Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L ² (R ⁺,t dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character e^–i(2n++1); under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.     

On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

Approximation Properties of the Operators ⋎n+2r(f) and of Their Discrete Analogs

This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier--Legendre sums of order n with 2r terms of the form _k=1 ^2r a_kP_n+k(x) added; here P _m(x) denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval [-1,1], which, in fact, for r= = 1 allows us to significantly improve the approximation properties of partial Fourier--Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions and A _q (B). With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.     

On the Denseness of Rational Systems

This note characterizes the denseness of rational systems

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in C[−1, 1], where the nonreal poles in {a_k}^∞_k=1 \[−1, 1] are paired by complex conjugation. This extends an Achiezer's result.     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

Complex Zeros of Algebraic Polynomial with Non-Zero Mean Random Coefficients

In this paper we obtain a formula for the average density of the distribution of complex zeros of an algebraic polynomial with random coefficients. The coefficients are assumed independent identical normally distributed random variables with mean and variance ². The value of the average density for the case of =0 and ²=1 was obtained previously. Some limits of the distribution of the complex zeros are provided using the presented formula.     

Elements of provable high orders in finite fields

A method is given for constructing elements in whose orders are larger than any polynomial in when becomes large. As a by-product a theorem on multiplicative independence of compositions of polynomials is proved.

     

Some rapidly converging series for

For a natural number , the author derives several families of series representations for the Riemann Zeta function . Each of these series representing converges remarkably rapidly with its general term having the order estimate:

Relevant connections of the results presented here with many other known series representations for are also pointed out.