On the summability of Jacobi series at Lebesgue points (Q1878608)

This paper studies matrix summability methods for Fourier-Jacobi series, using a triangular matrix \(\Lambda=\{\lambda_m^{(n)}\mid m,n=0,1,\ldots;\lambda_m^{(n)}=0 \text{ for } m\geq n+1\}\). Starting from the series \[ f(t)\sim \sum_{m=0}^{\infty}\,a_m{\widehat P}_m^{(\alpha,\beta)}(t),\quad h\rightarrow +0, \] where the \(\widehat P\)'s are the orthonormal system of Jacobi polynomials (\(\alpha,\beta>-1\)) with weight function \(\rho(t) = (1-t)^{\alpha}(1+t)^{\beta}\) and the function \(f\) belongs to \(L^1([-1,1],\rho)\), one constructs the linear means by \[ \tau_n^{(\alpha,\beta)}(f,t;\Lambda)= \sum_{m=0}^n\,\lambda_m^{(n)}a_m {\widehat P}_m^{(\alpha,\beta)}(t). \] The point \(t=1\) is called a Lebesgue point of \(f\) if there exists a number \(A\) with \[ \int_{1-h}^h\,| f(t)-A| dt=0(h),\quad h\rightarrow +0. \] The main results are then Theorem. Let \(-1/2<\alpha<1/2,\;\beta>-1/2\) and let \(\Lambda\) satisfy \[ \begin{gathered} \lambda_m^{(n)}\rightarrow 1\quad \text{as}\;n\rightarrow\infty\;\text{for every fixed}\;m, \tag{1} \\ \sum_{m=0}^{n-1}\,(m+1)\left({n-m \over n+1}\right)^{1/2-\alpha} | \Delta^2 \lambda_m^{(n)}| \leq C, \tag{2} \end{gathered} \] and for some \(\delta,\;0<\delta<\beta+1/2\) \[ \sum_{m=0}^n\,(m+1)^{1/2+\alpha} | \Delta^2 \lambda_m^{(n)}| \leq C n^{-\delta}, \tag{3} \] (in the case that \(\delta\geq \beta+1/2\), condition (3) can be omitted). A. Then the Jacobi-series of a function \(f\) is \(\Lambda\)-summable at the Lebesgue point \(t=1\) if \(f\) satisfies the antipole condition \[ \int_{-1}^0\,| f(t)| (1+t)^{(2\beta +2\delta-1)/4}dt<\infty. \] B. For \(-1/2<\alpha<1/2\), \(-1<\beta <-1/2\) and \(\Lambda\) satisfying (1) and (2) only, the Jacobi series of \(f\in L^1([-1,1],\rho)\) is \(\lambda\)-summable at the Lebesgue point \(t=1\). The paper is nicely written and contains the proofs of the theorems.