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

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 p(x) > 1 that guarantee the uniform boundedness of the sequence S ^α,α _n (f), n = 0,1,..., of Fourier sums with respect to the ultraspherical Jacobi polynomials P ^α,α _k (x) in the weighted Lebesgue space L ^{p( x)} _μ ([-1, 1]) with weight μ = μ(x) = (1 - x²)^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 ^{p( x)} _μ ([-1,1]) with μ(x) = (1 - x²)^-1/2, it suffices and, in a certain sense, necessary that the variable exponent p satisfy the Dini-Lipschitz condition of the form

$$\left| {p(x) - p(y)} \right| \leq \frac{d}{{ - \ln \left| {x - y} \right|}},\;\;\;\text{where}\;\left| {x - y} \right| \leq \frac{1}{2},\;\;x,y \in [ - 1,1],\;\;d > 0,$$

and the condition p(x) > 1 for all x ∈ [-1,1].