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Register a new userWeighted Sobolev orthogonal polynomials on the unit ball
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For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.
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In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product begin{equation*} leftlangle p,qrightrangle _{s}=int_{mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{prime }(0)q^{prime }(0), end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.
In this contribution we consider the sequence ${Q_{n}^{lambda}}_{ngeq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences begin{equation*} langle p,qrangle _{lambda}=int_{0}^{infty}pleft(xright) qleft(xright) dpsi ^{(a)}(x)+lambda ,Delta p(c)Delta q(c), end{equation*} where $lambda in mathbb{R}_{+}$, $Delta $ denotes the forward difference operator defined by $Delta fleft(xright) =fleft(x+1right) -fleft(xright) $, $psi ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% begin{equation*} dpsi ^{(a)}(x)=frac{e^{-a}a^{x}}{x!}quad text{at}x=0,1,2,ldots, end{equation*}% and $cin mathbb{R}$ is such that $psi ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two differe
Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in $mathbb{C}^n$. The estimates are in terms of the Bekolle-Bonami constant of the weight.
The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.
Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 <A, x_k >B,$ which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$
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