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

Let p be a trigonometric polynomial, non-negative on the unit circle . We say that a measure σ on belongs to the polynomial Szegő class, if , σ_s is singular, and

For the associated orthogonal polynomials {_n}, we obtain pointwise asymptotics inside the unit disc . Then we show that these asymptotics hold in L²-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.     

Monotone Jacobi parameters and non-Szegő weights

We relate asymptotics of Jacobi parameters to asymptotics of the spectral weights near the edges. Typical of our results is that for a_n≡1, b_n=−Cn^−β (), one has on (−2,2), and near x=2, where

     

Asymptotic Properties of Zeros of Hypergeometric Polynomials

In a paper by K. Driver and P. Duren (1999, Numer. Algorithms21, 147–156) a theorem of Borwein and Chen was used to show that for each k the zeros of the hypergeometric polynomials F(−n, kn+1; kn+2; z) cluster on the loop of the lemniscate {z: |z^k(1−z)|=k^k/(k+1)^k+1}, with Re{z}>k/(k+1) as n→∞. We now supply a direct proof which generalizes this result to arbitrary k>0, while showing that every point of the curve is a cluster point of zeros. Examples generated by computer graphics suggest some finer asymptotic properties of the zeros.     

Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .     

Relative asymptotic of multiple orthogonal polynomials for Nikishin systems

We prove the relative asymptotic behavior for the ratio of two sequences of multiple orthogonal polynomials with respect to the Nikishin systems of measures. The first Nikishin system is such that for each k, σ_k has a constant sign on its compact support consisting of an interval , on which almost everywhere, and a discrete set without accumulation points in . If denotes the smallest interval containing , we assume that Δ_k∩Δ_k+1=0/, k=1,…,m−1. The second Nikishin system is a perturbation of the first by means of rational functions r_k, k=1,…,m, whose zeros and poles lie in .     

First and second kind paraorthogonal polynomials and their zeros

Given a probability measure μ with infinite support on the unit circle , we consider a sequence of paraorthogonal polynomials h_n(z,λ) vanishing at z=λ where is fixed. We prove that for any fixed z₀supp(dμ) distinct from λ, we can find an explicit ρ>0 independent of n such that either h_n or h_n+1 (or both) has no zero inside the disk B(z₀,ρ), with the possible exception of λ.Then we introduce paraorthogonal polynomials of the second kind, denoted s_n(z,λ). We prove three results concerning s_n and h_n. First, we prove that zeros of s_n and h_n interlace. Second, for z₀ an isolated point in supp(dμ), we find an explicit radius such that either s_n or s_n+1 (or both) have no zeros inside . Finally, we prove that for such z₀ we can find an explicit radius such that either h_n or h_n+1 (or both) has at most one zero inside the ball .     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Linear interpolation and Sobolev orthogonality

There is a strong connection between Sobolev orthogonality and Simultaneous Best Approximation and Interpolation. In particular, we consider very general interpolatory constraints , defined by

where f belongs to a certain Sobolev space, a_ij() are piecewise continuous functions over [a,b], b_ijk are real numbers, and the points t_k belong to [a,b] (the nonnegative integer m depends on each concrete interpolation scheme). For each f in this Sobolev space and for each integer l greater than or equal to the number of constraints considered, we compute the unique best approximation of f in , denoted by p_f, which fulfills the interpolatory data , and also the condition that best approximates f⁽ⁿ⁾ in (with respect to the norm induced by the continuous part of the original discrete–continuous bilinear form considered).     

The maximal range problem for a bounded domain

Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let

where . We relate the maximal polynomial range

to the geometry of Ω.     

The statistical distribution of the zeros of random paraorthogonal polynomials on the unit circle

The orthogonal polynomials on the unit circle are defined by the recurrence relation

where for any k0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ₀,Φ₁,…,Φ_n. The polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α₀,α₁,…,α_n-1.We take α₀,α₁,…,α_n-2 i.i.d. random variables distributed uniformly in a disk of radius r<1 and α_n-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φ_n(z)=Φ_n(z;α₀,…,α_n-2,α_n-1). The zeros of Φ_n are n random points on the unit circle.We prove that for any the distribution of the zeros of Φ_n in intervals of size near e^iθ is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials.     

