Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Asymptotics for Sobolev orthogonal polynomials with coherent pairs: The Jacobi case, type 1


Author: K. Pan
Journal: Proc. Amer. Math. Soc. 126 (1998), 2377-2388
MSC (1991): Primary 42C05
DOI: https://doi.org/10.1090/S0002-9939-98-04300-7
MathSciNet review: 1443401
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Define $P_n(x)$ and $Q_n(x)$ as the $n$th monic orthogonal polynomials with respect to $d\mu$ and $d\nu$ respectively. The pair $\{d\mu,d\nu\}$ is called a coherent pair if there exist non-zero constants $D_n$ such that

\begin{displaymath}Q_n(x)=\frac{P_{n+1}^\prime(x)}{n+1}+D_n\frac{P_n^\prime(x)}{n},\qquad n\ge 1.\end{displaymath}

One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types for each case: type 1 and 2. We investigate the asymptotic properties and zero distribution of orthogonal polynomials with respect to Sobolev inner product

\begin{displaymath}\langle f,g\rangle =\int _a^b f(x)g(x)d\mu(x)+\lambda \int _a^b f'(x)g'(x)d\nu(x)\end{displaymath}

for the coherent pair $\{d\mu,d\nu\}$: the Jacobi case, type 1.


References [Enhancements On Off] (What's this?)

  • [BdM] K. J. Bruinsma, M. G. de Bruin and H. G. Meijer, Zero of Sobolev orthogonal polynomials following from coherent pairs, Report of FTMI 95-65, Delft University of Technology, 1995.
  • [BSS] H. P. Blatt, E. B. Saff and M. Simkani, Jentzsch-Szeg\H{o} type theorems for the zeros of best approximants, J. London Math. Soc. (2) 38 (1988) 307-316. MR 90a:30004
  • [C] E. A. Cohen, Jr., Zero distribution and behavior of orthogonal polynomials in the Sobolev space $W^{1,2}[-1,1]$, SIAM J. Math. Anal. 6 (1975), 105-116. MR 50:13664
  • [IKNS] A. Iserles, P. E. Koch, P. Norsett and J. M. Sanz-Serna, On polynomials with respect to certain Sobolev inner products, J. Approx. Th. 65 (1991), 151-175. MR 92b:42029
  • [M1] H. G. Meijer, Coherent pairs and zeros of Sobolev-type orthogonal polynomials, Indag. Math. N.S. 4 (1993), 163-176. MR 94i:42030
  • [M2] H. G. Meijer, Determination of all coherent pairs of functionals, Report of FTMI 95-41, Delft University of Technology, 1995.
  • [N] P. Nevai, Orthogonal Polynomials, Memoirs Amer. Math. Soc. 213, Amer. Math. Soc., Providence, RI, 1979. MR 80k:42025
  • [P] K. Pan, On Sobolev orthogonal polynomials with coherent pairs: the Jacobi case, J. Comp. and Appl. Math. 79 (1997), 249-262. CMP 97:13
  • [Sz] G. Szeg\H{o}, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, Providence, RI (1975). MR 51:8724

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C05

Retrieve articles in all journals with MSC (1991): 42C05


Additional Information

K. Pan
Affiliation: Department of Mathematics, Barry University, Miami Shores, Florida 33161
Email: pan@euclid.barry.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04300-7
Received by editor(s): July 24, 1996
Received by editor(s) in revised form: January 22, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society