Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Abstract

Purpose

In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.

Design/methodology/approach

To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.

Findings

Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.

Originality/value

In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

Citation

Molano, A. (2023), "Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials", Arab Journal of Mathematical Sciences, Vol. 29 No. 2, pp. 228-241. https://doi.org/10.1108/AJMS-07-2021-0164

1. Introduction

Within the framework of spectral approximation, and to recover values of smooth functions with exponential accurate, it is customary to use Fourier series for periodic problems and series of classical orthogonal polynomials for nonperiodic problems. Nevertheless, if it deals with piecewise smooth function, estimates by means of partial sums are unhealthy; oscillations do not decrease near discontinuities with partial sums of higher order; and far of them, convergence order is low. Thus, the global properties from Fourier coefficients are not enough to obtain local information. This lack of uniform convergence is known as Gibbs phenomenon. A priori, this is a serious issue considering the large number of applications modeled through piecewise smooth function. In literature, methods to face the Gibbs phenomenon in reconstruction of piecewise smooth functions from partial sums have been widely studied. For instance, in Refs. [1, 2], the problem to construct piecewise smooth function values with exponential accuracy at all points is solved by means of approximations with Fourier–Gegenbauer coefficients expansions. These are the so-called Gegenbauer reconstruction methods where the expansion of Gegenbauer polynomials in its Fourier series is crucial. In Ref. [3], the Gegenbauer reconstruction methods are revisited and analyzed in order to prove that Gegenbauer reconstruction is also effective for Fourier–Bessel series. To do that, the author obtains coefficients Fourier for Jacobi polynomials and also for classical orthogonal polynomials with unbounded support (Laguerre, Hermite).

On the other hand, consider a vector of Borel positive measures (μ₀, μ₁, …, μ_m), on the real line, with finite moments and μ₀ with continuous support. Then, we define the Sobolev inner product on the space of polynomials with real coefficients.

(1.1)f,gS=∫Rf(x)g(x)dμ0+∑k=1m∫Rf(k)(x)g(k)(x)dμk.

A sequence of polynomials Snn≥0,   deg  S_n = n, is orthogonal with respect to (1.1) if

Sn,SmS=Knδn,m, Kn>0.

The sequence Snn≥0 is said to be a sequence of Sobolev polynomials orthogonal with respect to (1.1). If μ_k is discrete, for k = 1, …, m, the above inner product and the sequence Snn≥0 are said to be of Sobolev type. Sobolev orthogonal polynomials have been widely studied in the last three decades. The first publication on Sobolev polynomials goes back to 1962 in Ref. [4], which deals with certain extremal problem related to smooth polynomial approximation whose solution is posed by means of Sobolev–Legendre polynomials. Such a problem is formulated previously in Ref. [5], although not in terms of Sobolev orthogonality. It has been documented as the approximations with Sobolev–Fourier series from smooth functions in the corresponding Sobolev space improve approximations made through standard families of orthogonal polynomials (see Ref. [6]). Additional applications include spectral methods in numerical analysis for ordinary differential equations and partial differential equations, and generalization of Gauss quadrature formulas, among others. The nice surveys [7, 8] are highly recommended, as well as the paper [9] and references therein. In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product (1.1) when μ₀ is the classical and absolutely continuous Laguerre measure on [0, ∞). In the next section, we propose the basic background with respect to Laguerre polynomials, and we present the particular Sobolev–Laguerre type families of polynomials to be discussed. In Section 3, we obtain the respective Fourier coefficients by using of similar techniques as the presented in Ref. [10]. Since the orthogonality interval for Laguerre polynomials is unbounded, we will turn special attention to oscillation regions for the Sobolev polynomials.

2. Preliminaries

Let P be the space of polynomials with real coefficients

2.1 Classical Laguerre polynomials and generalities

The classical Laguerre polynomials Lnαn≥0, with α > − 1, are orthogonal with respect to the inner product:

p,qα≔∫0∞p(x)q(x)e−xxαdx,p,q∈P.

For an arbitrary polynomial p, k(p) will denote the leading coefficient of p. In the sequel, to normalize Laguerre polynomials, we assume that k(Lnα)≔(−1)n/n. These polynomials satisfy the three terms recurrence relation (TTRR in short),

(2.1)(n+1)Ln+1α(x)=(2n+α+1−x)Lnα(x)−n+αLn−1α(x),

for n ≥ 0 with the initial conditions L−1α≔0 and L0α=1. For n ≥ 1, the zeros of every Lnα are all real, simple and are located in (0, ∞) (see Ref. [11]). In the sequel, xn,ii=1n will denote the zeros of Lnα ordered in increasing order.

