Estimates of the norm of a function orthogonal to piecewiseconstant functions by moduli of continuity of high order

Abstract

In the paper, we estimate the uniform norm of a function defined on the real line and having zero integrals between integer points by its modulus of continuity of arbitrary even order. Sharp estimates of such kind are known for periodic functions. The passage to non-periodic functions essentially complicates the problem. In general, the constant for non-periodic functions is greater than for periodic ones. The constants in the estimate are improved in comparison with those known earlier. The estimates under discussion have something in common with the problem of finding the Whitney constants, i.e. the constants in the inequalities between the best approximations and the moduli of continuity of a function defined on the segment. The proof is based on the representation of the error of the polynomial interpolation as a product of the influence polynomial and the integrated difference of high order. We also obtain pointwise estimates in terms of moduli of continuity. Refs 5.