Estimation of the remainder of a cubature formula on a Chebyshev grid

Abstract

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [−1, 1] × [−1, 1] with the norm

$\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $

On a Chebyshev grid, a cubature formula of the form

$\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $

is considered in some class H(r ₁, r ₂) of functions f ∈ C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate

$\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $

where r ₁, r ₂ > 1; λ⁻¹ ≤ n/m ≤ λ with λ > 0; and the constant in O(1) depends on λ.