Definition 1.

Let p be a polynomial with real zeros. An oscillatory region I for p is any bounded interval containing their zeros, in such a way that p is monotone outside I.

With respect to an oscillatory region of classical Laguerre polynomials, we get the next.

Proposition 1.

([11]). For α > − 1 and n > 0, the n zeros of Lnα are into [0, ζ_n,α], where

ζn,α=2n+α+1+2n+α+12+14−α2.

We consider, for a nonnegative integer m, the functions βma,b defined as (see Ref. [10]),

(2.2)βma,b(z)=∫abxme−zxdx=z−m−1m!e−azem(az)−e−bzem(bz),

where e_m is the m − th partial sum of the Maclaurin series for the exponential function and [a, b] is a bounded interval.

As a consequence of this definition, it is possible to show that if x = ɛξ + δ, ε=b−a2 and δ=b+a2, we get

(2.3)∫−11εξ+δme−iπkξdξ=1εeikδ/εβma,biπkε.

In this way, the next result for the Fourier series for Laguerre polynomials is presented in Ref. [10].

Theorem 1.

Let [a, b] be an interval with − ∞ < a < b < ∞ and ξ ∈ [ − 1, 1], ε=b−a2, δ=b+a2. The Fourier coefficients for Lnα(εξ+δ), in the local variable ξ, are given by

(2.4) L^nα(k)=12εeikδ/ε∑t=0n(−1)tt!n+αn−tβta,biπkε.

2.2 Quasi-orthogonality and zeros

Let Pnn≥0 be a sequence of polynomials orthogonal with respect to a positive Borel measure μ supported on [a, b], with − ∞ ≤ a < b ≤ ∞, i.e.

∫abPn(x)xkdμ=δn,kKn,Kn>0,   k=1,…,n.

Definition 2.

Let r be a nonnegative integer and R_n a polynomial with degree n ≥ r satisfying ∫abRn(x)xkdμ=0 for k = 0, 1, 2, …, n − r − 1, and ∫abRn(x)xn−rdμ≠0. Then, R_n is said to be quasi-orthogonal of order r on [a, b] and with respect to μ.

Of course, if r = 0, then the orthogonality is recovered. The next result describes a necessary and sufficient condition for quasi-orthogonality.

Proposition 2.

([12]). R_n is quasi-orthogonal of order r on [a, b] with respect to μ if and only if there exist numbers b_n,i, i = 0, 1, …, r, with b_n,0b_n,r ≠ 0, such that

(2.5) Rn(x)=∑k=0rbn,kPn−k.

With respect to zeros of quasi-orthogonal polynomials, the next result is well known.

Proposition 3.

([12]). If R_n is quasi-orthogonal of order r with respect to μ on [a, b], then R_n has n − r simple zeros on (a, b).

Suppose that R_n is quasi-orthogonal of order r with respect to μ on [a, b] and R_n and P_n are monic. It is well known that the monic orthogonal polynomials Pnn≥0 can be obtained by means of a TTRR:

(2.6)Pn+1(x)=(A^n+1x+B^n+1)Pn(x)−C^n+1Pn−1(x),n≥0,

and we define Bn+1≔k(Pn)B^n+1k(Pn+1) and Cn+1≔k(Pn−1)C^n+1k(Pn+1). In the particular case, when r = 2, from (2.5), we get R_n(x) = b_n,0P_n(x) + b_n,1P_n−1(x) + b_n,2P_n−2(x), and we also define an≔k(Pn−1)bn,1k(Rn) and bn≔k(Pn−2)bn,2k(Rn). The next results refer to behavior and localization of zeros of quasi-orthogonal polynomials for r = 2.

Theorem 2.

([13]). If b_n ≤ C_n, the n zeros of R_n are real and simples.

Theorem 3.

([13]). Suppose that xn,ii=1n and yn,ii=1n are the zeros of P_n and R_n, respectively, and ordered in increasing order.

1. b_n < C_n if and only if

   (2.7) yn,1<xn−1,1<yn,2<xn−1,2<⋯<yn,n−1<xn−1,n−1<yn,n.

2. 0 < b_n < C_n and an>−bnxn,1+BnCn if and only if

   (2.8) yn,1<xn,1<yn,2<xn,2<⋯<xn,n−1<yn,n<xn,n.

3. 0 < b_n < C_n and an<−bnxn,n+BnCn if and only if

   (2.9) xn,1<yn,1<xn,2<yn,2<⋯<yn,n−1<xn,n<yn,n.

Theorem 4.

([13]). Suppose that b_n < C_n and b_n+1 < C_n+1 and we define

(2.10) fn(x)=k(Pn−1)Pn(x)k(Pn)Pn−1(x).

For i = 1, 2, …, n

fn+1(yn,i)fn(yn,i)+an+1fn(yn,i)+bn+1<0,

if and only if

(2.11)yn+1,1<yn,1<yn+1,2<yn,2<⋯<yn+1,n<yn,n<yn+1,n+1,

2.3 Laguerre–Sobolev type orthogonal polynomials, nondiagonal case

If p∈P and P(x)≔(p(x), p′(x))^t, we define the Laguerre–Sobolev type inner product

(2.12)p,qS1=p,qα+P(0)tAQ(0),

where A=M0λλM1, with M₀M₁ ≥ 0 and λ such that det A ≥ 0. Let Snαn≥0 be the sequence of polynomials orthogonal with respect to (2.12) such that k(Snα)=k(Lnα) for n ≥ 0.

Theorem 5.

([14]). For every n∈N

(2.13) Snα(x)=Lnα+2(x)+An,αLn−1α+2(x)+Bn,αLn−2α+2(x),

where

An,α∼α+1α+2n−2,Bn,α∼1−α+1α+2n.

2.3.1 Laguerre–Sobolev type polynomials of higher order derivatives

Let Sn,mα,Wn≥0 be orthogonal with respect to Sobolev inner product

(2.14)p,qS,mα,W=p,qα+Wp(m)(0)q(m)(0),

with W > 0, m a nonnegative integer and p,q∈P. Moreover k(Sn,mα,W)=k(Lnα).

Theorem 6.

([15]). For n > m

(2.15) Sn,mα,W(x)=∑k=0m+1An,kLn−k(α+k)(x),

where

An,0=1+WΓ(m+1)Γ(α+m+1)∑k=1m+1(−1)k+1n+αn−m−kn−km+1−k,

and for k = 1, …, m + 1

An,k=(−1)kWΓ(m+1)Γ(α+m+1)n+αn−mn−km+1−k.

With respect to zeros of every Sn,mα,W, we enunciate the next results.

Theorem 7.

(See Ref. [16]). For every n, the zeros of Sn,mα,W are real, simple and at most one of them is outside (0, ∞). If Sn.mα,W has a zero in (−∞, 0] then n ≥ m + 1. In addition, if for n₀, Sn0,mα,W has a negative zero, then Sn,mα,W has a negative zero for n > n₀.

Theorem 8.

Assume that n ≥ m + 1. If {υn,i}i=1n are the zeros of Sn,mα,W, ordered in increasing order, then υ_n,i < x_n,i for i = 1, …, n.

Theorem 9.

If ρ_n is the negative zero of Sn,mα,W then −mυ̃n,m+1<ρn, where υ̃n,m+1 denotes the m − th positive zero of Sn,mα,W.

2.3.2 Christoffel transformations and Laguerre–Sobolev type inner product with mass outside support

Given ξ ≤ 0, and an integer k ≥ 1, we consider the weight ω_α,k(x) = (x − ξ)^ke^−xx^α, on [0, ∞). This is a Christoffel perturbation of the classical Laguerre measure (see Ref. [11]). Ln(α,k)n≥0 denotes the respective sequence of orthogonal polynomials, where k(Ln(α,k))=k(Lnα) for every n and Ln(α,0)≔Lnα. An algebraic connection between polynomials orthogonal with respect to the weight ω_α,k(x) is as follows (see Ref. [11]),

(2.16)(x−ξ)Ln(α,k)(x)=Ln+1(α,k−1)(x)−Ln+1(α,k−1)(ξ)Ln(α,k−1)(ξ)Ln(α,k−1)(x).

Assume xn,i[k]i=1n are the zeros of Ln(α,k) in increasing order, with xn,i[0]≔xn,i.

Proposition 4.

([17]). For i = 1, …, n

(2.17) xn,i[k−1]<xn,i[k]<xn+1,i+1[k−1].

