### Approximation theory

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

# Convergence of Fourier series in Jacobi polynomials in weighted Lebesgue space with variable exponent

### UDK: 517.538

### Pages: 27 - 47

### DOI: 10.31029/demr.8.4

The problem of basis property of the Jacobi polynomials system $P_n^{\alpha,\beta}(x)$ in the weighted Lebesgue space $L^{p(x)}_\mu([-1,1])$ with variable exponent $p(x)$ and $\mu(x) = (1-x)^\alpha(1+x)^\beta$ is considered. It is shown that if $\alpha,\beta>-1/2$ and $p(x)$ satisfies on $[-1,1]$ some natural conditions then the orthonormal Jacobi polynomials system $p_n^{\alpha,\beta}(x)=(h_n^{\alpha,\beta})^{-\frac12}P_n^{\alpha,\beta}(x)$ $(n=0,1,\ldots)$ is a basis of $L^{p(x)}_\mu([-1,1])$ as
$4\frac{\alpha+1}{2\alpha+3}<p(1)<4\frac{\alpha+1}{2\alpha+1}$, $4\frac{\beta+1}{2\beta+3}<p(-1)<4\frac{\beta+1}{2\beta+1}$.

**Keywords: **
Basis property of the Jacobi polynomials, Fourier-Jacobi sums, convergence in the weighted Lebesgue space with variable exponent, Dini-Lipshits condition.