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

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.