Now we consider the Sobolev–Laguerre type inner product:

(2.18)p,qα,M,N=p,qα+Mp(ξ)q(ξ)+Np′(ξ)q′(ξ),

with M, N ≥ 0, ξ ≤ 0. Let Snα,M,Nn≥0 be the respective sequence of orthogonal polynomials such that k(Snα,M,N)=k(Lnα) for n ≥ 0.

Theorem 10.

([17]). There exist constants D_n,0, D_n,1 and D_n,2 such that

(2.19) Snα,M,N(x)=Dn,0Lnα(x)+Dn,1(x−ξ)Ln−1(α,2)(x)+Dn,2(x−ξ)2Ln−2(α,4)(x),

where:

1. If M, N > 0, then Dn,0∼8ξnαMLα(ξ)2, Dn,1∼32(−ξ)3/2nα−1/2MLα(ξ)2 and Dn,2∼1n2.

2. If M = 0 and N > 0, then Dn,0∼14−ξn, Dn,1∼−1n and Dn,2∼14n2−ξn.

3. If N = 0 and M > 0, then Dn,0∼−ξMn1/2−αLα(ξ)2, Dn,1∼−1n and D_n,2 = 0.

Let υn,ii=1n be the zeros of Snα,M,N in increasing order. To describe results on zeros of every Snα,M,N, we present the next results.

Proposition 5.

([17]). The zeros of Snα,M,N are real, simple and at most one of them is outside [ξ, ∞).

Proposition 6.

([18]). If ξ < υ_n,1 then

(2.20) υn,1<xn,1<⋯<υn,n<xn,n.

Proposition 7.

([17]). Suppose that υ_n,1 < ξ. Then

(2.21) 2ξ−xn−1,1[2]<υn,1<ξ<υn,2<xn−1,2[2]<⋯<υn,n<xn−1,n−1[2].

3. Fourier coefficients for Laguerre–Sobolev type polynomials

In this section, we describe the Fourier coefficients associated to Laguerre–Sobolev type polynomials presented in the above section, computed on any finite interval [a, b]. For approximation purposes, we will find an oscillatory region for every family of Sobolev–Laguerre polynomials, in order to exhibit a reasonable choose for the interval [a, b].

3.1 Nondiagonal case

Let Snαn≥0 be the sequence of Sobolev polynomials orthogonal with respect to the inner product (2.12). From (2.13), this sequence is quasi-orthogonal of order 2 with respect to the classical Laguerre polynomials with parameter α + 2. Then, we consider (2.1) for α + 2, and from (2.13) we define

an=An,α,bn=Bn,α,

and

Bn+1=−(2n+α+3),Cn+1=n(n+α+2).

Then, in the language of Theorem 3, we get the next result.

Corollary 1.

If

(3.1) 0<Bn+1,α<n(n+α+2),An+1,α<−Bn+1,αxn+1,n+1α+2−(2n+α+3)n(n+α+2),

(3.2) 0<Bn,α<(n−1)(n+α+1),An,α>−Bn,αxn,1α+2−(2n+α+1)(n−1)(n+α+1)

and

(3.3) fn+1(yn,i)fn(yn,i)+An+1,αfn(yn,i)+Bn+1,α<0,

for i = 1, 2, …, n, then Snαhas n real and simple zeros in the interval 0,ζn,α+2. Here, yn,ii=1n and xn,iα+2i=1n represent the zeros of Snα and Lnα+2, respectively, ordered in increasing order.

Proof. According to Theorem 3, Part 3, inequalities in (3.1) are equivalent to

xn+1,1α+2<yn+1,1<xn+1,2α+2<yn+1,2<⋯<yn+1,n<xn+1,n+1α+2<yn+1,n+1.

and from Part 2, inequalities in (3.2), are equivalent to

yn,1<xn,1α+2<yn,2<xn,2α+2<⋯<xn,n−1α+2<yn,n<xn,nα+2.

In the other hand, from Theorem 4, (3.1) is equivalent to

yn+1,1<yn,1<yn+1,2<yn,2<⋯<yn+1,n<yn,n<yn+1,n+1.

Finally, since the zeros of Lnα+2 and Ln+1α+2 are interlaced, we obtain

xn,nα+2<xn+1,n+1α+2.

The above inequalities imply that

xn+1,1α+2<yn+1,1<yn,1<⋯<yn,n<xn,nα+2<xn+1,n+1α+2.

