The authors are interested in sequences of orthogonal polynomials which are determined by a three term recurrence relation xpn,N(x)=an+1,Npn+1,N(x)+bn,Npn,N(x)+an,Npn−1,N(x), where the coefficients, and hence the orthogonal polynomials, depend not only on n but on an additional parameter N. In particular, it is assumed that N=N(n) and the ratio n/N(n)→t as t→∞. The main result describes the asymptotic behavior of the distribution of the zeros of pn,N as n/N→t. The authors take advantage of the fact that the zeros can be characterized as the eigenvalues of a Jacobi matrix, which can be approximated by certain matrices whose spectral distribution can be computed explicitly.

From Toeplitz matrix sequences to zero distribution of orthogonal polynomials

SERRA CAPIZZANO, STEFANO
2003-01-01

Abstract

The authors are interested in sequences of orthogonal polynomials which are determined by a three term recurrence relation xpn,N(x)=an+1,Npn+1,N(x)+bn,Npn,N(x)+an,Npn−1,N(x), where the coefficients, and hence the orthogonal polynomials, depend not only on n but on an additional parameter N. In particular, it is assumed that N=N(n) and the ratio n/N(n)→t as t→∞. The main result describes the asymptotic behavior of the distribution of the zeros of pn,N as n/N→t. The authors take advantage of the fact that the zeros can be characterized as the eigenvalues of a Jacobi matrix, which can be approximated by certain matrices whose spectral distribution can be computed explicitly.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1490613
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact