)∫-∞∞‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω=2∫0∞‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω(213)=2{∫01-ε‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω+∫1-ε1+ε‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω}(214)+2{∫1+ε∞‍|P~n(q;ω)-0|dω},
and we proceed to bound the three resulting integrals in (213) and (214).
From (130), there is a Q4>1 such that one obtains |P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|<ε1/6 for all ω∈[0,1-ε] and 1<q<Q4. This gives
(215)2∫01-ε‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω≤2∫01-ε‍ε16dω=2(1-ε)ε16<ε13
for all 1<q<Q4.
Next, by choice ofε, one has
(216)2∫1-ε1+ε‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω≤2∫1-ε1+ε‍|P~n(q;ω)|+|Pn(ω)χ~[-1,1](ω)|dω≤2∫1-ε1+ε‍Bn+Dndω=2(2ε)(Bn+Dn)(217)<2(2ε112(Bn+Dn))(Bn+Dn)=ε13
for all 1<q<Q1.
Finally, from (134), one has the existence of a q3>1 such that
(218)2∫1+ε∞‍|P~n(q;ω)-0|dω(219)≤2[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×π2qn(n+2)/4An∫1+ε∞‍e-[ln⁡(|ω|)-(n+1)ln⁡(q)/2]2/ln⁡(q)dω
for all 1<q<q3. Now, from (151), one has
(220)∫1+ε∞‍e-[ln⁡(|ω|)-(n+1)ln⁡(q)/2]2/ln⁡(q)dω≤π2ln⁡(q)eln⁡(q)/4+(n+1)ln⁡(q)/2×e-[ln⁡(1+ε)-(n+1)ln⁡(q)/2-ln⁡(q)/2]2/ln⁡(q)(221)=π2ln⁡(q)q1/4q(n+1)/2×e-[ln⁡(1+ε)-(n+2)ln⁡(q)/2]2/ln⁡(q)
for 1+ε>e(n+2)ln⁡(q)/2=q(n+2)/2. Thus, one uses (221) to bound (219) for 1<q<(1+ε)2/(n+2), obtaining
(222)2∫1+ε∞‍|P~n(q;ω)-0|dω(223)≤2[2(μq2)3ln⁡(q)πf0,q(0)]{π-ln⁡(q)}-1×πAn22ln⁡(q)q(n+1)(n+3)/4×e-[ln⁡(1+ε)-(n+2)ln⁡(q)/2]2/ln⁡(q).
Now, from (17), we have that (223) approaches 0 as q→1+. Thus, there is a Q5>1 such that, for 1<q<Q5,
(224)2∫1+ε∞‍|P~n(q;ω)-0|dω≤ε13.
Finally, using (215), (217), and (224) to bound (213) and (214) gives that
(225)∥P~n(q;ω)-Pn(ω)χ~[-1,1](ω)∥1=∫-∞∞‍|P~n(q;ω)-Pn(ω)χ~[-1,1](ω)|dω≤ε13+ε13+ε13=ε1
for all 1<q<min⁡{1+ε,q3,q6,Q1,Q4,Q5}. Thus, (206) holds in the case p=1.
Next, observe that iff and g are both bounded on R, then, for 1≤p<∞, one has
(226)∫-∞∞‍|f(x)-g(x)|pdx=∫-∞∞‍|f(x)-g(x)|p-1|f(x)-g(x)|dx(227)≤∥f-g∥∞p-1∫-∞∞‍|f(x)-g(x)|dx=∥f-g∥∞p-1∥f-g∥1.
Thus, in this bounded setting, L1 convergence is sufficient for Lp convergence.
We have observed that|P~n(q;ω)| is bounded by Bn for 1<q<q6. Also |Pn(ω)χ~[-1,1](ω)|≤Dn. By the above remarks, it follows that (206) now also holds in the cases 1<p<∞, and the theorem is proven.


## 11. ApproximatingP~n(q;ω)

The goal of this section is to provide and analyze the two approximations forP~n(q;ω) given by (228) and (229) below. These two expressions are related to the exact expression (109) for P~n(q;ω), but they are computationally simpler approximations of P~n(q;ω). We will also show that as q→1+ the difference of P~n(q;ω) 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):
(228)P~n(q;ω)≈P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1).