From the Proposition 1, we get the result. □

On the other hand, we suppose that x ∈ [a, b], and we make the transformation x = ɛξ + δ, where ξ ∈ [ − 1, 1], ε=b−a2 and δ=b+a2.Then

Snα(εξ+δ)=∑k=0∞Snα^(k)eikπξ,

and by using of (2.13) we get

Snα^(k)=12∫−11Lnα+2(εξ+δ)+An,αLn−1α+2(εξ+δ)+Bn,αLn−2α+2(εξ+δ)e−ikπξdξ=12∫−11Lnα+2(εξ+δ)e−ikπξdξ+12An,α∫−11Ln−1α+2(εξ+δ)e−ikπξdξ+12Bn,α∫−11Ln−2α+2(εξ+δ)e−ikπξdξ,=12L^nα+2(k)+12An,αL^n−1α+2k+12Bn,αL^n−2α+2(k),

and by using of (2.4) we obtain

Snα^(k)=12L^nα+2(k)+12An,αL^n−1α+2k+12Bn,αL^n−2α+2(k),=14εeikδ/ε∑t=0n(−1)tt!n+α+2n−tβta,biπkε+An,α∑t=0n−1(−1)tt!n+α+1n−1−tβta,biπkε+Bn,α∑t=0n−2(−1)tt!n+αn−2−tβta,biπkε=14εeikδ/ε(−1)nn!βna,biπkε−nn+α+2βn−1a,biπkε−An,αnβn−1a,biπkε+∑t=0n−2(−1)tt!βta,biπkεn+α+2n−t+An,αn+α+1n−1−t+Bn,αn+αn−2−t.

We summarize in the next.

Proposition 8.

Let [a, b] be a bounded interval. Assume x in [a, b], and x = ɛξ + δ, where ξ ∈ [ − 1, 1], ε=b−a2 and δ=b+a2. The coefficients of Fourier for Snα(εξ+δ), in the local variable ξ, are giving by

Snα^(k)=14εeikδ/ε(−1)nn!βna,biπkε−nn+α+2βn−1a,biπkε−An,αnβn−1a,biπkε+∑t=0n−2(−1)tt!βta,biπkεn+α+2n−t+An,αn+α+1n−1−t+Bn,αn+αn−2−t.

3.2 Higher order derivatives

Sn,mα,Wn≥0 represents the sequence of polynomials orthogonal with respect to (2.14). As before, we propose a bounded interval that containing the n zeros of Sn,mα,W for n large enough.

Corollary 2.

For n ≥ m + 1, the zeros of Sn,mα,W are located in [ − mx_n,m+1, ζ_n,α].

Proof. From Theorem 8, υ̃n,m+1<xn,m+1, and from Theorem 9, the result is

−mxn,m+1<ρn,

□

As before, we assume x ∈ [a, b] and x = ɛξ + δ, where ξ ∈ [ − 1, 1], ε=b−a2 and δ=b+a2.

From (2.15) we have

S^n,mα,W(k)=12∫−11Sn,mα,W(εξ+δ)e−ikπξdξ=12∑q=0m+1An,q∫−11Ln−q(α+q)(εξ+δ)e−ikπξdξ=12∑q=0m+1An,qL^n−qα+q(k),

and from (2.4) we arrive to the next.

Proposition 9.

Let [a, b] a bounded interval and n ≥ m + 1. Consider the transformation x = ɛξ + δ, where x ∈ [a, b], ξ ∈ [ − 1, 1], ε=b−a2 and δ=b+a2. The Fourier series for Sn,mα,W, with the local variable ξ, is given by

Sn,mα,W(εξ+δ)=∑k=−∞∞S^n.mα,W(k)eikπξ,

where

S^n,mα,W(k)=14εeikδ/ϵ∑q=0m+1∑t=0n−q(−1)tt!An,qn+αn−q−tβta,biπkε.

3.3 Mass outside support

Assume that

(3.4)Ln(α,k)(x)=∑j=0nan,jα,[k]xj,

with an,jα,[0]≔an,jα=(−1)jj!n+αn−j (see Ref. [11]). From (2.16), for k ≥ 1,

(3.5)(x−ξ)Ln(α,k)(x)=Ln+1(α,k−1)(x)−dn+1,ξ(α,k−1)Ln(α,k−1)(x),

with

dn+1,ξ(α,k−1)=Ln+1(α,k−1)(ξ)Ln(α,k−1)(ξ).

