On Approximation Properties of Fourier Series in Jacobi Polynomials $P_n^{\alpha-r,-r}(x)$ Orthogonal in the Sense of Sobolev

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 $\{\varphi_n(x)\}_{n=0}^\infty$.