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1.
Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let where . We relate the maximal polynomial range to the geometry of Ω. 相似文献
2.
Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:where δ-1<δ1 are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving thatin the region . 相似文献
3.
Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σs is singular, andFor the associated orthogonal polynomials {n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators. 相似文献
4.
Let K be an arbitrary field of characteristic zero, Pn:=K[x1,…,xn] be a polynomial algebra, and , for n2. Let σ′AutK(Pn) be given by It is proved that the algebra of invariants, , is a polynomial algebra in n−1 variables which is generated by quadratic and cubic (free) generators that are given explicitly.Let σAutK(Pn) be given by It is well known that the algebra of invariants, , is finitely generated (theorem of Weitzenböck [R. Weitzenböck, Über die invarianten Gruppen, Acta Math. 58 (1932) 453–494]), has transcendence degree n−1, and that one can give an explicit transcendence basis in which the elements have degrees 1,2,3,…,n−1. However, it is an old open problem to find explicit generators for Fn. We find an explicit vector space basis for the quadratic invariants, and prove that the algebra of invariants is a polynomial algebra over in n−2 variables which is generated by quadratic and cubic (free) generators that are given explicitly.The coefficients of these quadratic and cubic invariants throw light on the ‘unpredictable combinatorics’ of invariants of affine automorphisms and of SL2-invariants. 相似文献
5.
Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L2 space. 相似文献
6.
Let , with
-1=x0n<x1n<<xnn<xn+1,n=1