### Approximation theory

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

# On the uniform boundedness of the family of shifts of Steklov functions in weighted Lebesgue spaces with variable exponent

### UDK: 517.5

### Pages: 93 - 99

### DOI: 10.31029/demr.8.9

The problem of the uniform boundedness of the Steklov functions shifts families of the form
$
S_{\lambda,\tau}(f)=S_{\lambda}(f)(x+\tau)=\lambda\int_{x+\tau-\frac 1{2\lambda}}^{x+\tau+\frac 1{2\lambda}}f(t)dt
$
was considered. It was shown that these shifts are uniformly bounded in weighted variable exponent Lebesgue spaces $L^{p(x)}_{2\pi,w}$, where $w=w(x)$ is the weight function satisfying the analogue of Muckenhoupt's condition.

**Keywords: **
Lebesgue spaces with variable exponent, Dini -- Lipschitz condition, Steklov operators.