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Register a new userTuran inequalities and zeros of orthogonal polynomials
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Authors
Ilia Krasikov
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We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence ${c_k}$. As a special case they include a non-asymptotic version of Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=k^{delta} (1+ o(k^{-2/3})).$
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Let ${cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := {z in mathbb{C}: |z| leq 1, , , Im(z) geq 0}$$ be the closed upper half-disk of the complex plane. For integers $0 leq k leq n$ let ${mathcal F}_{n,k}^c$ be the set of all polynomials $P in {mathcal P}_n^c$ having at least $n-k$ zeros in $D^+$. Let $$|f|_A := sup_{z in A}{|f(z)|}$$ for complex-valued functions defined on $A subset {Bbb C}$. We prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that $$c_1 left(frac{n}{k+1}right)^{1/2} leq inf_{P}{frac{|P^{prime}|_{[-1,1]}}{|P|_{[-1,1]}}} leq c_2 left(frac{n}{k+1}right)^{1/2}$$ for all integers $0 leq k leq n$, where the infimum is taken for all $0 otequiv P in {mathcal F}_{n,k}^c$ having at least one zero in $[-1,1]$. This is an essentially sharp reverse Markov-type inequality for the classes ${mathcal F}_{n,k}^c$ extending earlier results of Turan and Komarov from the case $k=0$ to the cases $0 leq k leq n$.
We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family ${F_i }$. The most important example is a polynomial with $c=1.$ It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm.
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, $O(k^{-2/3}).$
This paper deals with monic orthogonal polynomial sequences (MOPS in short) generated by a Geronimus canonical spectral transformation of a positive Borel measure $mu$, i.e., begin{equation*} frac{1}{(x-c)}dmu (x)+Ndelta (x-c), end{equation*} for some free parameter $N in mathbb{R}_{+}$ and shift $c$. We analyze the behavior of the corresponding MOPS. In particular, we obtain such a behavior when the mass $N$ tends to infinity as well as we characterize the precise values of $N$ such the smallest (respectively, the largest) zero of these MOPS is located outside the support of the original measure $mu$. When $mu$ is semi-classical, we obtain the ladder operators and the second order linear differential equation satisfied by the Geronimus perturbed MOPS, and we also give an electrostatic interpretation of the zero distribution in terms of a logarithmic potential interaction under the action of an external field. We analyze such an equilibrium problem when the mass point of the perturbation $c$ is located outside of the support of $mu$.
Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ($q=1$) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont-Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials.
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