The second expression, (229), is obtained by setting q=1 in the summation for (228):
(229)P~n(q;ω)≈P~0(q;ω)Pn(ω)=P~0(q;ω)∑k=0⌊n/2⌋Cn,n-2kωn-2k.
The convergence is as follows. Given any compact set C⊂R, there is an ω0 with C⊂[-ω0,ω0]⊂R. Given such C and corresponding ω0 and given ε>0, there is a q^>1 such that for all 1<q<q^ one has
(230)|P~n(q;ω)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)|<ε,|P~n(q;ω)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2k|<ε,
for all ω∈[-ω0,ω0] and therefore for all ω∈C. Note first that, for n=0,1 there is equality in (228) and (229) for all q>1 and all ω∈R. So, we assume that n≥2 and proceed as follows.To obtain convergence for (228), note first that, for each p≥0, one has
(231)|P~0(q;ω)-P~0(q;ωqp)|=2(μq2)3πf0,q(0)×|∫-∞ω‍vθ(q2;v2)dv-∫-∞ω/qp‍vθ(q2;v2)dv|=2(μq2)3πf0,q(0)|∫ω/qpω‍vθ(q2;v2)dv|=2(μq2)3πf0,q(0)∫|ω|/qp|ω|‍vθ(q2;v2)dv(232)≤2(μq2)3πf0,q(0){|ω|-|ω|qp}max⁡(vθ(q2;v2))=2(μq2)3πf0,q(0)|ω|{1-1qp}1θ(q2;12),
where the maximum of v/θ(q2;v2) occurring at 1 is shown in [4]. From (26), we bound 1/θ(q2;12) in (232) from the above to give
(233)|P~0(q;ω)-P~0(q;ωqp)|≤2(μq2)3πf0,q(0)|ω|{1-1qp}q-1/4π/ln⁡⁡(q)-1=2ln⁡⁡(q)(μq2)3πf0,q(0)1ln⁡⁡(q)|ω|{1-1qp}q-1/4π-ln⁡⁡(q)=2ln⁡(q)(μq2)3πf0,q(0)|ω|(q-1)ln⁡(q)[∑k=0p-1‍qkqp]×q-1/4π-ln⁡⁡(q)(234)≤2ln⁡(q)(μq2)3πf0,q(0)|ω0|(q-1)ln⁡(q)ln⁡⁡(q)[p]q-1/4π
for all ω with |ω|≤|ω0|.From L’Hopital’s rule, one has that(235)lim⁡q→1+q-1ln⁡(q)=1,
which combines with (138) to imply that the coefficients in (234) have the following limit:
(236)lim⁡q→1+2ln⁡(q)(μq2)3πf0,q(0)q-1ln⁡(q)q-1/4π=2π·π2·1·1π=1π<1.
Thus, from (234) and (236), one has that there is a q7 such that, for all 1<q<q7 and all |ω|≤|ω0|,
(237)|P~0(q;ω)-P~0(q;ωqp)|≤1·|ω0|pln⁡(q).
