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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 S10044 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:
 00219045(1998)94:1<128:GPATKF>2.0.ZU;2P
 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. 128143
Abstract
Let {phi(k)}(k=0)(n), n<m, be a family of polynomials orthogonal withrespect to the positive semidefinite 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 longstanding conjectured inequality for the confluent hypergeometric function F1(1), also known as Kummer's function, i.e., that F1(1)((1  a)/2, 1, t(2)) less than or equal to F1(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.
ASDD Area Sistemi Dipartimentali e Documentali, Università di Bologna, Catalogo delle riviste ed altri periodici
Documento generato il 22/01/21 alle ore 11:48:59