Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Molano, Alejandro

doi:10.1108/AJMS-07-2021-0164

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.

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.

{⟨f, g⟩}_{S} = \int_{R} f (x) g (x) d μ_{0} + \sum_{k = 1}^{m} \int_{R} f^{(k)} (x) g^{(k)} (x) d μ_{k} .

(1.1)

A sequence of polynomials ${\{S_{n}\}}_{n \geq 0}$ ⁠, deg S_n = n, is orthogonal with respect to (1.1) if

{⟨S_{n}, S_{m}⟩}_{S} = K_{n} δ_{n, m}, K_{n} > 0 .

The sequence ${\{S_{n}\}}_{n \geq 0}$ 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 ${\{S_{n}\}}_{n \geq 0}$ 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 ${\{L_{n}^{α}\}}_{n \geq 0}$ ⁠, with α > − 1, are orthogonal with respect to the inner product:

{⟨p, q⟩}_{α} ≔ \int_{0}^{\infty} p (x) q (x) e^{- x} x^{α} d x, p, q \in 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 $k (L_{n}^{α}) ≔ {(- 1)}^{n} / n$ ⁠. These polynomials satisfy the three terms recurrence relation (TTRR in short),

(n + 1) L_{n + 1}^{α} (x) = (2 n + α + 1 - x) L_{n}^{α} (x) - (n + α) L_{n - 1}^{α} (x),

(2.1)

for n ≥ 0 with the initial conditions $L_{- 1}^{α} ≔ 0$ and $L_{0}^{α} = 1$ ⁠. For n ≥ 1, the zeros of every $L_{n}^{α}$ are all real, simple and are located in (0, ∞) (see Ref. [11]). In the sequel, ${\{x_{n, i}\}}_{i = 1}^{n}$ will denote the zeros of $L_{n}^{α}$ 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 $L_{n}^{α}$ are into [0, ζ_n,α], where

ζ_{n, α} = 2 n + α + 1 + \sqrt{{(2 n + α + 1)}^{2} + \frac{1}{4} - α^{2}} .

We consider, for a nonnegative integer m, the functions $β_{m}^{a, b}$ defined as (see Ref. [10]),

β_{m}^{a, b} (z) = \int_{a}^{b} x^{m} e^{- z x} d x = z^{- m - 1} m! (e^{- a z} e_{m} (a z) - e^{- b z} e_{m} (b z)),

(2.2)

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, $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2}$ ⁠, we get

\int_{- 1}^{1} {(ε ξ + δ)}^{m} e^{- i π k ξ} d ξ = \frac{1}{ε} e^{i k δ / ε} β_{m}^{a, b} (\frac{i π k}{ε}) .

(2.3)

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], $ε = \frac{b - a}{2}$ , $δ = \frac{b + a}{2}$ ⁠. The Fourier coefficients for $L_{n}^{α} (ε ξ + δ)$ , in the local variable ξ, are given by

{\hat{L}}_{n}^{α} (k) = \frac{1}{2 ε} e^{i k δ / ε} \sum_{t = 0}^{n} \frac{{(- 1)}^{t}}{t!} (\binom{n + α}{n - t}) β_{t}^{a, b} (\frac{i π k}{ε}) .

(2.4)

2.2 Quasi-orthogonality and zeros

Let ${\{P_{n}\}}_{n \geq 0}$ be a sequence of polynomials orthogonal with respect to a positive Borel measure μ supported on [a, b], with − ∞ ≤ a < b ≤ ∞, i.e.

\int_{a}^{b} P_{n} (x) x^{k} d μ = δ_{n, k} K_{n}, K_{n} > 0, k = 1, \dots, n .

Definition 2.

Let r be a nonnegative integer and R_n a polynomial with degree n ≥ r satisfying $\int_{a}^{b} R_{n} (x) x^{k} d μ = 0$ for k = 0, 1, 2, …, n − r − 1, and $\int_{a}^{b} R_{n} (x) x^{n - r} d μ \neq 0$ ⁠. 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

R_{n} (x) = \sum_{k = 0}^{r} b_{n, k} P_{n - k} .

(2.5)

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 ${\{P_{n}\}}_{n \geq 0}$ can be obtained by means of a TTRR:

P_{n + 1} (x) = ({\hat{A}}_{n + 1} x + {\hat{B}}_{n + 1}) P_{n} (x) - {\hat{C}}_{n + 1} P_{n - 1} (x), n \geq 0,

(2.6)

and we define $B_{n + 1} ≔ \frac{k (P_{n}) {\hat{B}}_{n + 1}}{k (P_{n + 1})}$ and $C_{n + 1} ≔ \frac{k (P_{n - 1}) {\hat{C}}_{n + 1}}{k (P_{n + 1})}$ ⁠. 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 $a_{n} ≔ \frac{k (P_{n - 1}) b_{n, 1}}{k (R_{n})}$ and $b_{n} ≔ \frac{k (P_{n - 2}) b_{n, 2}}{k (R_{n})}$ ⁠. 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 ${\{x_{n, i}\}}_{i = 1}^{n}$ and ${\{y_{n, i}\}}_{i = 1}^{n}$ are the zeros of P_n and R_n, respectively, and ordered in increasing order.

