Approximation with Singular Integrals of the Jackson Type

Abstract

Assuming f(x) ϵ C _2π we consider a sequence L _n,p (n = 1,2,...) of linear positive operators of the form

$$ {L_{n,p}}(f;x) = \frac{1}{{A(n,p)}}\int\limits_{ - \pi }^\pi {f(x + t)} \frac{{\sin \frac{1}{2}nt}}{{\sin \frac{1}{2}}}2{p_{dt;}} $$

((1))

here p denotes an arbitrary positive integer and

$$ A(n,p) = \int\limits_{ - \pi }^\pi {(\frac{{\sin \frac{1}{2}nt}}{{\sin \frac{1}{2}t}})2{P_{dt.}}}$$

.