Estimates of the norm of a function orthogonal to the piecewise-constant functions in terms of higher-order moduli of continuity

Abstract

The uniform norm of a function that is defined on the real line and has zero integrals between integer points is estimated in terms of its modulus of continuity of arbitrary even order. Sharp bounds of this kind are known for periodic functions. The passage to nonperiodic functions significantly complicates the problem. In general, the constant for nonperiodic functions is greater than that for periodic functions. The constants in the bound are improved compared with those known earlier. The proof is based on a representation of the error of the polynomial interpolation as the product of the influence polynomial and an integrated difference of higher order.