b_n < C_n if and only if
$y_{n, 1} < x_{n - 1,1} < y_{n, 2} < x_{n - 1,2} < \dots < y_{n, n - 1} < x_{n - 1, n - 1} < y_{n, n} .$
(2.7)
0 < b_n < C_n and $a_{n} > - \frac{b_{n} (x_{n, 1} + B_{n})}{C_{n}}$ if and only if
$y_{n, 1} < x_{n, 1} < y_{n, 2} < x_{n, 2} < \dots < x_{n, n - 1} < y_{n, n} < x_{n, n} .$
(2.8)
0 < b_n < C_n and $a_{n} < - \frac{b_{n} (x_{n, n} + B_{n})}{C_{n}}$ if and only if
$x_{n, 1} < y_{n, 1} < x_{n, 2} < y_{n, 2} < \dots < y_{n, n - 1} < x_{n, n} < y_{n, n} .$
(2.9)

Theorem 4.

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

f_{n} (x) = \frac{k (P_{n - 1}) P_{n} (x)}{k (P_{n}) P_{n - 1} (x)} .

(2.10)

For i = 1, 2, …, n

f_{n + 1} (y_{n, i}) f_{n} (y_{n, i}) + a_{n + 1} f_{n} (y_{n, i}) + b_{n + 1} < 0,

if and only if

y_{n + 1,1} < y_{n, 1} < y_{n + 1,2} < y_{n, 2} < \dots < y_{n + 1, n} < y_{n, n} < y_{n + 1, n + 1},

(2.11)

2.3 Laguerre–Sobolev type orthogonal polynomials, nondiagonal case

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

{⟨p, q⟩}_{S_{1}} = {⟨p, q⟩}_{α} + P {(0)}^{t} A Q (0),

(2.12)

where $A = (\begin{matrix} M_{0} & λ \\ λ & M_{1} \end{matrix})$ ⁠, with M₀M₁ ≥ 0 and λ such that det A ≥ 0. Let ${\{S_{n}^{α}\}}_{n \geq 0}$ be the sequence of polynomials orthogonal with respect to (2.12) such that $k (S_{n}^{α}) = k (L_{n}^{α})$ for n ≥ 0.

Theorem 5.

([14]). For every $n \in N$

S_{n}^{α} (x) = L_{n}^{α + 2} (x) + A_{n, α} L_{n - 1}^{α + 2} (x) + B_{n, α} L_{n - 2}^{α + 2} (x),

(2.13)

where

A_{n, α} \sim \frac{(α + 1) (α + 2)}{n} - 2, B_{n, α} \sim 1 - \frac{(α + 1) (α + 2)}{n} .

2.3.1 Laguerre–Sobolev type polynomials of higher order derivatives

Let ${\{S_{n, m}^{α, W}\}}_{n \geq 0}$ be orthogonal with respect to Sobolev inner product

{⟨p, q⟩}_{S, m}^{α, W} = {⟨p, q⟩}_{α} + W p^{(m)} (0) q^{(m)} (0),

(2.14)

with W > 0, m a nonnegative integer and $p, q \in P$ ⁠. Moreover $k (S_{n, m}^{α, W}) = k (L_{n}^{α})$ ⁠.

Theorem 6.

([15]). For n > m

S_{n, m}^{α, W} (x) = \sum_{k = 0}^{m + 1} A_{n, k} L_{n - k}^{(α + k)} (x),

(2.15)

where

A_{n, 0} = 1 + W \frac{Γ (m + 1)}{Γ (α + m + 1)} \sum_{k = 1}^{m + 1} {(- 1)}^{k + 1} (\binom{n + α}{n - m - k}) (\binom{n - k}{m + 1 - k}),

and fork = 1, …, m + 1

A_{n, k} = {(- 1)}^{k} W \frac{Γ (m + 1)}{Γ (α + m + 1)} (\binom{n + α}{n - m}) (\binom{n - k}{m + 1 - k}) .

With respect to zeros of every $S_{n, m}^{α, W}$ ⁠, we enunciate the next results.

Theorem 7.

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

Theorem 8.

Assume that n ≥ m + 1. If ${υ_{n, i}}_{i = 1}^{n}$ are the zeros of $S_{n, m}^{α, W}$ , 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 $S_{n, m}^{α, W}$ then $- m {\tilde{υ}}_{n, m + 1} < ρ_{n}$ , where ${\tilde{υ}}_{n, m + 1}$ denotes the m − th positive zero of $S_{n, m}^{α, W}$ ⁠.

