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

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.

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