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

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## Abstract

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
and the condition

*p(x)*> 1 that guarantee the uniform boundedness of the sequence*S*_{ n}^{α,α}(*f*),*n*= 0,1,..., of Fourier sums with respect to the ultraspherical Jacobi polynomials*P*_{ k}^{α,α}(*x*) in the weighted Lebesgue space*L*_{ μ}^{ p( x)}([-1, 1]) with weight*μ*=*μ*(*x*) = (1 -*x*^{2})^{α}, where*α*>-1/2. The case*α*= -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,..., of Fourier—Chebyshev sums in the space*L*_{ μ}^{ p( x)}([-1,1]) with*μ*(*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$$\left| {p(x) - p(y)} \right| \leq \frac{d}{{ - \ln \left| {x - y} \right|}},\;\;\;\text{where}\;\left| {x - y} \right| \leq \frac{1}{2},\;\;x,y \in [ - 1,1],\;\;d > 0,$$

*p*(*x*) > 1 for all*x*∈ [-1,1].## Keywords

the basis property of ultraspherical polynomials Fourier—Jacobi sums Fourier— Chebyshev sums convergence in a weighted Lebesgue space with variable exponent Dini—Lipschitz condition## Preview

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## Notes

### Funding

This work was supported by the Russian Foundation for Basic Research under grant 16-01-00486.

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