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]). ${\{L_{n}^{(α, k)}\}}_{n \geq 0}$ denotes the respective sequence of orthogonal polynomials, where $k (L_{n}^{(α, k)}) = k (L_{n}^{α})$ for every n and $L_{n}^{(α, 0)} ≔ L_{n}^{α}$ ⁠. An algebraic connection between polynomials orthogonal with respect to the weight ω_α,k(x) is as follows (see Ref. [11]),

(x - ξ) L_{n}^{(α, k)} (x) = L_{n + 1}^{(α, k - 1)} (x) - \frac{L_{n + 1}^{(α, k - 1)} (ξ)}{L_{n}^{(α, k - 1)} (ξ)} L_{n}^{(α, k - 1)} (x) .

(2.16)

Assume ${\{x_{n, i}^{[k]}\}}_{i = 1}^{n}$ are the zeros of $L_{n}^{(α, k)}$ in increasing order, with $x_{n, i}^{[0]} ≔ x_{n, i}$ ⁠.

Proposition 4.

([17]). For i = 1, …, n

x_{n, i}^{[k - 1]} < x_{n, i}^{[k]} < x_{n + 1, i + 1}^{[k - 1]} .

(2.17)

Now we consider the Sobolev–Laguerre type inner product:

{⟨p, q⟩}_{α, M, N} = {⟨p, q⟩}_{α} + M p (ξ) q (ξ) + N p^{'} (ξ) q^{'} (ξ),

(2.18)

with M, N ≥ 0, ξ ≤ 0. Let ${\{S_{n}^{α, M, N}\}}_{n \geq 0}$ be the respective sequence of orthogonal polynomials such that $k (S_{n}^{α, M, N}) = k (L_{n}^{α})$ for n ≥ 0.

Theorem 10.

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

S_{n}^{α, M, N} (x) = D_{n, 0} L_{n}^{α} (x) + D_{n, 1} (x - ξ) L_{n - 1}^{(α, 2)} (x) + D_{n, 2} {(x - ξ)}^{2} L_{n - 2}^{(α, 4)} (x),

(2.19)

where:

If M, N > 0, then $D_{n, 0} \sim \frac{8 ξ n^{α}}{M {(L^{α} (ξ))}^{2}}$ , $D_{n, 1} \sim \frac{32 {(- ξ)}^{3 / 2} n^{α - 1 / 2}}{M {(L^{α} (ξ))}^{2}}$ and $D_{n, 2} \sim \frac{1}{n^{2}}$ ⁠.
If M = 0 and N > 0, then $D_{n, 0} \sim \frac{1}{4 \sqrt{- ξ n}}$ , $D_{n, 1} \sim - \frac{1}{n}$ and $D_{n, 2} \sim \frac{1}{4 n^{2} \sqrt{- ξ n}}$ ⁠.
If N = 0 and M > 0, then $D_{n, 0} \sim \frac{\sqrt{- ξ}}{M n^{1 / 2 - α} {(L^{α} (ξ))}^{2}}$ , $D_{n, 1} \sim - \frac{1}{n}$ and D_n,2 = 0.

Let ${\{υ_{n, i}\}}_{i = 1}^{n}$ be the zeros of $S_{n}^{α, M, N}$ in increasing order. To describe results on zeros of every $S_{n}^{α, M, N}$ ⁠, we present the next results.

Proposition 5.

([17]). The zeros of $S_{n}^{α, M, N}$ are real, simple and at most one of them is outside [ξ, ∞).

Proposition 6.

([18]). If ξ < υ_n,1 then

υ_{n, 1} < x_{n, 1} < \dots < υ_{n, n} < x_{n, n} .

(2.20)

Proposition 7.

([17]). Suppose that υ_n,1 < ξ. Then

2 ξ - x_{n - 1,1}^{[2]} < υ_{n, 1} < ξ < υ_{n, 2} < x_{n - 1,2}^{[2]} < \dots < υ_{n, n} < x_{n - 1, n - 1}^{[2]} .

(2.21)

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 ${\{S_{n}^{α}\}}_{n \geq 0}$ 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

a_{n} = A_{n, α}, b_{n} = B_{n, α},

and

B_{n + 1} = - (2 n + α + 3), C_{n + 1} = n (n + α + 2) .

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

Corollary 1.

