Abstract:
The paper presents sufficient conditions for the generalized absolute convergence of series of Fourier–Jacobi coefficients (with factors) of functions in $L^p[0,\pi]$, $1<p\leq 2$, with Jacobi weight, where the factors satisfy the reverse Hölder inequality. A proof is given that this result cannot be improved in the case of $p=2$ under certain restrictions. An analog of the Titchmarsh equivalence theorem on the relationship between the smoothness of a function and the behavior of the remainder of a series of its Fourier–Jacobi coefficients with a power weight is also given.
Keywords:Fourier–Jacobi series, generalized absolute convergence, generalized modulus of smoothness.
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