zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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<p<, defined by

𝒥 n,p [f](x)=nk=1λ kn f(x kn )|K n (x,x kn )| p k=1 n λ kn |K n (x,x kn )| p

for weights W 2 (x)=exp(-2Q(x)), are introduced. Here K 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 2 , and {x kn } and {λ 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<p2 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

41A36Approximation by positive operators
42C05General theory of orthogonal functions and polynomials