Polynomials In L2 Research Articles

A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM.

It is known that if (p n ) n ∈ℕ is a sequence of orthogonal polynomials in L 2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution π -1(1 - x 2)-1 dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if f(x)(1 - x 2)1/4 ∈ L 2(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x 2)1/4 ∈ L2(-1, 1) and has mean value 0 on (-1, 1), then .

Formula omitted]-widths of certain function classes defined by the modulus of continuity

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various n-widths in L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson–Stechkin type over all subspaces of dimension N is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φ is a non-negative integrable weight function on (0,h](0<h≤π/n).

On the density of polynomials in some spaces L 2(M)

The question of the density of polynomials in some spaces L2(M) is studied. The following two variants of the measure M and the polynomials are considered: (1) an N × N matrix-valued nonnegative Borel measure on ℝ and vector-valued polynomials p(x) = (p0(x), p1(x), …, pN−1(x)), where the pj(x) are complex polynomials and N ∈ ℕ (2) a scalar nonnegative Borel measure on the strip Π = {(x, φ): x ∈ ℝ, ϕ ∈ [−π, π)}, and power-trigonometric polynomials $$p(x,\phi ) = \sum\limits_{m = 0}^\infty {\sum\limits_{n = - \infty }^\infty {\alpha _{m,n} x^m e^{in\phi } } } ,\alpha _{m,n} \in \mathbb{C}$$ , where only finitely many αm,n are nonzero. We show that the polynomials are dense in L2(M) if and only if M is the canonical solution of the corresponding moment problem. It should be stressed that we do not impose any additional constraints on the measure, except the existence of moments. Using the known descriptions of the canonical solutions,, we obtain conditions on the density of polynomials in L2(M). Simultaneously, we establish a model for commuting self-adjoint and unitary operators with spectrum of finite multiplicity.

On contractions with compact defects

In 2005, the following question was posed by Duggal, Djordjevic, and Kubrusly: Assume that T is a contraction of the class C 10 such that I − T * T is compact and the spectrum of T is the unit disk. Can the isometric asymptote of T be a reductive unitary operator? In this paper, we give a positive answer to this question. We construct two kinds of examples. One of them are the operators of multiplication by independent variable in the closure of analytic polynomials in L 2(ν),where ν is an appropriate positive finite Borel measure on the closed unit disk. The second kind of examples is based on a theorem by Chevreau, Exner, and Pearcy. We obtain a contraction T satisfying all the needed conditions and such that I − T * T belongs to the Schatten–von Neumann classes $$ {\mathfrak{S}_p} $$ for all p > 1. We give an example of a contraction T such that I − T * T belongs to $$ {\mathfrak{S}_p} $$ for all p > 1, T is quasisimilar to a unitary operator and has “more” invariant subspaces than this unitary operator. Also, following Bercovici and Kerchy, we show that if a subset of the unit circle is the spectrum of a contraction quasisimilar to a given absolutely continuous unitary operator, then this contraction T can be chosen so that I − T*T is compact. Bibliography: 29 titles.

Best Multi-degree Reduction of Bernstein Polynomial in L2-norm Based on an Explicit Termination Criterion

AbstractBased on the properties of orthogonal polynomials, we derive an explicit constrained degree reduction criterion for Bernstein-Bezier polynomials in L2 -norm. The criterion can be used to determine whether a further degree reduction can be applied to the polynomial in advance with a given tolerance e . An efficient algorithm is also presented for obtaining the best Bernstein-Bezier polynomial after degree reduction. With the proposed algorithm, one can avoid the blind procedure for degree reduction and terminate the procedure in advance when the estimated error is larger than the given tolerance.

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set P of all probability measures σ on the unit circle T splits into three disjoint subsets depending on properties of the derived set of {|ϕn|2dσ}n⩾0, denoted by Lim(σ). Here {ϕn}n⩾0 are orthogonal polynomials in L2(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ∈P with {m}=Lim(σ), m being the normalized (m(T)=1) Lebesgue measure on T. The second subset Mar(T) consists of Markoff measures, i.e., of σ∈P with m∉Lim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar(T) iff there are ε>0 and l>0 such that sup{|an+j|:0⩽j⩽l}>ε, n=0,1,2,…,{an} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ∈P with {m}⫋Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σ∈Mar(T)). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zn=λ, λ∈T, n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zn, n=1,2,…, of a closed arc J (including J=T) with removed open concentric arc J0 (including J0=∅). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).

Quasisimilar operators in the commutant of a cyclic subnormal operator

Compactly supported positive regular Borel measures on the complex plane that share "certain" properties with normalized arclength measure on the boundary of the unit disk are called m m -measures. Let μ \mu be an m m -measure and let S μ {S_\mu } be the cyclic subnormal operator of multiplication by z z on the closure of the polynomials in L 2 ( μ ) {L^2}(\mu ) . Necessary and sufficient conditions for an operator in the commutant of S μ {S_\mu } to be quasisimilar to S μ {S_\mu } are investigated. In particular it is shown that if the Bergman shift and an operator in its commutant are quasisimilar, then they are unitarily equivalent.

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L2 (0, 2π) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality En−1(f)L2<(C 2m m ωm(2π/n;f)L2(1≠ const) is proved, where the constant (C 2m m )−1/2 is unimprovable for the whole space L2 (0, 2π). Two titles are cited in the bibliography.