Polynomial approximation with doubling weights having finitely many zeros and singularities
نویسنده
چکیده
We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and not too “rapidly changing” away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian–Totik weights. We approximate in the weighted Lp (quasi) norm ∥ f ∥p,w with 0 < p < ∞, where ∥ f ∥p,w := 1 −1 | f (u) | p w(u)du 1/p . Equivalence type results involving related realization functionals are also discussed. c ⃝ 2015 Elsevier Inc. All rights reserved. MSC: 41Axx
منابع مشابه
Zeros of exceptional Hermite polynomials
Exceptional orthogonal polynomials were introduced by Gomez-Ullate, Kamran and Milson as polynomial eigenfunctions of second order differential equations with the remarkable property that some degrees are missing, i.e., there is not a polynomial for every degree. However, they do constitute a complete orthogonal system with respect to a weight function that is typically a rational modification ...
متن کاملConsequences of a Factorization Theorem for Generalized Exponential Polynomials with Infinitely Many Integer Zeros
A factorization theorem is proved for a class of generalized exponential polynomials having all but finitely many of integer zeros belong to a finite union of arithmetic progressions. This theorem extends a similar result for ordinary exponential polynomials due to H. N. Shapiro in 1959. The factorization makes apparent those factors corresponding to all zeros in such a union.
متن کاملMulticomplexes and polynomials with real zeros
We show that each polynomial a(z)=1+a1z+· · ·+adzd inN[z] having only real zeros is the f-polynomial of a multicomplex. It follows that a(z) is also the h-polynomial of a Cohen–Macaulay ring and is the g-polynomial of a simplicial polytope.We conjecture that a(z) is also the f-polynomial of a simplicial complex and show that the multicomplex result implies this in the special case that the zero...
متن کاملOn the polar derivative of a polynomial
For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...
متن کاملOn the $s^{th}$ derivative of a polynomial
For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Journal of Approximation Theory
دوره 198 شماره
صفحات -
تاریخ انتشار 2015