### Approximation theory

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

# Overlapping transformations for approximation of continuous functions by means of repeated mean Valle Poussin

### UDK: 517.538

### Pages: 70 - 92

### DOI: 10.31029/demr.8.8

On the basis of trigonometric sums of Fourier $S_n(f,x)$ and classical means of Valle Poussin $_1V_{n,m}(f,x)= \frac1n\sum_{l=m}^{m+n-1}S_l(f,x)$
in this paper, repeated mean Valle Poussin is introduced as follows $_2V_{n,m}(f,x)= \frac1n\sum_{k=m}^{m+n-1}{}_1V_{n,k}(f,x),$ ${}_{l+1}V_{n,m}(f,x)= \frac1n\sum_{k=m}^{m+n-1} {}_{l}V_{n,k}(f,x)\quad(l\ge1).$
On the basis of the mean $_2V_{n,m}(f,x)$ and overlapping transforms, operators that approximate continuous (in general, nonperiodic) functions are constructed and their approximative properties are investigated.

**Keywords: **
the repeated mean Valle Poussin, overlapping transforms, local approximative properties.