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Titolo:
GRAM POLYNOMIALS AND THE KUMMER FUNCTION
Autore:
BARNARD RW; DAHLQUIST G; PEARCE K; REICHEL L; RICHARDS KC;
Indirizzi:
TEXAS TECH UNIV,DEPT MATH LUBBOCK TX 79409 ROYAL INST TECHNOL,DEPT COMP SCI S-10044 STOCKHOLM SWEDEN KENT STATE UNIV,DEPT MATH & COMP SCI KENT OH 44242 SOUTHWESTERN UNIV,DEPT MATH GEORGETOWN TX 78626
Titolo Testata:
Journal of approximation theory (Print)
fascicolo: 1, volume: 94, anno: 1998,
pagine: 128 - 143
SICI:
0021-9045(1998)94:1<128:GPATKF>2.0.ZU;2-P
Fonte:
ISI
Lingua:
ENG
Tipo documento:
Article
Natura:
Periodico
Settore Disciplinare:
CompuMath Citation Index
Science Citation Index Expanded
Citazioni:
22
Recensione:
Indirizzi per estratti:
Citazione:
R.W. Barnard et al., "GRAM POLYNOMIALS AND THE KUMMER FUNCTION", Journal of approximation theory (Print), 94(1), 1998, pp. 128-143

Abstract

Let {phi(k)}(k=0)(n), n<m, be a family of polynomials orthogonal withrespect to the positive semi-definite bilinear form (g, h)(d) := 1/m Sigma(j=1)(m) g(x(j))h(x(j)), x(j) ;= -1 + (2j - 1)/m. These polynomials are known as Gram polynomials. The present paper investigates the growth of \phi(k)(x)\ as a function of k and m for fixed x is an element of [ -1, 1]. We show that when n less than or equal to 2.5m(1/2), the polynomials in the family {phi(k)}(k=0)(n) are of modest size on [ -1, 1], and they are therefore well suited for the approximation of functions on this interval. We also demonstrate that if the degree k is close to m, and m greater than or equal to 10, then phi(k)(x) oscillates with large amplitude for values of x near the endpoints of [ -1, 1],and this behavior makes phi(k) poorly suited for the approximation offunctions on [ -1, 1]. We study the growth properties of \phi(k)(x)\ by deriving a second order differential equation, one solution of which exposes the growth. The connection between Gram polynomials and thissolution to the differential equation suggested what became a long-standing conjectured inequality for the confluent hypergeometric function F-1(1), also known as Kummer's function, i.e., that F-1(1)((1 - a)/2, 1, t(2)) less than or equal to F-1(1)(1/2, 1, t(2)) for all a greater than or equal to 0. In this paper we completely resolve this conjecture by verifying a generalization of the conjectured inequality with sharp constants. (C) 1998 Academic Press.

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Documento generato il 25/11/20 alle ore 18:51:33