### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 8 (2017)

# Approximation of piecewise linear functions by discrete Fourier sums

### UDK: 517.521.2

### Pages: 21 - 26

### DOI: 10.31029/demr.8.3

Let be a natural number greater than . We select uniformly distributed points on . Denote by the trigonometric polynomial of order possessing the least quadratic deviation from with respect to the system . In the present article the problem of function approximation by the polynomials is considered. Special attention is paid to approximation of -periodic functions and by the polynomials , where and for . For the first function we show that instead of the estimate which follows from well-known Lebesgue inequality for the polynomials we found an exact order estimate () which is uniform with respect to . Moreover, we found a local estimate () which is also uniform with respect to . For the second function we found only a local estimate () which is uniform with respect to . The proofs of these estimations based on comparing of approximating properties of discrete and continuous finite Fourier series.

**Keywords: **
function approximation, trigonometric polynomials, Fourier series.