Thus, for 0≤k≤p, one has
(238)|P~0(q;ω)-P~0(q;ωqk)|≤|ω0|kln⁡(q)≤|ω0|pln⁡(q)
for all |ω|≤|ω0| and all 1<q<q7.Now, from (109) coupled with (238), one has that
(239)|P~n(q;ω)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)|=|∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)(P~0(q;ωqk)-P~0(q;ω))|(240)≤(∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2kq7k(k+1))|ω0|⌊n2⌋ln⁡(q)
for all |ω|≤|ω0| and all 1<q<q7. Placing (240) below ε and solving the resulting inequality for q gives that, given ε>0, there is a qε>1 given by
(241)qε=min⁡{q7,exp⁡(|ω0|2⌊n2⌋2((∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2kq7k(k+1))2)-1ε2[(∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2kq7k(k+1))2×|ω0|2⌊n2⌋2((∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2kq7k(k+1))2]-1)}
such that
(242)|P~n(q;ω)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)|<ε
for all 1<q<qε and ω∈[-ω0,ω0]. This gives convergence for (228).We turn next to (229). Observe that, for |ω|≤|ω0|, one has
(243)|P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2k-P~0(q;ω)∑k=0⌊n/2⌋Cn,n-2kωn-2kqk(k+1)|=|P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2k(1-qk(k+1))|≤max⁡{|P~0(q;ω)|}∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2k(qk(k+1)-1)(244)≤P~0(q;0)·∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2k(q⌊n/2⌋(⌊n/2⌋+1)-1)(245)≤2·∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2k(q⌊n/2⌋(⌊n/2⌋+1)-1),
where (244) follows from (118) and (245) holds for 1<q<Q with Q chosen as in Proposition 20 (for ε set equal to 1). Placing (245) below ε and solving the resulting inequality for q gives that, given ε>0, there is a q~ε>1, given by
(246)q~ε=min⁡{Q,(1+ε2∑k=0⌊n/2⌋‍|Cn,n-2k||ω0|n-2k)1/{⌊n/2⌋(⌊n/2⌋+1)}}
such that
(247)|P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2k-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)|<ε
for |ω|≤|ω0| and 1<q<q^ε. Thus, by setting ε=ε~/2 in (242) and (247) and relying on the triangle inequality, one has
(248)|P~n(q;ω)-P~0(q;ω)Pn(ω)|≤|P~n(q;ω)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)|+|P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2kqk(k+1)-P~0(q;ω)∑k=0⌊n/2⌋‍Cn,n-2kωn-2k|≤ε~2+ε~2=ε~
for |ω|≤|ω0| and 1<q<min⁡{qε~/2,q^ε~/2}. This gives convergence for (229). Now let q^=min⁡{qε,qε~/2,q^ε~/2} and the result holds.We remark here thatP~0(q;ω) in (229) is a smooth Schwartz analogue of the truncation function χ~[-1,1](ω) in (19). Note also that the approximations (228) and (229) need not be wavelets, while each P~n(q;ω) for n≥1 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 Section4. 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 ω=0, because its inverse Fourier transform will be a wavelet solution to a corresponding MADE.Consider a differential equation of the form(249)a(t,Dt)f(t)=[b(tq,Dt)f](qt).
Here a(t,τ) and b(t/q,τ) are polynomials in the two variables t/q and τ as in (250). More explicitly, (249) is given in expanded form by
(250)∑‍aijtiDtjf(t)=∑‍bmktm·[Dtkf](qt).
Now, suppose that there are nonnegative integers Na and Mb so that as t→0(251)a(t,τ)=∑‍aijtiτj⟶∑j=0Na‍a0jτj≡A(τ),b(tq,τ)=∑‍bmk(tq)mτk⟶∑k=0Mb‍b0kτk≡B(τ),
where the degrees deg⁡⁡A(τ) and deg⁡⁡B(τ) are not both -∞ (i.e., a0Na≠0 or b0Mb≠0). Then, letting t→0 in the operators in (249) (but not in the arguments for the function f), one obtains A(τ)f(t)=[B(τ)f](qt) or equivalently A(Dt)f(t)=[B(Dt)f](qt). That is, as t→0 in the operators, (249) and (250) formally converge to the equation
(252)a0Na[τNaf](t)+∑j=0Na-1‍a0j[τjf](t)=b0Mb[τMbf](qt)+∑k=0Mb-1‍b0k[τkf](qt),
which is equivalent to
(253)a0Na[DtNaf](t)+∑j=0Na-1‍a0j[Dtjf](t)=b0Mb[DtMbf](qt)+∑k=0Mb-1‍b0k[Dtkf](qt).