Then, replacing (3.4) in (3.5) we obtain

∑j=1n+1an,j−1α,[k]xj−∑j=0nξan,jα,[k]xj=an+1,n+1α,[k−1]xn+1+∑j=0nan+1,jα,[k−1]−dn+1,ξ(α,k−1)an,jα,[k−1]xj,

or equivalently

an,nα,[k]xn+1+∑j=1nan,j−1α,[k]−ξan,jα,[k]xj−ξan,0α,[k]=an+1,n+1α,[k−1]xn+1+∑j=1nan+1,jα,[k−1]−dn+1,ξ(α,k−1)an,jα,[k−1]xj+an+1,0α,[k−1]−dn+1,ξ(α,k−1)an,0α,[k−1],

then we get the equations

an,nα,[k]=an+1,n+1α,[k−1],

ξan,0α,[k]=dn+1,ξ(α,k−1)an,0α,[k−1]−an+1,0α,[k−1],

and

ξan,jα,[k]=an,j−1α,[k]+dn+1,ξ(α,k−1)an,jα,[k−1]−an+1,jα,[k−1],j=1,…,n.

Lemma 1.

If

Ln(α,k)(x)=∑j=0nan,jα,[k]xj,an,jα,[0]≔an,jα,

and ξ < 0, then

an,nα,[k]=an+1,n+1α,[k−1],

and the coefficients an,jα,[k], with j = 1, …, n − 1, can be obtained recurrently by means of

an,jα,[k]=ξ−1an,j−1α,[k]+dn+1,ξ(α,k−1)an,jα,[k−1]−an+1,jα,[k−1],

with the initial condition

an,0α,[k]=ξ−1dn+1,ξ(α,k−1)an,0α,[k−1]−an+1,0α,[k−1].

If ξ = 0 then

an,nα,[k]=an+1,n+1α,[k−1],

and for j = 1, …, n

an,j−1α,[k]=an+1,jα,[k−1]−dn+1,0(α,k−1)an,jα,[k−1].

According to (2.17), we can deduce that

xn,1<xn,1[1]<xn,1[2]<xn,1[3]<⋯<xn,1[k]<⋯,

and

xn,n[k]<xn+1,n+1[k−1]<xn+2,n+2[k−2]<⋯<xn+k,n+k,

and as consequence we get the next.

Lemma 2.

The zeros of Ln(α,k) are located in [x_n,1, β_n + k,α].

Since the Fourier series for Ln(α,k)(εη+δ) in the local variable η is determined by the coefficients

L^n(α,k)(k)=12∫−11Ln(α,k)(εη+δ)e−ikπηdη,

by using of Lemma 1, (2.2) and (2.3) we have the next.

Proposition 10.

Fourier coefficients for Ln(α,k) on a finite interval [a, b], in the local variable η, with x = ɛη + δ, η ∈ [ − 1, 1], ε=b−a2 and δ=b+a2, are defined by

(3.6) L^n(α,k)(k)=eikδ/ε2ε∑j=0nan,jα,[k]βja,biπkε.

Let Snα,M,Nn≥0 be the sequence of polynomials orthogonal with respect to (2.18).

Corollary 3.

For n ≥ 2, the zeros of Snα,M,N are into [2ξ − x_n+1,3, x_n+1,n+1].

Proof. From (2.17) we get

xn−1,1[1]<xn−1,1[2]<xn,2[1],    and      xn,2<xn,2[1]<xn+1,3,

thus

(3.7) xn−1,1[2]<xn+1,3.

In the same way, xn−1,n−1[1]<xn−1,n−1[2]<xn,n[1] and xn,n<xn,n[1]<xn+1,n+1, thus

(3.8)xn−1,n−1[2]<xn+1,n+1.

