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

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 [− 1, 1]2. And, we prove that the growth order of the Lebesgue constant is O((n + 2)2).)is result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O( � n √ ). 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 [− 1, 1]2, the growth order of which is O((lnn)2).

Let K ⊂ R d be a nonempty compact set and V be a subspace of Π d n , where Π d n denotes the space of polynomials with d variables whose degrees do not exceed n and the dimension dimV � N.
en, based on the nodes X ≔ x k N k�1 ⊂ K, the Lagrange interpolation problem related to V and X can be described as follows: for any function f ∈ C(K), where C(K) represents the continuous function space on K, we can find a unique polynomial p ∈ V to satisfy the equation is polynomial is the so-called Lagrange interpolation polynomial and can be expressed as where l k (x) are the Lagrange interpolation basis functions that satisfy the following formula: e mapping f ⟶ L n f can be regarded as an operator from C(K) to itself, and the norm of the operator is defined as which is called the Lebesgue constant. We know that the uniform convergence of L n (f, x) for f ∈ C(K) is closely related to the Lebesgue constant. e univariate Lagrange interpolation polynomial and its Lebesgue constant have been extensively studied (cf. [2,16]). Specially, for K � [− 1, 1] and V � Π 1 n , the Lebesgue constant ‖L n ‖ ≥ C logn and the order of the Lebesgue constant is O(logn) when the Chebyshev points are taken as the nodes (cf. [16]).
ere are relatively few research results on multivariate Lagrange interpolation polynomials. In [3], from Berman's eorem, it is shown that for K � B d , the unit ball in R d , d ≥ 2, and V � d n , the order of the Lebesgue constant is O(n (d− 1)/2 ).
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 satisfies and 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 K � [− 1, 1] 2 , 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 0 < p ≤ ∞, 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 (ln n) 2 , 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 (ln n) 2 .
In [10], for K � [− 1, 1] 2 , 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 0 < p ≤ 2, 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.

The Lebesgue Constant of Xu-Type Lagrange Interpolation Polynomial on the Second Kind Chebyshev Polynomial
In order to prove eorem 1, by using reproducing kernel, we give the expression of the Lebesgue constant λ n in this section. First, we briefly introduce the Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial in [10].
is called Lebesgue constant of Xu-type Lagrange interpolation polynomial on the second kind Chebyshev polynomial. Writing then e expression of K * n (x, y) given in [10] is

Journal of Mathematics
By Lemmas 2 and 3, the following result can be obtained. 6

Proof of Theorem 1
e proof of eorem 1 is given in this section. And, since the estimates of Λ 1 n (x) and Λ 2 n (x) are similar, we need to only estimate Λ 1 n (x). (11) i,j A (12) i,j + B (11) i,j B (12) i,j , where To prove eorem 1, we first prove some lemmas.
And, we have so we obtain (37).
Similar to Lemma 8, we can obtain the following conclusion.