On \(W^{1,p}\)-convergence of Fourier-Sobolev expansions
The paper considers the convergence problem for the Fourier-Sobolev expansion in weighted Sobolev spaces on the interval \([-1,1]\). The considered weighted Sobolev space \(W^{1,p}([-1,1],(w_0,w_1))\) is defined as the set of all real functions \(f\) on \([-1,1]\) such that \(f\in L^p([-1,1],w_0)\) and \(f'\in L^p([-1,1],w_1)\), where \(w_0\in L^\infty([-1,1])\) and \(w_1\) is a Kufner-Opic type weight. Let \(\langle \cdot, \cdot\rangle_S\) denote the corresponding weighted Sobolev inner product, and \(\{q_n\}_{n\geq 0}\) be the sequence of orthonormal polynomials with respect to the weighted Sobolev inner product. Let \(S_n(f)(x):=\sum_{k=0}^n \langle f, q_k\rangle_S q_k\) be the \(n\)-th Fourier-Sobolev partial sum and \(S(f)(x):=\sum_{k=0}^\infty \langle f, q_k\rangle_S q_k\). The authors prove that, if \(w_0\in L^\infty([-1,1])\) and \(w_1\) is a Muckenhoupt \(A_p\) weight on \([-1,1]\), then \(s_nf\) converges to \(f\) in the space \(W^{1,p}([-1,1],(w_0,w_1))\) if and only if the \(W^{1,p}([-1,1],(w_0,w_1))\) norms of \(S_n(f)\) for all \(n\geq0\) are not more than a constant multiple of \(\|f\|_{W^{1,p}([-1,1],(w_0,w_1))}\). Some necessary conditions for the inequality \(\|S_n(f)\|_{W^{1,p}([-1,1],(w_0,w_1))}\leq C \|f\|_{W^{1,p}([-1,1],(w_0,w_1))}\) are also obtained.
scientific article; zbMATH DE number 6117544