Positive periodic solutions of periodic neutral Lotka–Volterra system with state dependent delays

By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays

where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively.     

An upper bound on Jacobi polynomials

Let be an orthonormal Jacobi polynomial of degree k. We will establish the following inequality:

where δ_-1<δ₁ are appropriate approximations to the extreme zeros of . As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [T. Erdélyi, A.P. Magnus, P. Nevai, Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994) 602–614] by proving that

in the region .     

Intersective polynomials and the polynomial Szemerédi theorem

Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p₁(n),…,a+p_r(n)}. We prove that a polynomial family P={p₁,…,p_r} has the PSZ property if and only if the polynomials p₁,…,p_r are jointly intersective, meaning that for any there exists such that the integers p₁(n),…,p_r(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nE_i such that {a,a+p₁(n),…,a+p_r(n)}E_i.     

Density results for Gabor systems associated with periodic subsets of the real line

The well-known density theorem for one-dimensional Gabor systems of the form , where , states that a necessary and sufficient condition for the existence of such a system whose linear span is dense in , or which forms a frame for , is that the density condition is satisfied. The main goal of this paper is to study the analogous problem for Gabor systems for which the window function g vanishes outside a periodic set which is -shift invariant. We obtain measure-theoretic conditions that are necessary and sufficient for the existence of a window g such that the linear span of the corresponding Gabor system is dense in L²(S). Moreover, we show that if this density condition holds, there exists, in fact, a measurable set with the property that the Gabor system associated with the same parameters a,b and the window g=χ_E, forms a tight frame for L²(S).     

Some two color, four variable Rado numbers

There exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromatic solution to x+y+kz=ℓw for , where N depends on k and ℓ. We determine N when ℓ−k{0,1,2,3,4,5}, for all k,ℓ for which , as well as for arbitrary k when ℓ=2.     

Periodic Schur functions and slit discs

A simply connected domain is called a slit disc if minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π. The conformal mapping of onto , (0)=0, ^′(0)>0, extends to a continuous function on mapping it onto . A finite union E of closed non-intersecting arcs e_k on is called rational if for every k, ν_E(e_k) being the harmonic measures of e_k at ∞ for the domain . A compact E is rational if and only if there is a rational slit disc such that . A compact E essentially supports a measure with periodic Verblunsky parameters if and only if for a rationally placed . For any tuple (α₁,…,α_g+1) of positive numbers with ∑_kα_k=1 there is a finite family of closed non-intersecting arcs e_k on such that ν_E(e_k)=α_k. For any set and any >0 there is a rationally placed compact such that the Lebesgue measure |EE^*| of the symmetric difference EE^* is smaller than .     

On the nonlinear wave equation with the mixed nonhomogeneous conditions: Linear approximation and asymptotic expansion of solutions

In this paper, we consider the following nonlinear wave equation

(1)

where , , μ, f, g are given functions. To problem (1), we associate a linear recursive scheme for which the existence of a local and unique weak solution is proved by applying the Faedo–Galerkin method and the weak compact method. In the case of , , μ(z)≥μ₀>0, μ₁(z)≥0, for all , and , , , a weak solution u_ε1,ε2(x,t) having an asymptotic expansion of order N+1 in two small parameters ε₁, ε₂ is established for the following equation associated to (1)_2,3:

(2)

     

Stability of the notion of approximating class of sequences and applications

Given an approximating class of sequences {{B_n,m}_n}_m for {A_n}_n, we prove that (X⁺ being the pseudo-inverse of Moore–Penrose) is an approximating class of sequences for , where {A_n}_n is a sparsely vanishing sequence of matrices A_n of size d_n with d_k>d_q for k>q,k,qN. As a consequence, we extend distributional spectral results on the algebra generated by Toeplitz sequences, by including the (pseudo) inversion operation, in the case where the sequences that are (pseudo) inverted are distributed as sparsely vanishing symbols. Applications to preconditioning and a potential use in image/signal restoration problems are presented.     

Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .