Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

Abstract

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square . And, we prove that the growth order of the Lebesgue constant is . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square , the growth order of which is .

1. Introduction

Chebyshev polynomials play an important role in modern developments, including orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods for partial differential equations (cf. [1]). Especially, the zeros of Chebyshev polynomials are often used in the studies of one-variable Lagrange interpolation polynomials. Many good approximation properties have been obtained over the past decades (cf. [2]). Since multivariate Lagrange interpolation polynomials are difficult to express concretely, many scholars are interested to study them (cf. [3–15]).

Let be a nonempty compact set and be a subspace of , where denotes the space of polynomials with variables whose degrees do not exceed and the dimension . Then, based on the nodes , the Lagrange interpolation problem related to and can be described as follows: for any function , where represents the continuous function space on , we can find a unique polynomial to satisfy the equationThis polynomial is the so-called Lagrange interpolation polynomial and can be expressed aswhere are the Lagrange interpolation basis functions that satisfy the following formula:

The mapping can be regarded as an operator from to itself, and the norm of the operator is defined aswhich is called the Lebesgue constant. We know that the uniform convergence of for is closely related to the Lebesgue constant.

The univariate Lagrange interpolation polynomial and its Lebesgue constant have been extensively studied (cf. [2, 16]). Specially, for and , the Lebesgue constant and the order of the Lebesgue constant is when the Chebyshev points are taken as the nodes (cf. [16]).

There are relatively few research results on multivariate Lagrange interpolation polynomials. In [3], from Berman’s Theorem, it is shown that for , the unit ball in , and , the order of the Lebesgue constant is .

It is well known that the Lagrange interpolation polynomial is closely related to cubature formula. Möller (cf. [4]) stated that for centrally symmetric weight functions, the node number of cubature formula satisfiesand it is the so-called minimal cubature formula if the number of nodes reaches the lower bound. In [5], Xu studied the relationship between the compact cubature formula and the Lagrange interpolation polynomial. By using this relationship, Xu in [6] established the quadrature formula and the Lagrange interpolation polynomial on , based on the common zeros of the product Chebyshev polynomial of the first kind, which are called minimal cubature formula and Xu-type Lagrange interpolation polynomial on the first kind Chebyshev polynomial. Moreover, for , the mean convergence of the interpolation polynomial is also obtained.

Bos et al. [7] gave the numerical study of the upper bound of Lebesgue constant of the Xu-type Lagrange interpolation polynomial on the first kind Chebyshev polynomial, the order of which lies in , and they gave detailed proof of the order in [8]. And, Vecchia et al. [9] gave that the order of the lower bound estimate is .

In [10], for , we gave the compact formulae of the cubature formula and the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind, which are called minimal cubature formula and Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial. Furthermore, for , we studied the mean convergence of the Lagrange interpolation polynomials.

In this paper, we study the growth order of the Lebesgue constant and provide a direct elementary proof.

Theorem 1. For , the upper bound estimate of the Lebesgue constant of Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial in [10] is

Our result gives that the growth order of the Lebesgue constant of Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial on the square is . Obviously, it is different from the Lebesgue constant on the disk , the growth order of which is , and is different from the Lebesgue constant of Xu-type Lagrange interpolation polynomial on the first kind Chebyshev polynomial on , the growth order of which is .

2. The Lebesgue Constant of Xu-Type Lagrange Interpolation Polynomial on the Second Kind Chebyshev Polynomial

In order to prove Theorem 1, by using reproducing kernel, we give the expression of the Lebesgue constant in this section.

First, we briefly introduce the Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial in [10].

Let denote the set of nonnegative integers. For , Chebyshev polynomial of the second kind (cf. [17]) is defined byand they are orthogonal polynomials with respect to the second kind Chebyshev weight .

The product Chebyshev polynomial of the second kind of degree on (cf. [5]) is defined byand correspondingly, the product Chebyshev weight function of the second kind is

For , the reproducing kernel of the product Chebyshev polynomials is defined by

Let , where , , be nodes; the Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial iswhere , ,

Obviously, the node number of formula (11) is . When , , which reaches the lower bound . When , , which is one more than the lower bound (cf. [4]).is called Lebesgue constant of Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial. Writingthen

The expression of given in [10] iswhere . Then,

Lemma 1. If , thenIf , thenwhere .
We only prove formula (19); other formulae can be proved similarly. For , we haveHence,

Lemma 2. If , thenIf , thenwhere .
By (18), we haveFrom (19) and (20), we can obtain (24). And, we can similarly prove (25).

Lemma 3. The following relation holds:where . We haveBy Lemmas 2 and 3, the following result can be obtained.

Lemma 4. If , thenIf , thenFurthermore, we can obtain the following lemma.

Lemma 5. Let . If , thenIf , then

3. Proof of Theorem 1

The proof of Theorem 1 is given in this section. And, since the estimates of and are similar, we need to only estimate .

Setting , we havewhere

To prove Theorem 1, we first prove some lemmas.

Lemma 6. If and , we have the following:(1)For or , For or ,(2)For , For , For or ,(1)We first consider the case of . For , since , we obtain Noticing that and is a convex function on , we have For , then By (40)–(42), we can obtain (35). For , since , then For , we have and combining (41), we can obtain (35). For or , it is easy to prove that . When , by setting , we can similarly prove (35) and (36).(2)If , for , since , then And, we have so we obtain (37). For , considering we have (38). For or , it is easy to prove that .When , setting , similar to the case of , the estimation of can be obtained.
In the same way, we can obtain the following estimates of and .

Lemma 7. If and , then we have the following:(1)For or ,