The Basis Property of Ultraspherical Jacobi Polynomials in a Weighted Lebesgue Space with Variable Exponent

The problem of the basis property of ultraspherical Jacobi polynomials in a Lebesgue space with variable exponent is studied. We obtain sufficient conditions on the variable exponent $p(x)>1$ that guarantee the uniform boundedness of the sequence $S_n^{\alpha,\alpha}(f)$, $n=0,1,\dots$, of Fourier sums with respect to the ultraspherical Jacobi polynomials $P_k^{\alpha,\alpha}(x)$ in the weighted Lebesgue space $L_\mu^{p(x)}([-1,1])$ with weight $\mu=\mu(x)=(1-x^2)^\alpha$, where $\alpha>-1/2$. The case $\alpha=-1/2$ is studied separately. It is shown that, for the uniform boundedness of the sequence $S_n^{-1/2,-1/2}(f)$, $n=0,1,\dots$, of Fourier–Chebyshev sums in the space $L_\mu^{p(x)}([-1,1])$ with $\mu(x)=(1-x^2)^{-1/2}$, it suffices and, in a certain sense, necessary that the variable exponent $p$ satisfy the Dini–Lipschitz condition of the form
$$ |p(x)-p(y)|\le \frac{d}{-\ln|x-y|}\mspace{2mu}, \qquad\text{where}\quad |x-y|\le \frac{1}{2},\quad x,y\in[-1,1],\quad d>0, $$
and the condition $p(x)>1$ for all $x\in[-1,1]$.