Vinogradov, O. L. Upper bounds of the Lebesgue constants of summation methods for Fourier-Jacobi series defined by a multiplier function. (English. Russian original) Zbl 1092.42018 J. Math. Sci., New York 120, No. 5, 1662-1671 (2004); translation from Zap. Nauchn. Semin. POMI 282, 34-50 (2001). Summary: Let \(P_k^{(\alpha,\beta)}\) be the Jacobi polynomials and let \(C[a,b]\) be the space of continuous functions on \([a,b]\) with the uniform norm. In this paper, we study sequences of Lebesgue constants, i.e., of the norms of linear operators \({\mathcal U}_n^\Lambda: C[-1,1]\to C[-1,1]\) generated by a multiplier matrix \(\Lambda= \{\lambda_k^{(n)}\}\) defined by the following relations: \[ f\sim \sum_{k=0}^\infty a_k P_k^{(\alpha,\beta)}, \qquad {\mathcal U}_n^\Lambda f\sim \sum_{k=0}^\infty \lambda_k^{(n)} a_k P_k^{(\alpha,\beta)}, \] and \[ {\mathfrak L}_n^{(\alpha,\beta)}(\Lambda)= \sum_{y\in[-1,1]}\, \sum_{\|f\|\leq 1} |{\mathcal U}_n^\Lambda f(y)|. \] In the case \(|\alpha|= |\beta|= 1/2\), we prove the following statements for the Jacobi polynomials (these statements are similar to known results for the trigonometrical system). Consider the cases \[ \begin{aligned} \alpha = \beta= -1/2 &\quad\text{and}\quad \lambda_k^{(n)}= \varphi(k/n); \tag{1}\\ \alpha = \beta= 1/2 &\quad\text{and}\quad \lambda_k^{(n)}= \varphi((k+1)/n); \tag{2}\\ \text{and } \alpha = -\beta= \pm1/2 &\quad\text{and}\quad \lambda_k^{(n)}= \varphi((k+1/2)/n). \tag{3}\end{aligned} \] Under some conditions on a function \(\varphi\), the values \(\sup_{n\in\mathbb N}{\mathfrak L}_n^{(\alpha,\beta)}(\Lambda)\) and \(\lim_{n\to\infty}{\mathfrak L}_n^{(\alpha,\beta)}(\Lambda)\) equal \[ \begin{aligned} \frac 2\pi \int_0^\infty \Biggl| \int_0^\infty \varphi(t)\cos zt\,dt\Biggr|\,dz &\qquad \text{(case (1))}\\ \text{and} \frac 2\pi \int_0^\infty z\Biggl| \int_0^\infty t\varphi(t)\sin zt\,dt\Biggr|\,dz &\qquad \text{(cases (2) and (3))}. \end{aligned} \] In addition, we show that for the summation methods for Fourier-Legendre series \((\alpha= \beta= 0)\) generated by the multiplier function \(\lambda_k^{(n)}= \varphi(k/n)\), the limit and supremum of the sequence of Lebesgue constants may differ. Cited in 1 Document MSC: 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42A24 Summability and absolute summability of Fourier and trigonometric series Keywords:Fourier-Jacobi series; Fourier-Legendre series; summation; Lebesgue constants × Cite Format Result Cite Review PDF Full Text: DOI