On Some Problems of P. Turán Concerning Lm Extremal Polynomials and Quadrature Formulas
✍ Scribed by Ying Guang Shi
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 139 KB
- Volume
- 100
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
The L m extremal polynomials in an explicit form with respect to the weights (1&x) &1Â2 (1+x) (m&1)Â2 and (1&x) (m&1)Â2 (1+x) &1Â2 for even m are given. Also, an explicit representation for the Cotes numbers of the corresponding Tura n quadrature formulas and their asymptotic behavior is provided. 1999 Academic Press 1. INTRODUCTION AND MAIN RESULTS Throughout this paper let m be an even integer, w a weight (function) on [&1, 1], and P N the set of polynomials of degree N. In what follows we denote by c, c 1 , ... the positive constants independent of variables and indices, unless otherwise indicated; their value may be different at different occurrences, even in subsequent formulas. Let w :, ; (x) :=(1&x) : (1+x) ; , :, ;> &1, and v(x) :=w (&1Â2, &1Â2) (x),
{ u m (x) :=w ((m&1)Â2, (m&1)Â2) (x), (1.1) v m (x) :=w (&1Â2, (m&1)Â2) (x), w m (x) :=w ((m&1)Â2, &1Â2) (x).