We refer to (249) as a MADE if it formally converges to (253) where deg⁡⁡B(τ)≤deg⁡⁡A(τ) as t→0 (i.e., Mb≤Na). 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 t of f not multiplicatively advanced or delayed by q and some lower or equal order term with constant coefficient does contain an argument of form qt.Next, take the inverse Fourier transform of (250) and denote F-1[f]=f^. Then, from (46) and (58), one obtains
(254)a(-iDω,-iω)f^(ω)=(1q)b(-iDω,-iωq)f^(ωq),
where the polynomials a and b are identical to those in (249). Now, using the derivation property of Dω to reorder the ω terms to precede the Dω terms in (254), collecting powers of i, and absorbing powers of q into coefficients, one obtains two resulting polynomials a^(ω,w) and b^(qω,w) in the two variables ω and w with
(255)a^(ω,Dω)f^(ω)=[b^(qω,Dω)f^](ωq).
Now, suppose that as ω→0 in the operators of (255) one obtains
(256)a^(ω,w)⟶∑j=0N^a‍a^jwj≡A^(w),b^(ω,w)⟶∑k=0M^b‍b^kwk≡B^(w),
where deg⁡⁡A^ and deg⁡⁡B^ are not both -∞. (i.e., a^N^a≠0 or b^N^b≠0). Then, similar to the process for (253), one has (255) that formally converges to the equation
(257)a^N^a^[DωN^a^f^](ω)+∑j=0N^a^-1‍a^j[Dωjf^](ω)=b^M^b^[DωM^b^f^](ωq)+∑k=0M^b^-1‍b^k[Dωkf^](ωq).
Now, when deg⁡⁡B^≥deg⁡⁡A^ (i.e., M^b^≥N^a^), (257) will not have its highest order term with constant coefficients satisfying that its argument is neither advanced nor delayed by q>1. Thus, when deg⁡⁡B^≥deg⁡⁡A^ (i.e., M^b^≥N^a^), we change variables so that u=ω/q to give
(258)b^M^b^[DuM^b^f^](u)+∑k=0M^b^-1‍b^k[Dukf^](u)=qa^N^a^[DuN^a^f^](qu)+q∑j=0N^a^-1‍a^j[Dujf^](qu),
yielding a MADE. Similarly, one now relies on the change of variable u=ω/q to convert (255) into a MADE. Thus, (249) starts as a MADE when deg⁡⁡B(τ)≤deg⁡⁡A(τ) (i.e., Mb≤Na) and converts to a MADE under inverse Fourier transform when deg⁡⁡B^≥deg⁡⁡A^ (i.e., M^b^≥N^a^).Example 22.
Equation (57) in the proof of Theorem 6 is
(259)t2jn′′(q;t)+2tjn′(q;t)-n(n+1)jn(q;t)=-qn+3t2jn(q;qt),
which can be rewritten as
(260)a(t,Dt)jn(q;t)=[b(tq,Dt)jn](q;qt),
where
(261)a(t,τ)=t2τ2+2tτ-n(n+1),b(tq,τ)=-qn+3(tq)2.
Note that substituting qt into (t/q)2 in (260) gives t2 in (259). For the case n≥1, taking limits as t→0 gives
(262)lim⁡t→0a(t,τ)=-n(n+1)≡A(τ),lim⁡t→0b(tq,τ)=0≡B(τ).
Thus, (259) is appropriately considered a MADE as deg⁡⁡A=0≥deg⁡⁡B=-∞. That is the highest order term in (259) with constant coefficients, namely, -n(n+1)jn(q;t), does not have its argument t advanced or delayed by q>1.
Taking inverse Fourier transforms of both sides of (259), relying on (46) and (58), and multiplying through by (-i)n2/π yields (64); namely,
(263)ω2Dω2P~n(q;ω)+2ωDωP~n(q;ω)-n(n+1)P~n(q;ω)=qn+2Dω2{P~n(q;ωq)}.
This is equivalent to
(264)a^(ω,Dω)P~n(q;ω)=[b^(qω,Dω)P~n](q;ωq),
where
(265)a^(ω,w)=ω2w2+2ωw-n(n+1),b^(qω,w)=qnw2.
