Geodesic
On Approximation Properties of Fourier Series in Jacobi Polynomials $P_n^{\alpha-r,-r}(x)$ Orthogonal in the Sense of Sobolev
Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 803-818

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The article deals with the problem of the deviation from a function $f$ belonging to the space $W^r$ of partial sums of Fourier series with respect to the system of polynomials $\{\varphi_n(x)\}_{n=0}^\infty$, orthogonal with respect to an inner product of Sobolev type. Here $\varphi_n(x)=(x+1)^n/n!$ for $0\le n\le r-1$ and $$ \varphi_n(x)=\frac{2^r}{(n+\alpha-r)^{[r]} \sqrt{h_{n-r}^{\alpha,0}}}\,P_n^{\alpha-r,-r}(x)\qquad\text{for}\quad n\ge r, $$ where $P_n^{\alpha-r,-r}(x)$ is the Jacobi polynomial of degree $n$. The main attention is paid to obtaining an upper bound for a Lebesgue-type function of partial sums of the Fourier series with respect to the system $\{\varphi_n(x)\}_{n=0}^\infty$.
Keywords: Sobolev-type inner product, Jacobi polynomials, Fourier series, approximation properties. 
@article{MZM_2022_111_6_a0,
     author = {R. M. Gadzhimirzaev},
     title = {On {Approximation} {Properties} of {Fourier} {Series} in {Jacobi} {Polynomials} $P_n^{\alpha-r,-r}(x)$ {Orthogonal} in the {Sense} of {Sobolev}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {803--818},
     publisher = {mathdoc},
     volume = {111},
     number = {6},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a0/}
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R. M. Gadzhimirzaev. On Approximation Properties of Fourier Series in Jacobi Polynomials $P_n^{\alpha-r,-r}(x)$ Orthogonal in the Sense of Sobolev. Matematičeskie zametki, Tome 111 (2022) no. 6, pp. 803-818. http://geodesic.mathdoc.fr/item/MZM_2022_111_6_a0/