If

0 < B_{n + 1, α} < n (n + α + 2), A_{n + 1, α} < - \frac{B_{n + 1, α} (x_{n + 1, n + 1}^{α + 2} - (2 n + α + 3))}{n (n + α + 2)},

(3.1)

0 < B_{n, α} < (n - 1) (n + α + 1), A_{n, α} > - \frac{B_{n, α} (x_{n, 1}^{α + 2} - (2 n + α + 1))}{(n - 1) (n + α + 1)}

(3.2)

and

f_{n + 1} (y_{n, i}) f_{n} (y_{n, i}) + A_{n + 1, α} f_{n} (y_{n, i}) + B_{n + 1, α} < 0,

(3.3)

fori = 1, 2, …, n, then

S_{n}^{α}

hasnreal and simple zeros in the interval

[0, ζ_{n, α + 2}]

⁠. Here,

{\{y_{n, i}\}}_{i = 1}^{n}

and

{\{x_{n, i}^{α + 2}\}}_{i = 1}^{n}

represent the zeros of

S_{n}^{α}

and

L_{n}^{α + 2}

⁠, respectively, ordered in increasing order.

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

x_{n + 1,1}^{α + 2} < y_{n + 1,1} < x_{n + 1,2}^{α + 2} < y_{n + 1,2} < \dots < y_{n + 1, n} < x_{n + 1, n + 1}^{α + 2} < y_{n + 1, n + 1} .

and from Part 2, inequalities in , are equivalent to

y_{n, 1} < x_{n, 1}^{α + 2} < y_{n, 2} < x_{n, 2}^{α + 2} < \dots < x_{n, n - 1}^{α + 2} < y_{n, n} < x_{n, n}^{α + 2} .

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

y_{n + 1,1} < y_{n, 1} < y_{n + 1,2} < y_{n, 2} < \dots < y_{n + 1, n} < y_{n, n} < y_{n + 1, n + 1} .

Finally, since the zeros of $L_{n}^{α + 2}$ and $L_{n + 1}^{α + 2}$ are interlaced, we obtain

x_{n, n}^{α + 2} < x_{n + 1, n + 1}^{α + 2} .

The above inequalities imply that

x_{n + 1,1}^{α + 2} < y_{n + 1,1} < y_{n, 1} < \dots < y_{n, n} < x_{n, n}^{α + 2} < x_{n + 1, n + 1}^{α + 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], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2} .$ Then

S_{n}^{α} (ε ξ + δ) = \sum_{k = 0}^{\infty} \hat{S_{n}^{α}} (k) e^{i k π ξ},

and by using of (2.13) we get

\begin{array}{l} \hat{S_{n}^{α}} (k) & = \frac{1}{2} \int_{- 1}^{1} (L_{n}^{α + 2} (ε ξ + δ) + A_{n, α} L_{n - 1}^{α + 2} (ε ξ + δ) + B_{n, α} L_{n - 2}^{α + 2} (ε ξ + δ)) e^{- i k π ξ} d ξ \\ = \frac{1}{2} \int_{- 1}^{1} L_{n}^{α + 2} (ε ξ + δ) e^{- i k π ξ} d ξ + \frac{1}{2} A_{n, α} \int_{- 1}^{1} L_{n - 1}^{α + 2} (ε ξ + δ) e^{- i k π ξ} d ξ \\ + \frac{1}{2} B_{n, α} \int_{- 1}^{1} L_{n - 2}^{α + 2} (ε ξ + δ) e^{- i k π ξ} d ξ, \\ = \frac{1}{2} {\hat{L}}_{n}^{α + 2} (k) + \frac{1}{2} A_{n, α} {\hat{L}}_{n - 1}^{α + 2} k) + \frac{1}{2} B_{n, α} {\hat{L}}_{n - 2}^{α + 2} (k), \end{array}

and by using of (2.4) we obtain

\begin{array}{l} \hat{S_{n}^{α}} (k) \\ = \frac{1}{2} {\hat{L}}_{n}^{α + 2} (k) + \frac{1}{2} A_{n, α} {\hat{L}}_{n - 1}^{α + 2} k) + \frac{1}{2} B_{n, α} {\hat{L}}_{n - 2}^{α + 2} (k), \\ = \frac{1}{4 ε} e^{i k δ / ε} [\sum_{t = 0}^{n} \frac{{(- 1)}^{t}}{t!} (\binom{n + α + 2}{n - t}) β_{t}^{a, b} (\frac{i π k}{ε}) \\ + A_{n, α} \sum_{t = 0}^{n - 1} \frac{{(- 1)}^{t}}{t!} (\binom{n + α + 1}{n - 1 - t}) β_{t}^{a, b} (\frac{i π k}{ε}) \\ + B_{n, α} \sum_{t = 0}^{n - 2} \frac{{(- 1)}^{t}}{t!} (\binom{n + α}{n - 2 - t}) β_{t}^{a, b} (\frac{i π k}{ε})] \\ = \frac{1}{4 ε} e^{i k δ / ε} (\frac{{(- 1)}^{n}}{n!} [β_{n}^{a, b} (\frac{i π k}{ε}) - n (n + α + 2) β_{n - 1}^{a, b} (\frac{i π k}{ε}) - A_{n, α} n β_{n - 1}^{a, b} (\frac{i π k}{ε})] \\ + \sum_{t = 0}^{n - 2} \frac{{(- 1)}^{t}}{t!} β_{t}^{a, b} (\frac{i π k}{ε}) [(\binom{n + α + 2}{n - t}) + A_{n, α} (\binom{n + α + 1}{n - 1 - t}) + B_{n, α} (\binom{n + α}{n - 2 - t})]) . \end{array}

