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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 ${\cal J}\sb{n,p}$, where $0<p<\infty$, defined by $$ {\cal J}\sb{n,p}[f](x)=\frac{\sum {n}{k=1}\lambda\sb{kn}f(x\sb{kn})\vert K\sb n(x,x\sb{kn})\vert\sp p}{\sum\sp{n}\sb{k=1}\lambda\sb{kn}\vert K\sb n(x,x\sb{kn})\vert\sp p} $$ for weights $W\sp 2(x)=\exp (-2Q(x))$, are introduced. Here $K\sb n(x,t)$ is the kernel of degree at most n-1 in x, t for the partial sums of the orthogonal expansions with respect to $W\sp 2$, and $\{x\sb{kn}\}$ and $\{\lambda\sb{kn}\}$ are the abcissas and weights in the Gaussian quadrature of order n. Their basic properties are established, and their convergence is proved for $1<p\le 2$ and a certain class of weights on the whole real line. {\it 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

41A36Approximation by positive operators
42C05General theory of orthogonal functions and polynomials
Full Text: DOI
[1] Freud, G.: Orthogonal polynomials. (1966) · Zbl 0265.42010
[2] Freud, G.: On the greatest zero of an orthogonal polynomial, II. Acta sci. Math. (Szeged) 36, 49-54 (1974) · Zbl 0285.33012
[3] Freud, G.: On Markov-Bernstein-type inequalities and their applications. J. approx. Theory 19, 22-37 (1977) · Zbl 0356.41003
[4] D. S. Lubinsky, Gaussian quadrature, weights on the whole real line and even entire functions with non-negative even order derivatives, J. Approx. Theory, in press. · Zbl 0608.41017
[5] Nevai, P. G.: Orthogonal polynomials. Mem. amer. Math. soc. 18, No. 213 (1979) · Zbl 0405.33009
[6] P. G. Nevai, Exact bounds for orthogonal polynomials with exponential weights, J. Approx. Theory, in press. · Zbl 0605.42019
[7] Szegö, G.: Orthogonal polynomials. (1967) · Zbl 65.0278.03