On the other hand, from (2.20), (2.21), (3.7) and (3.8), we obtain

2ξ−xn+1,3<2ξ−xn−1,1[2]<υn,1<⋯<υn,n<xn+1,n+1

□

On the one hand, and on a finite interval [a, b], we compute the Fourier coefficients for every Snα,M,N and in terms of the local variable η. Indeed, if

Snα,M,N(εη+δ)=∑k=0∞S^nα,M,N(k)eikπη,

by using of (2.19)

S^nα,M,N(k)=12Dn,0L^nα(k)+Dn,112∫−11(x−ξ)Ln−1(α,2)(x)e−ikπξdξ+Dn,212∫−11(x−ξ)2Ln−2(α,4)(x)e−ikπξdξ

Then, from (3.5), for k = 2 and k = 4, we obtain

(x−ξ)Ln−1(α,2)(x)=Ln(α,1)(x)−dn,ξ(α,1)Ln−1(α,1)(x),

and

(x−ξ)2Ln−2(α,4)(x)=(x−ξ)Ln−1(α,3)(x)−dn−1,ξ(α,3)(x−ξ)Ln−2(α,3)(x)=Ln(α,2)(x)−dn,ξ(α,2)+dn−1,ξ(α,3)Ln−1(α,2)(x)+dn−1,ξ(α,3)dn−1,ξ(α,2)Ln−2(α,2)(x),

respectively.

As a consequence, (2.19) can be written as

Snα,M,N(x)=Dn,0Lnα(x)+Dn,1Ln(α,1)(x)−dn,ξ(α,1)Ln−1(α,1)(x)+Dn,2Ln(α,2)(x)−dn,ξ(α,2)+dn−1,ξ(α,3)Ln−1(α,2)(x)+dn−1,ξ(α,3)dn−1,ξ(α,2)Ln−2(α,2)(x)=Dn,0Lnα(x)+Dn,1Ln(α,1)(x)−Dn,1dn,ξ(α,1)Ln−1(α,1)(x)+Dn,2Ln(α,2)(x)+γn,ξ,1α,2Ln−1(α,2)(x)+γn,ξ,2α,2Ln−2(α,2)(x),

where

(3.9)γn,ξ,1α,2=−dn,ξ(α,2)+dn−1,ξ(α,3),γn,ξ,2α,2=Dn,2dn−1,ξ(α,3)dn−1,ξ(α,2).

Then, Fourier coefficients are given by

S^nα,M,N(m)=12∫−11Dn,0Lnα(εη+δ)+Dn,1Ln(α,1)(εη+δ)−Dn,1dn,ξ(α,1)Ln−1(α,1)(εη+δ)+Dn,2Ln(α,2)(εη+δ)+γn,ξ,1α,2Ln−1(α,2)(εη+δ)+γn,ξ,2α,2Ln−2(α,2)(εη+δ)e−imπηdη=Dn,0L^nα(m)+Dn,1L^n(α,1)(m)−Dn,1dn,ξ(α,1)L^n−1(α,1)(m)+Dn,2L^n(α,2)(m)+γn,ξ,1α,2L^n−1(α,2)(m)+γn,ξ,2α,2L^n−2(α,2)(m)

and if we use (2.4) and (3.6) we get

S^nα,M,N(m)=Dn,0eikδ/ϵ2ε∑j=0nan,jαβja,biπmϵ+Dn,1eikδ/ϵ2ε∑j=0nan,jα,[1]βja,biπmε−Dn,1dn,ξ(α,1)eikδ/ϵ2ε∑j=0n−1an−1,jα,[1]βja,biπmε+Dn,2eikδ/ϵ2ε∑j=0nan,jα,[2]βja,biπmε+γn,ξ,1α,2eikδ/ϵ2ε∑j=0n−1an−1,jα,[2]βja,biπmε+γn,ξ,2α,2eikδ/ϵ2ε∑j=0n−2an−2,jα,[2]βja,biπmε.

We summarize in the next.

Theorem 11.

Consider x = ɛη + δ, where η ∈ [ − 1, 1], ε=b−a2 and δ=b+a2. For every n ≥ 2, the Fourier coefficients for the polynomial Snα,M,N defined in (2.19), and in the local variable η, are given by

S^nα,M,N(m)=eikδ/ϵ2εΦn,n,ξα,[1,2]+Φn,n−1,ξα,[1,2]+Ωn−1,n−1,ξα,[1,2]+∑j=0n−2Φn,j,ξα,[1,2]+Ωn−1,j,ξα,[1,2]+γn,ξ,2α,2an−2,jα,[2]βja,biπmε,

where

Φn,j,ξα,[1,2]=Dn,0an,jα+Dn,1an,jα,[1]+Dn,2an,jα,[2]βja,biπmε,

Ωn−1,j,ξα,[1,2]=−Dn,1dn,ξ(α,1)an−1,jα,[1]+γn,ξ,1α,2an−1,jα,[2]βja,biπmε,

and γn,ξ,1α,2, γn,ξ,2α,2 are given in (3.9).