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], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2} .$ The coefficients of Fourier for $S_{n}^{α} (ε ξ + δ)$ , in the local variable ξ, are giving by

\begin{array}{l} \hat{S_{n}^{α}} (k) \\ = \frac{1}{4 ε} e^{i k δ / ε} [\frac{{(- 1)}^{n}}{n!} [β_{n}^{a, b} (\frac{i π k}{ε}) - n (n + α + 2) β_{n - 1}^{a, b} (\frac{i π k}{ε}) - A_{n, α} n β_{n - 1}^{a, b} (\frac{i π k}{ε})] \\ + \sum_{t = 0}^{n - 2} \frac{{(- 1)}^{t}}{t!} β_{t}^{a, b} (\frac{i π k}{ε}) [(\binom{n + α + 2}{n - t}) + A_{n, α} (\binom{n + α + 1}{n - 1 - t}) + B_{n, α} (\binom{n + α}{n - 2 - t})]] . \end{array}

3.2 Higher order derivatives

${\{S_{n, m}^{α, W}\}}_{n \geq 0}$ represents the sequence of polynomials orthogonal with respect to (2.14). As before, we propose a bounded interval that containing the n zeros of $S_{n, m}^{α, W}$ for n large enough.

Corollary 2.

For n ≥ m + 1, the zeros of $S_{n, m}^{α, W}$ are located in [ − mx_n,m+1, ζ_n,α].

Proof. From Theorem 8, ${\tilde{υ}}_{n, m + 1} < x_{n, m + 1}$ ⁠, and from Theorem 9, the result is

- m x_{n, m + 1} < ρ_{n},

□

As before, we assume x ∈ [a, b] and x = ɛξ + δ, where ξ ∈ [ − 1, 1], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2} .$

From (2.15) we have

\begin{array}{l} {\hat{S}}_{n, m}^{α, W} (k) & = \frac{1}{2} \int_{- 1}^{1} S_{n, m}^{α, W} (ε ξ + δ) e^{- i k π ξ} d ξ \\ = \frac{1}{2} \sum_{q = 0}^{m + 1} A_{n, q} (\int_{- 1}^{1} L_{n - q}^{(α + q)} (ε ξ + δ) e^{- i k π ξ} d ξ) \\ = \frac{1}{2} \sum_{q = 0}^{m + 1} A_{n, q} {\hat{L}}_{n - q}^{α + q} (k), \end{array}

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], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2}$ ⁠. The Fourier series for $S_{n, m}^{α, W}$ , with the local variable ξ, is given by

S_{n, m}^{α, W} (ε ξ + δ) = \sum_{k = - \infty}^{\infty} {\hat{S}}_{n . m}^{α, W} (k) e^{i k π ξ},

where

{\hat{S}}_{n, m}^{α, W} (k) = \frac{1}{4 ε} e^{i k δ / ϵ} \sum_{q = 0}^{m + 1} \sum_{t = 0}^{n - q} \frac{{(- 1)}^{t}}{t!} A_{n, q} (\binom{n + α}{n - q - t}) β_{t}^{a, b} (\frac{i π k}{ε}) .

3.3 Mass outside support

Assume that

L_{n}^{(α, k)} (x) = \sum_{j = 0}^{n} a_{n, j}^{α, [k]} x^{j},

(3.4)

with $a_{n, j}^{α, [0]} ≔ a_{n, j}^{α} = \frac{{(- 1)}^{j}}{j!} (\binom{n + α}{n - j})$ (see Ref. [11]). From (2.16), for k ≥ 1,

(x - ξ) L_{n}^{(α, k)} (x) = L_{n + 1}^{(α, k - 1)} (x) - d_{n + 1, ξ}^{(α, k - 1)} L_{n}^{(α, k - 1)} (x),

(3.5)

with

d_{n + 1, ξ}^{(α, k - 1)} = \frac{L_{n + 1}^{(α, k - 1)} (ξ)}{L_{n}^{(α, k - 1)} (ξ)} .

