Polynomial projections in \(C[-1,1]\) and \(L^1(-1,1)\) with growth \(n^Y\), \(0<Y\leq 1/2\)
Let \(X\) be either the space \(C[-1,1]\) or the space \(L^1_{\omega_{\alpha,\beta}}(-1,1),\) where \(\omega_{\alpha, \beta}\) is the weight function of the Jacobi polynomials \(P_m^{\alpha,\beta}\) and \((\alpha, \beta) \in\{0,0), (-{1\over 2}, -{1\over{2}}), (-{1\over 2}, 0), (0, -{1\over 2})\}.\) If \(Y\) is a subspace in \(X,\) than by \(\lambda(Y,X)\) \(:=\inf\{| | P| | ;\) \(P\) is a continuous linear projection from \(X\) onto \(Y\}\) is denoted the relative projection constant of \(Y\) in \(X.\) It is proved that for each \(j\in(0, {1\over 2})\) there exists a set of integers \(0\leq k_1<\ldots<k_n,\) that \(c_1n^\gamma\leq\lambda(Y_n,X)\leq c_2n^\gamma,\) where \(Y_n:=span\{P_{k_j}^{2\alpha+{1\over 2}, 2\beta+{1\over 2}}(x);\) \(1\leq j\leq n\}\) and \(c_1, c_2>0\) are certain constants that do not depend on \(n.\) It should be noted from the more comparison of the growth rate of two sequences \(\{k_j\}\) it is impossible to establish which of the projection constants increases faster.
scientific article; zbMATH DE number 1271287
scientific article