Taking limits as ω→0 gives
(266)lim⁡ω→0a^(ω,w)=-n(n+1)≡A^(w),lim⁡ω→0b^(ω,w)=qnw2≡B^(w)
with deg⁡⁡A^=0≤deg⁡⁡B^=2. Since B^ acts on a term with a delayed argument in (264), one makes the substitution u=ω/q in (263) to obtain the MADE equivalent to (56)
(267)(qu)2{1qDu}2{P~n(q;qu)}+2(qu){1qDu}{P~n(q;qu)}-n(n+1)P~n(q;qu)=qn+2{1qDu}2P~n(q;u).Example 23.
Forq>1, consider the MADE
(268)DttDt2tDtf(t)=f(qt).
Equation (268) is appropriately considered a MADE as follows. After using derivation properties to have the t variables precede the Dt operators, (268) becomes
(269)2Dt2f(t)+4tDt3f(t)+t2Dt4f(t)=f(qt).
With the corresponding polynomials
(270)a(t,τ)=2τ2+4tτ3+t2τ4,b(tq,τ)=1.
Letting t→0 yields a(0,τ)=2τ2=A(τ) and b(0,τ)=1. Since deg⁡⁡A=2>deg⁡⁡B=0, (269) and (268) are MADEs.
Taking inverse Fourier transform of (268) and relying on (46) and (58), one obtains
(271)(-iω)(-iDω)(-iω)2(-iDω)(-iω)F-1[f(t)](ω)=1qF-1[f(t)](ωq),
which simplifies to
(272)ωDωω2Dωωf^(ω)=-q-1f^(ωq).
After using derivation properties to have the ω variables precede the Dω operators, (272) becomes
(273)2ω2f^(ω)+4ω3Dωf^(ω)+ω4Dωf^(ω)=-q-1f^(ωq)
with the corresponding polynomials
(274)a^(ω,w)=2ω2+4ω3w+ω4w2,b^(qω,w)=-q-1.
Letting ω→0 yields a^(0,w)=0=A^(w) and b(0,w)=-q-1. Since deg⁡⁡B^=0>deg⁡⁡A^=-∞, we convert (273) by letting u=ω/q, 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 -q-1f^(ω/q), we make the substitution u=ω/q with Dω=q-1Du to reexpress (272) as the MADE
(275)quq-1Du(qu)2q-1Du(qu){f^(qu)}=-q-1f^(u),
which simplifies to the following equation:
(276)f^(u)=-q3uDuu2Duu{f^(qu)}.
We exhibit a functionf^ with f^(ω) a solution of (272) (and with f^(u) a solution of (276) as well) satisfying the condition that f^(0)=0. Set
(277)f^(ω)=Sin⁡q(1/[qω])ω.
To verify (272), one proceeds as follows:
(278)ωDωω2Dωωf^(ω)=ωDωω2DωSin⁡q(1qω)=ωDωω2(-qqω2Cos⁡q(1ω))=-ωDωCos⁡q(1ω)=ω(1ω2(-1)Sin⁡q(1ω))=-q-1Sin⁡q(1/[q(ω/q)])ω/q=-q-1f^(ωq).
We have relied on the facts that(279)Sin⁡′q(t)=qCos⁡q(qt),Cos⁡′q(t)=-Sin⁡q(t),
where Sin⁡q(t) is given by (9) and Cos⁡q(t) is given by (28). These facts are proven in [5].
Note that sinceSin⁡q(ω) is Schwartz, one has f^(0)=lim⁡ω→0Sin⁡q(1/[qω])/ω=0. And thus the solution to the MADE (268) given by
(280)f(t)=F[f^(ω)](t)=12π∫-∞∞‍e-itωSin⁡q(1/[qω])ωdω=12π∫-∞∞‍cos⁡(ωt)Sin⁡q(1/[qω])ωdω=-12π∫-∞∞‍cos⁡(tqv)Sin⁡q(v)vdv=-12π∫-∞∞‍cos⁡(tqv)j0(q;v)dv
exhibits wavelet properties, since its 0th moment vanishes. A change of variables v=1/[qω] was made to obtain the last two equalities and express the solution in terms of the 0th order q-advanced spherical Bessel function.



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*Source: 890456-2014-10-16.xml*