Integral approximation of the characteristic function of an interval and the Jackson inequality in C( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC % vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz % ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb % L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe % pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam % aaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv39ga % iyaacqWFtcpvaaa!46CC! $$ \mathbb{T} $$ )

Abstract

We apply the results on the integral approximation of the characteristic function of an interval by the subspace \( \mathcal{T}_{n - 1} \) of trigonometric polynomials of order at most n − 1, which were obtained by the authors earlier, to investigate the Jackson inequality between the best uniform approximation of a continuous periodic function by the subspace \( \mathcal{T}_{n - 1} \) and its modulus of continuity of the second order. The corresponding method of the uniform approximation of continuous periodic functions by trigonometric polynomials is constructed.