zbMATH — the first resource for mathematics

Examples
 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.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 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)
Uniform spacing of zeros of orthogonal polynomials. (English) Zbl 1205.42027

If $w$ is a weight function (id est a non-negative integrable function) on $\left[-1,1\right]$, then the doubling property means that for some constant $L$,

${\int }_{2I}w\le L{\int }_{I}w$

for all intervals $I\subset \left[-1,1\right]$, 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 ${\theta }_{m,k}\in \left[0,\pi \right]$ 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.

MSC:
 42C05 General theory of orthogonal functions and polynomials
References:
 [1] Findley, M.: Universality for locally Szego measures. J. Approx. Theory 155, 136–154 (2008) · Zbl 1171.42015 · doi:10.1016/j.jat.2008.03.013 [2] Freud, G.: Orthogonal Polynomials. Akadémiai Kiadó, Budapest (1971) [3] Levin, A.L., Lubinsky, D.S.: Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials. J. Approx. Theory 150, 69–95 (2008) · Zbl 1138.33006 · doi:10.1016/j.jat.2007.05.003 [4] Levin, A.L., Lubinsky, D.: Orthogonal polynomials for exponential weights. In: CMS Books in Mathematics. Ouvrages de Mathématiques de la SMC, vol. 4. Springer, New York (2001) [5] Mastroianni, G., Totik, V.: Polynomial inequalities with doubling and A weights. Constr. Approx. 16, 37–71 (2000) · Zbl 0956.42001 · doi:10.1007/s003659910002