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

\sum_{j = 1}^{n + 1} a_{n, j - 1}^{α, [k]} x^{j} - \sum_{j = 0}^{n} ξ a_{n, j}^{α, [k]} x^{j} = a_{n + 1, n + 1}^{α, [k - 1]} x^{n + 1} + \sum_{j = 0}^{n} (a_{n + 1, j}^{α, [k - 1]} - d_{n + 1, ξ}^{(α, k - 1)} a_{n, j}^{α, [k - 1]}) x^{j},

or equivalently

\begin{array}{l} a_{n, n}^{α, [k]} x^{n + 1} + \sum_{j = 1}^{n} (a_{n, j - 1}^{α, [k]} - ξ a_{n, j}^{α, [k]}) x^{j} - ξ a_{n, 0}^{α, [k]} \\ = a_{n + 1, n + 1}^{α, [k - 1]} x^{n + 1} + \sum_{j = 1}^{n} (a_{n + 1, j}^{α, [k - 1]} - d_{n + 1, ξ}^{(α, k - 1)} a_{n, j}^{α, [k - 1]}) x^{j} + (a_{n + 1,0}^{α, [k - 1]} - d_{n + 1, ξ}^{(α, k - 1)} a_{n, 0}^{α, [k - 1]}), \end{array}

then we get the equations

a_{n, n}^{α, [k]} = a_{n + 1, n + 1}^{α, [k - 1]},

ξ a_{n, 0}^{α, [k]} = (d_{n + 1, ξ}^{(α, k - 1)} a_{n, 0}^{α, [k - 1]} - a_{n + 1,0}^{α, [k - 1]}),

and

ξ a_{n, j}^{α, [k]} = a_{n, j - 1}^{α, [k]} + d_{n + 1, ξ}^{(α, k - 1)} a_{n, j}^{α, [k - 1]} - a_{n + 1, j}^{α, [k - 1]}, j = 1, \dots, n .

Lemma 1.

If

L_{n}^{(α, k)} (x) = \sum_{j = 0}^{n} a_{n, j}^{α, [k]} x^{j}, a_{n, j}^{α, [0]} ≔ a_{n, j}^{α},

andξ < 0, then

a_{n, n}^{α, [k]} = a_{n + 1, n + 1}^{α, [k - 1]},

and the coefficients

a_{n, j}^{α, [k]}

, withj = 1, …, n − 1, can be obtained recurrently by means of

a_{n, j}^{α, [k]} = ξ^{- 1} (a_{n, j - 1}^{α, [k]} + d_{n + 1, ξ}^{(α, k - 1)} a_{n, j}^{α, [k - 1]} - a_{n + 1, j}^{α, [k - 1]}),

with the initial condition

a_{n, 0}^{α, [k]} = ξ^{- 1} (d_{n + 1, ξ}^{(α, k - 1)} a_{n, 0}^{α, [k - 1]} - a_{n + 1,0}^{α, [k - 1]}) .

If ξ = 0 then

a_{n, n}^{α, [k]} = a_{n + 1, n + 1}^{α, [k - 1]},

and forj = 1, …, n

a_{n, j - 1}^{α, [k]} = a_{n + 1, j}^{α, [k - 1]} - d_{n + 1,0}^{(α, k - 1)} a_{n, j}^{α, [k - 1]} .

According to (2.17), we can deduce that

x_{n, 1} < x_{n, 1}^{[1]} < x_{n, 1}^{[2]} < x_{n, 1}^{[3]} < \dots < x_{n, 1}^{[k]} < \dots,

and

x_{n, n}^{[k]} < x_{n + 1, n + 1}^{[k - 1]} < x_{n + 2, n + 2}^{[k - 2]} < \dots < x_{n + k, n + k},

and as consequence we get the next.

Lemma 2.

The zeros of $L_{n}^{(α, k)}$ are located in [x_n,1, β_n ₊ _k,α].

Since the Fourier series for $L_{n}^{(α, k)} (ε η + δ)$ in the local variable η is determined by the coefficients

{\hat{L}}_{n}^{(α, k)} (k) = \frac{1}{2} \int_{- 1}^{1} L_{n}^{(α, k)} (ε η + δ) e^{- i k π η} d η,

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

Proposition 10.

Fourier coefficients for $L_{n}^{(α, k)}$ on a finite interval [a, b], in the local variable η, with x = ɛη + δ, η ∈ [ − 1, 1], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2}$ , are defined by

{\hat{L}}_{n}^{(α, k)} (k) = \frac{e^{i k δ / ε}}{2 ε} \sum_{j = 0}^{n} a_{n, j}^{α, [k]} β_{j}^{a, b} (\frac{i π k}{ε}) .

(3.6)

Let ${\{S_{n}^{α, M, N}\}}_{n \geq 0}$ be the sequence of polynomials orthogonal with respect to (2.18).

Corollary 3.

For n ≥ 2, the zeros of $S_{n}^{α, M, N}$ are into [2ξ − x_n+1,3, x_n+1,n+1].

Proof. From (2.17) we get

x_{n - 1,1}^{[1]} < x_{n - 1,1}^{[2]} < x_{n, 2}^{[1]}, and x_{n, 2} < x_{n, 2}^{[1]} < x_{n + 1,3},

thus

x_{n - 1,1}^{[2]} < x_{n + 1,3} .

