Approximation of functions by Fourier-Laguerre sums
Let \(L_n^\alpha(x)\) \((n= 0, 1,\dots)\) be an orthogonal system of Laguerre polynomials and \[ f(x)= \sum_{k= 0}^\infty b_k(f) L_n^\alpha(x) \] the Fourier-Laguerre series for \(f\in L_2= L_2(R; \exp(- x) x^\alpha)\) \((\{\alpha\}>- \frac 12)\). Using estimates of the partial sums of Fourier-Laguerre series and the generalized shift operator \(f_h\), two sided estimates of the Kolmogorov \(n\)-diameter of \(D_n(W_\omega^r(D), L_2)\), where \(\omega(t)\) is an arbitrary modulus of continuity and \(D\) is the Laguerre differential operator, are given.
scientific article; zbMATH DE number 846473