Inequalities Concerning theLp-Norm of a Polynomial
โ Scribed by K.K. Dewan; Aijaz Ahmad Bhat; Mohammad Sayeed Pukhta
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 111 KB
- Volume
- 224
- Category
- Article
- ISSN
- 0022-247X
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
In this paper we give a new characterization of the classical orthogonal polynomials (Hermite, generalized Laguerre, Jacobi) by extremal properties in some weighted polynomial inequalities in \(L^{2}\)-norm. 1994 Academic Press. Inc.
## Abstract Let equation image where equation image In 1958, Vietoris proved that __ฯ~n~__ (__x__) is positive for all __n__ โฅ 1 and __x__ โ (0, __ฯ__). We establish the following refinement. The inequalities equation image hold for all natural numbers __n__ and real numbers __n__ โฅ 1 and __x
Let p n z be a polynomial of degree n and D ฮฑ p n z its polar derivative. It has been proved that if p n z has no zeros in z < 1, then for ฮด โฅ 1 and ฮฑ โฅ 1, 2ฯ 0 D ฮฑ p n e iฮธ ฮด dฮธ 1/ฮด โค n ฮฑ + 1 F ฮด 2ฯ 0 p n e iฮธ ฮด dฮธ 1/ฮด where F ฮด = 2ฯ/ 2ฯ 0 1 + e iฮธ ฮด dฮธ 1/ฮด . We also obtain analogous inequalities
Let C \* n , n=0, 1, ..., \*>&1ร2 be the ultraspherical (Gegenbauer) polynomials, orthogonal on (&1, 1) with respect to the weight (1&x 2 ) \*&1ร2 . Denote by `n, k (\*), k=1, ..., [nร2] the positive zeros of C \* n enumerated in decreasing order. The problem of finding the ``extremal'' function f f