(3.7)

In the same way, $x_{n - 1, n - 1}^{[1]} < x_{n - 1, n - 1}^{[2]} < x_{n, n}^{[1]}$ and $x_{n, n} < x_{n, n}^{[1]} < x_{n + 1, n + 1}$ ⁠, thus

x_{n - 1, n - 1}^{[2]} < x_{n + 1, n + 1} .

(3.8)

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

2 ξ - x_{n + 1,3} < 2 ξ - x_{n - 1,1}^{[2]} < υ_{n, 1} < \dots < υ_{n, n} < x_{n + 1, n + 1}

□

On the one hand, and on a finite interval [a, b], we compute the Fourier coefficients for every $S_{n}^{α, M, N}$ and in terms of the local variable η. Indeed, if

S_{n}^{α, M, N} (ε η + δ) = \sum_{k = 0}^{\infty} {\hat{S}}_{n}^{α, M, N} (k) e^{i k π η},

by using of (2.19)

\begin{array}{l} {\hat{S}}_{n}^{α, M, N} (k) \\ = \frac{1}{2} D_{n, 0} {\hat{L}}_{n}^{α} (k) + D_{n, 1} \frac{1}{2} \int_{- 1}^{1} (x - ξ) L_{n - 1}^{(α, 2)} (x) e^{- i k π ξ} d ξ \\ + D_{n, 2} \frac{1}{2} \int_{- 1}^{1} {(x - ξ)}^{2} L_{n - 2}^{(α, 4)} (x) e^{- i k π ξ} d ξ \end{array}

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

(x - ξ) L_{n - 1}^{(α, 2)} (x) = L_{n}^{(α, 1)} (x) - d_{n, ξ}^{(α, 1)} L_{n - 1}^{(α, 1)} (x),

and

\begin{array}{l} {(x - ξ)}^{2} L_{n - 2}^{(α, 4)} (x) & = (x - ξ) L_{n - 1}^{(α, 3)} (x) - d_{n - 1, ξ}^{(α, 3)} (x - ξ) L_{n - 2}^{(α, 3)} (x) \\ = L_{n}^{(α, 2)} (x) - (d_{n, ξ}^{(α, 2)} + d_{n - 1, ξ}^{(α, 3)}) L_{n - 1}^{(α, 2)} (x) + d_{n - 1, ξ}^{(α, 3)} d_{n - 1, ξ}^{(α, 2)} L_{n - 2}^{(α, 2)} (x), \end{array}

respectively.

As a consequence, (2.19) can be written as

\begin{array}{l} S_{n}^{α, M, N} (x) & = D_{n, 0} L_{n}^{α} (x) + D_{n, 1} (L_{n}^{(α, 1)} (x) - d_{n, ξ}^{(α, 1)} L_{n - 1}^{(α, 1)} (x)) \\ + D_{n, 2} (L_{n}^{(α, 2)} (x) - (d_{n, ξ}^{(α, 2)} + d_{n - 1, ξ}^{(α, 3)}) L_{n - 1}^{(α, 2)} (x) + d_{n - 1, ξ}^{(α, 3)} d_{n - 1, ξ}^{(α, 2)} L_{n - 2}^{(α, 2)} (x)) \\ = D_{n, 0} L_{n}^{α} (x) + D_{n, 1} L_{n}^{(α, 1)} (x) - D_{n, 1} d_{n, ξ}^{(α, 1)} L_{n - 1}^{(α, 1)} (x) \\ + D_{n, 2} L_{n}^{(α, 2)} (x) + γ_{n, ξ, 1}^{α, 2} L_{n - 1}^{(α, 2)} (x) + γ_{n, ξ, 2}^{α, 2} L_{n - 2}^{(α, 2)} (x), \end{array}

where

γ_{n, ξ, 1}^{α, 2} = - (d_{n, ξ}^{(α, 2)} + d_{n - 1, ξ}^{(α, 3)}), γ_{n, ξ, 2}^{α, 2} = D_{n, 2} d_{n - 1, ξ}^{(α, 3)} d_{n - 1, ξ}^{(α, 2)} .

(3.9)

Then, Fourier coefficients are given by

\begin{array}{l} {\hat{S}}_{n}^{α, M, N} (m) \\ = \frac{1}{2} \int_{- 1}^{1} [D_{n, 0} L_{n}^{α} (ε η + δ) + D_{n, 1} L_{n}^{(α, 1)} (ε η + δ) - D_{n, 1} d_{n, ξ}^{(α, 1)} L_{n - 1}^{(α, 1)} (ε η + δ) \\ + D_{n, 2} L_{n}^{(α, 2)} (ε η + δ) + γ_{n, ξ, 1}^{α, 2} L_{n - 1}^{(α, 2)} (ε η + δ) + γ_{n, ξ, 2}^{α, 2} L_{n - 2}^{(α, 2)} (ε η + δ)] e^{- i m π η} d η \\ = D_{n, 0} {\hat{L}}_{n}^{α} (m) + D_{n, 1} {\hat{L}}_{n}^{(α, 1)} (m) - D_{n, 1} d_{n, ξ}^{(α, 1)} {\hat{L}}_{n - 1}^{(α, 1)} (m) \\ + D_{n, 2} {\hat{L}}_{n}^{(α, 2)} (m) + γ_{n, ξ, 1}^{α, 2} {\hat{L}}_{n - 1}^{(α, 2)} (m) + γ_{n, ξ, 2}^{α, 2} {\hat{L}}_{n - 2}^{(α, 2)} (m) \end{array}

