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

摘要

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~(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 |p(x) -p(y)| ≤ d/(-ln|x-y|),where |x - y| ≤1/2, x,y∈[-1,1], d ＞ 0, and the condition p(x) ＞ 1 for all x ∈ [-1,1].