Approximation of classes of periodic multivariable functions by linear positive operators

Abstract

In an N-dimensional space, we consider the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is positive. We establish that the least upper bound of this approximation does not exceed the sum of properly chosen least upper bounds in m- and (( N ‐ m ))-dimensional spaces. We also consider the cases where the inequality obtained turns into the equality. It is known that the determination of least upper bounds for the approximation of classes of periodic functions by linear operators in an N-dimensional space is not always reducible to the calculation of the corresponding quantities in a space of lower dimension. In the present work, we indicate conditions under which this reduction is possible. In what follows, we prove that the least upper bound of the approximation of classes of translation-invariant periodic functions by a linear operator whose kernel is the product of two kernels one of which is m-dimensional and positive does not exceed the sum of properly chosen least upper bounds in m- and ( N ‐ m )-dimensional spaces. We establish that the inequality obtained turns into the equality in the case of centrally symmetric classes of continuous or essentially bounded functions that satisfy an additional condition. The results obtained yield, as a consequence, Theorem 1 from [1]. We show that the approximation by a linear positive operator with arbitrary kernel depends only on the one-dimensional components of this kernel, i.e., it coincides with the approximation by a linear positive operator whose kernel is the product of one-dimensional kernels. Let C N , L N , and L p be the spaces of functions f ( x ) = f ( x 1 , … , x N ) that are 2π-periodic in each of N variables and are, respectively, continuous, essentially bounded, and summable to the p th power ( 1 ≤ p < ∞ ) with the norms