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

\begin{array}{l} {\hat{S}}_{n}^{α, M, N} (m) \\ = D_{n, 0} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n} a_{n, j}^{α} β_{j}^{a, b} (\frac{i π m}{ϵ}) + D_{n, 1} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n} a_{n, j}^{α, [1]} β_{j}^{a, b} (\frac{i π m}{ε}) \\ - D_{n, 1} d_{n, ξ}^{(α, 1)} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n - 1} a_{n - 1, j}^{α, [1]} β_{j}^{a, b} (\frac{i π m}{ε}) \\ + D_{n, 2} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n} a_{n, j}^{α, [2]} β_{j}^{a, b} (\frac{i π m}{ε}) + γ_{n, ξ, 1}^{α, 2} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n - 1} a_{n - 1, j}^{α, [2]} β_{j}^{a, b} (\frac{i π m}{ε}) \\ + γ_{n, ξ, 2}^{α, 2} \frac{e^{i k δ / ϵ}}{2 ε} \sum_{j = 0}^{n - 2} a_{n - 2, j}^{α, [2]} β_{j}^{a, b} (\frac{i π m}{ε}) . \end{array}

We summarize in the next.

Theorem 11.

Consider x = ɛη + δ, where η ∈ [ − 1, 1], $ε = \frac{b - a}{2}$ and $δ = \frac{b + a}{2}$ ⁠. For every n ≥ 2, the Fourier coefficients for the polynomial $S_{n}^{α, M, N}$ defined in (2.19), and in the local variable η, are given by

\begin{array}{l} {\hat{S}}_{n}^{α, M, N} (m) & = \frac{e^{i k δ / ϵ}}{2 ε} [Φ_{n, n, ξ}^{α, [1,2]} + Φ_{n, n - 1, ξ}^{α, [1,2]} + Ω_{n - 1, n - 1, ξ}^{α, [1,2]} \\ + \sum_{j = 0}^{n - 2} (Φ_{n, j, ξ}^{α, [1,2]} + Ω_{n - 1, j, ξ}^{α, [1,2]} + γ_{n, ξ, 2}^{α, 2} a_{n - 2, j}^{α, [2]} β_{j}^{a, b} (\frac{i π m}{ε}))], \end{array}

where

Φ_{n, j, ξ}^{α, [1,2]} = (D_{n, 0} a_{n, j}^{α} + D_{n, 1} a_{n, j}^{α, [1]} + D_{n, 2} a_{n, j}^{α, [2]}) β_{j}^{a, b} (\frac{i π m}{ε}),

Ω_{n - 1, j, ξ}^{α, [1,2]} = (- D_{n, 1} d_{n, ξ}^{(α, 1)} a_{n - 1, j}^{α, [1]} + γ_{n, ξ, 1}^{α, 2} a_{n - 1, j}^{α, [2]}) β_{j}^{a, b} (\frac{i π m}{ε}),

and

γ_{n, ξ, 1}^{α, 2}

,

γ_{n, ξ, 2}^{α, 2}

are given in.

The authors thank the referees for the careful revision of the manuscript. Their suggestions have contributed to improving the presentation. This work has been partially supported by Dirección de Investigaciones of Universidad Pedagógica y Tecnológica de Colombia.

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Published in the Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) license. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this license may be seen at http://creativecommons.org/licences/by/4.0/legalcode

Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

1. Introduction

2. Preliminaries

2.1 Classical Laguerre polynomials and generalities

2.2 Quasi-orthogonality and zeros

2.3 Laguerre–Sobolev type orthogonal polynomials, nondiagonal case

2.3.1 Laguerre–Sobolev type polynomials of higher order derivatives

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

3. Fourier coefficients for Laguerre–Sobolev type polynomials

3.1 Nondiagonal case

3.2 Higher order derivatives

3.3 Mass outside support

References

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Cited By

Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

1. Introduction

2. Preliminaries

2.1 Classical Laguerre polynomials and generalities

2.2 Quasi-orthogonality and zeros

2.3 Laguerre–Sobolev type orthogonal polynomials, nondiagonal case

2.3.1 Laguerre–Sobolev type polynomials of higher order derivatives

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

3. Fourier coefficients for Laguerre–Sobolev type polynomials

3.1 Nondiagonal case

3.2 Higher order derivatives

3.3 Mass outside support

References

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