Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1205.42027
Mastroianni, Giuseppe; Totik, Vilmos
Uniform spacing of zeros of orthogonal polynomials. (English)
[J] Constr. Approx. 32, No. 2, 181-192 (2010). ISSN 0176-4276; ISSN 1432-0940/e

If $w$ is a weight function (id est a non-negative integrable function) on $[-1,1]$, then the doubling property means that for some constant $L$, $$\int_{2I}w\leq L\int_Iw$$ for all intervals $I\subset[-1,1]$, where $2I$ denotes the ``doubled" interval $I$ (twice enlarged from its center). The constant $L$ is referred to as the doubling constant of $w$. It is shown that for those doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced, which means that if $\cos\theta_{m,k}$ with $\theta_{m,k}\in[0,\pi]$ are the zeros of the $m$-th orthogonal polynomial associated with $w$, then $\theta_{m,k}-\theta_{m,k+1}\sim\frac{1}{m}$. It is also shown that for doubling weights, neighbouring Cotes numbers are of the same order. In fact, it is shown that these two properties are actually equivalent to the doubling property of the weight function.
[Roelof Koekoek (Delft)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials

Keywords: orthogonal polynomials; spacing of zeros; doubling weights

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]