Jackson-type theorem on approximation by algebraic polynomials in the uniform metric with Laguerre weight (Q6636266)

Let \(\alpha >-1\), let \(r \in \mathbb{N}\), and let \(f\in C[0, \infty)\) be such that the derivatives \(f^{(k)}(0); k=1, \dots, r-1\) exist; in [Sib. Math. J. 58, No. 2, 338--362 (2017; Zbl 1371.33023); translation from Sib. Mat. Zh. 58, No. 2, 440--467 (2017)] \textit{I. I. Sharapudinov} studied series in Laguerre polynomials \(L_k^{\alpha}\) of the form\N\[\Nf(x) \backsim P_{r-1}(f,x)+ x^r\sum\widehat{f}^{\alpha,r}_kL_k^{\alpha}(x),\N\]\Nwhere \(P_{r-1}(f,x)\) denotes the Maclaurin polynomial of degree \(r-1\) of \(f(x)\),\N\[\N\widehat{f}^{\alpha,r}_k=h^{-\alpha}_k\int_0^{\infty} f_r(t)\, \rho(t)\, L_k^{\alpha}(t)\, dt, \quad \rho(t)=t^{\alpha}e^{-t}, \quad f_r(t)=\frac{f(t)-P_{r-1}(f,t)}{t^{r}},\N\]\Nand \(h^{\alpha}_k\) is a normalizing factor. In particular, Sharapudinov studied the approximation properties of the polynomials \(S_n^{\alpha}(x)\), where \(S_n^{\alpha}(x)\) denotes the truncation of the above displayed series at \(k=n-r\), and obtained results that were expressed in terms of \(E_n(f, u_r)\), where\N\[\NE_n(f, u_r)=\inf_{p_n}\sup_{x>0}|p_n(x)-f(x)|u_r, \quad u_r(x)=e^{-x/2}x^{-r/2+1/4},\N\]\Nand the lower bound is taken over all algebraic polynomials \(p_n\) of degree not exceeding \(n\) such that \(f^k(0)=p_n(0),\ 0 \le k \le r-1\).\N\NThe main objective of the paper under review is to find lower and upper bounds for \(E_n(f, u_1)\). To find these bounds the author uses Vallée-Poussin means of partial sums of the Fourier series of \(f\) in the system of Laguerre-Sobolev polynomials introduced by \textit{D. S. Lubinsky} and \textit{V. Totik} [SIAM J. Math. Anal. 25, No. 2, 555--570 (1994; Zbl 0799.41009)].