Summability of product Jacobi expansions
The authors study expansions using orthogonal polynomials in \(d\) variables on the hypercube \([-1,1]^d\), where the weight is the product of Jacobi weights \[ W_{\alpha,\beta}(\mathbf{x})=\prod_{i=1}^d w_{\alpha_i,\beta_i}(x_i) = \prod_{i=1}^d c_{\alpha_i,\beta_i}(1-x_i)^{\alpha_i}(1+x_i)^{\beta_i}, \] with multi-indices \(\alpha=(\alpha_1,\ldots,\alpha_d),\;\beta=(\beta_1,\ldots,\beta_d)\) satisfying \(\alpha_i, \beta_i>-1\;(1\leq i\leq d)\) and \(\mathbf{x}=(x_1,\ldots,x_d)\in [-1,1]^d\); the constants are used to make the integral of the weights equal to \(1\). Using the standard orthonormal Jacobi polynomials \(p_n^{(\alpha_i,\beta_i)}\), the product orthogonal polynomials are given by \[ P^{(\alpha,\beta)}_{\mathbf{k}}(\mathbf{x}) = \prod_{i=1}^d p_{k_i}^{(\alpha_i,\beta_i)}(x_i),\;\text\textbf{k}=(k_1,\ldots,k_d)\in\text\textbf{N}_0^d. \] They form an orthonormal basis for \(L^2(W_{\alpha,\beta},[-1,1]^d)\). For an arbitrary function \(f\), Lebesgue integrable w.r.t. \(W_{\alpha,\beta}\) on \([-1,1]^d\), the Fourier expansion w.r.t. the product weight is then given by \[ f\sim\sum_{m=0}^{\infty} \sum_{|\mathbf{k}|=0}^m a_{\mathbf{k}}(f) P^{(\alpha,\beta)}_{\mathbf{k}} \text{ with } a_{\mathbf{k}}(f)=\int_{[-1,1]^d} f(\mathbf{y}) P^{(\alpha,\beta)}_{\text\textbf{k}}(\text\textbf{y})W_{\alpha,\beta}(\text\textbf{y})d\text\textbf{y}. \] The partial sums \(S_n(W_{\alpha,\beta};f,\mathbf{x})\) (use in the first summation \(\sum_{m=0}^n\)) are used to define for \(\delta>0\) the Cesàro \((C,\delta)\) means: \[ S^{\delta}_n(W_{\alpha,\beta};f)={n+\delta\choose n}^{-1} \sum_{k=0}^n {n-k+\delta-1\choose n-k}S_k(W_{\alpha,\beta};f). \] The main results, generalizing the one variable Jacobi case, are then \textbf{Case 1.} Put \(\alpha_j, \beta_j>-1, \alpha_j+\beta_j\geq -1\). \textbf{(a)} The \((C,\delta)\) means are uniformly bounded in the norm of \(L^p(W_{\alpha,\beta};[-1,1]^d),\;1\leq p<\infty\), and in the norm of \(C([-1,1]^d)\) if \[ \delta>\delta_0:=\sum_{j=1}^d \max \{\alpha_j,\beta_j\}+ {d\over 2} + \max \Biggl\{0,-\sum_{j=1}^d \min\{\alpha_j,\beta_j\}- {d+2\over 2} \Biggr\}. \] \textbf{(b)} If \(-\sum_{j=1}^d \min\{\alpha_j,\beta_j\}- {d+2\over 2}>0\) and \(\alpha_i=\beta_i=-{1\over 2}\) does not hold for any \(i\), then the conclusion in \text\textbf{(a)} holds for \(\delta\geq \delta_0\). In particular, for each \(f\) in \(L^p(W_{\alpha,\beta};[-1,1]^d),\;1 \leq p<\infty\) or in \(C([-1,1]^d)\), the \((C,\delta)\) means converge to \(f\) in the accompanying norm for \(n\rightarrow\infty\). \textbf{Case 2.} Put \(\alpha_j,\beta_j\geq -{1\over 2}\). Then the \((C,\delta)\) means define a \textit{positive} linear approximation identity on \(C([-1,1]^d)\) if \(\delta\geq\sum_{i=1}^d (\alpha_i+\beta_i)+3d-1\). Moreover, the order of summability is best possible in the sense that the means do \textit{not} define a positive identity, id \(\delta\) drops below the bound given. The proofs are quite technical, using estimates on certain \({}_2F_1\)'s and on special integrals of rational functions in sines and cosines.
scientific article; zbMATH DE number 1496280
scientific article