Polynomial approximation on infinite intervals with weights having inner zeros
The authors generalize Languerre weights on \(\mathbb R^+\) by multiplying them by translations of finitely many Freud type weights which have singularities, and prove polynomial approximation theorems in the corresponding weighted spaces. Let \(\mathcal V\) denote the set of continuous functions \(v(x)\) which are even on \(\mathbb R=(-\infty,\infty)\), \(v(0)=0\), \(v'(x)>0\) on \(\mathbb R^+:=(0,\infty)\), and \[ A(v):=\lim_{x\to 0} \frac{xv'(x)}{v(x)} <\infty . \] Further let (1) \(u(x)=w_\alpha(x)V(x)\), \(V(x)=\prod^s_{k=1}v_k(x-t_k)\) with \(w_\alpha(x)=x^\alpha e^{-x}\), \(\alpha>-1\), \(x>0\), \(0<t_1<t_2<\dots <t_s<\infty\), \(v_k\in \mathcal V\), \(k=1,\dots,s\). Let \(L^p(A)\), \(A\subset \mathbb R\), \(1\leq p<\infty\), denote the set of all measurable functions \(f\) with the usual norm \(\|f\|_p(\int_A|f(x)|^pdx)^{1/p}\). If \(\sigma\) is a weight function then \(f\in L^p_\sigma(A)\) if \(f\sigma\in L^p(A)\). In order to give the central result of the paper to all functions that have singular points at the zeros \(t_i\) of the weight \(u\), the authors introduce some new classes of functions. For \(1\leq p<\infty\), \(L^p_u:=L^p_u(\mathbb R^+)\) and \(L^\infty_u\) denote the set of all continuous functions in \(\mathbb R^+\setminus \{t_1,t_2,\dots,t_s\}=:\tilde{\mathbb R}^+\) such that \(\lim_{x\to t_i}(fu)(x)=0\), \(i=1,\dots,s\) and \(\lim_{x\to 0}(fu)(x)=\lim_{x\to \infty}(fu)(x)=0\). Finally is defined \(W^p_r(u)=\{f\in L^p_u:f^{(r-1)}\in AC(\tilde{\mathbb R}^+)\), \(\|f^{(r)}\varphi^r u\|_p<\infty\}\) where \(r\geq 1\), \(1\leq p\leq \infty\), \(\varphi(x)=\sqrt x\) and \(AC(\tilde{\mathbb R}^+)\) is the set of all absolutely continuous functions in \(\tilde{\mathbb R}^+\). Theorem. Assume that for \(v_k\in \mathcal V\) we have \(0<A(v_k)\neq j-\frac 1p\), \(k=1,\dots,s\); \(j=1,\dots,r\) with \(1\leq p\leq \infty\), or \(A(v_k)=0\) for some \(1\leq k\leq s\) with \(p=\infty\). Then for each function \(f\in W^p_r(u)\), we have \[ E_n(f)_{u,p}\leq \frac C{n^{r/2}}\big(\|f^{(r)}\varphi^r u\|_p+\|fu\|_p\big), \] where \(C>0\) is an absolute constant, \(E_n(f)_{u,p}=\text{inf}_{p\in \mathbb P_n}\|(f-P)u\|_p\) is the error of the best approximation and \(\mathbb P_n\) is the set of all polynomials of degree at most \(n\). The other polynomial approximation theorems are given with the aid of a \(K\)-functional and the corresponding modulus of smoothness.
scientific article; zbMATH DE number 1912066