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We prove that if a multiplier sequence for the Legendre basis can be interpolated by a polynomial, then the polynomial must have the form {h(k2+k)}k=0∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidema...
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Citations
... This paper characterizes a continuation of several works, including [1], [2], [4], [5], [6], [8], and [14]. These works analyze the multiplier sequences of Jacobi polynomials and/or special cases of these, such as the Legendre polynomials. ...
... [5] studies complex zero decreasing operators and demonstrates the existence of a class of multiplier sequences for the Chebyshev basis. [6] characterizes polynomial multiplier sequences as having the form {h(k 2 + k)} ∞ k=0 for the Legendre basis, and begins work on the extension of this research to other bases and multiplier sequences. [8] proves the nonexistence of multiplier sequences for the Legendre basis of cubic form, additionally developing a partial-methodology that may be applied for higher orders of multiplier sequences, to which we refer in deriving our own general proof. ...
... The Symbol. In this subsection, we apply an argument similar to that in section 3 ("Form and Order") of [6], which we outline here for the convenience of the reader. For a general linear operatorK, the symbol ofK is given by ...
We demonstrate that a multiplier sequence for the Chebyshev basis cannot be interpolated by a polynomial of odd degree and that any such interpolating polynomial must be an even function. We exhibit this by direct algebraic proof and by utilizing various computational methods for both preliminary proofs and for the symbolic interpretation of required sequences. Additionally, we provide a foundation for continued research on even degree multiplier sequences for the Chebyshev basis and extensions of this work to the Gegenbauer and Jacobi basis sets.
