Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

Inequalities for derivative of polynomial approximation with Erdos type weight

H. N. Mhaskar obtained the inequalities for derivative of polynomial approximation with Freud type weight. In this paper, we showed the counterpart of the inequalities with Erdos type weight.

The de la Vallee Poussin mean and polynomial approximation for exponential weight

We study the de la Vallee Poussin mean for exponential weights and give a polynomial approximation on real line. H. N. Mhasker proved the corresponding result for Freud-type weights. Our proof is valid for Erdos-type weights.

Continuity of weighted operators, Muckenhoupt $A_p$ weights, and Steklov problem for orthogonal polynomials

We consider weighted operators acting on $L^p(\mathbb{R}^d)$ and show that they depend continuously on the weight $w\in A_p(\mathbb{R}^d)$ in the operator topology. Then, we use this result to estimate $L^p_w(\mathbb{T})$ norm of polynomials orthogonal on the unit circle when the weight $w$ belongs to Muckenhoupt class $A_2(\mathbb{T})$ and $p>2$. The asymptotics of the polynomial entropy is obtained as an application.

Polynomial Sequences Associated with the Moments of Hypergeometric Weights

We present some families of polynomials related to the moments of weight functions of hypergeometric type. We also consider different types of generating functions, and give several examples.

Orthogonal polynomials for the weight $x^ν \exp(-x - t/x)$

Orthogonal polynomials for the weight $x^ν \exp(-x - t/x),\ x, t > 0, ν\in \mathbb{R}$ are investigated. Differential-difference equations, recurrence relations, explicit representations, generating functions and Rodrigues-type formula are obtained.

Orthogonal polynomials on the unit circle, $q$-Gamma weights, and discrete Painlevé equations

We consider orthogonal polynomials on the unit circle with respect to a weight which is a quotient of $q$-gamma functions. We show that the Verblunsky coefficients of these polynomials satisfy discrete Painlevé equations, in a Lax form, which correspond to an $A_3^{(1)}$ surface in Sakai's classification.

The growth of polynomials orthogonal on the unit circle with respect to a weight w that satisfies $w$,$1/w \in L^\infty(\mathbb{T})$

We consider the weight w: 1<w<T on the unit circle and prove that the corresponding orthonormal polynomials can grow.

Uniform polynomial approximation with $A^*$ weights having finitely many zeros

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly changing" away from these zeros. This class of weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. Main part and complete weighted moduli of smoothness are introduced, their properties are investigated, and equivalence type results involving related realization functiona…

Asymptotics of the orthogonal polynomials for the Szego class with a polynomial weight

Let p(t) be a trigonometric polynomial, non-negative on the unit circle. We say that a measure σbelongs to a polynomial Szego class, if the logarithm of its density is summable over the circle with the weight p(t). For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that these asymptotics holds in L^2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.

Approximation by polynomials in a weighted space of infinitely differentiable functions

The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result for description of strong dual for weighted spaces of infinitely differentiable functions on real line and weighted spaces of sequences of infinitely differentiable functions on real line in terms of the Fourier-Laplace transform of functionals.

On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq \infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is well known and corresponds to weighted Chebyshev polynomials; the cas…

四次Freud权下的斜正交多项式：两类准正交多项式

Skew-orthogonal polynomials for a quartic Freud weight: two classes of quasi-orthogonal polynomials

研究四次Freud权下斜正交多项式的评估方法，提出通过递推关系求解系数，并发现偶次和奇次多项式分别构成两种不同半经典拉盖尔权下的准正交多项式。