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Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line. (English) Zbl 0604.41025

Positive interpolation operators ${𝒥}_{n,p}$, where $0, defined by

${𝒥}_{n,p}\left[f\right]\left(x\right)=\frac{\sum nk=1{\lambda }_{kn}f\left({x}_{kn}\right){|{K}_{n}\left(x,{x}_{kn}\right)|}^{p}}{{\sum }_{k=1}^{n}{\lambda }_{kn}{|{K}_{n}\left(x,{x}_{kn}\right)|}^{p}}$

for weights ${W}^{2}\left(x\right)=exp\left(-2Q\left(x\right)\right)$, are introduced. Here ${K}_{n}\left(x,t\right)$ is the kernel of degree at most n-1 in x, t for the partial sums of the orthogonal expansions with respect to ${W}^{2}$, and $\left\{{x}_{kn}\right\}$ and $\left\{{\lambda }_{kn}\right\}$ are the abcissas and weights in the Gaussian quadrature of order n. Their basic properties are established, and their convergence is proved for $1 and a certain class of weights on the whole real line. P. G. Nevai [Orthogonal polynomials, Mem. Am. Math. Soc. 213 (1979; Zbl 0405.33009)] has considered the special case $p=2$ and weights on [-1,1].

Reviewer: H.R.Dowson

##### MSC:
 41A36 Approximation by positive operators 42C05 General theory of orthogonal functions and polynomials