Perturbation of Orthogonal Polynomials on an Arc of the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the are {e^iθ : α ≤ θ ≤ 2 π − α) where cos(α/2) [formula] with α (0, π). We analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the are. In addition, we also prove the unit circle analogue of M. G. Krein′s characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.     

Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for

In the paper, we discuss Voronovskaya-type theorem and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0<q<1. We give explicit formulas of Voronovskaya-type for the q-Bernstein polynomials for 0<q<1. If , we show that the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only ifWe also prove that if f is convex on [0,1] or analytic on (-ε,1+ε) for some ε>0, then the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only if f is linear.     

A Two-Parameter Family of Orthogonal Polynomials with Respect to a Jacobi-Type Weight on the Unit Circle

A two-parameter family of polynomials is introduced by a recursion formula. The polynomials are orthogonal on the unit circle with respect to the weight ω_{α, β}(θ) = |(1 − z)^α(1 + z)^β|², α, β > − , z = e^iθ. Explicit representation, norm estimates, shift identities, and explicit connection to Jacobi polynomials on the real interval [−1, 1] is presented.     

Müntz-type theorem on the segments emerging from the origin

A necessary and sufficient condition is obtained for the linear span of a system of monomials {z^λ:λΛ} to be dense in the space of all continuous functions defined on the line segments emerging from the origin, where Λ is a set of nonnegative integers. The result is a generalization of the Müntz theorem to the segments emerging from the origin and an extension of the Mergelyan theorem to lacunary polynomials.     

On Sobolev Orthogonality for the Generalized Laguerre Polynomials

The orthogonality of the generalized Laguerre polynomials, {L^(α)_n(x)}_n0, is a well known fact when the parameterαis a real number but not a negative integer. In fact, for −1<α, they are orthogonal on the interval [0, +∞) with respect to the weight functionρ(x)=x^αe^−x, and forα<−1, but not an integer, they are orthogonal with respect to a non-positive definite linear functional. In this work we will show that, for every value of the real parameterα, the generalized Laguerre polynomials are orthogonal with respect to a non-diagonal Sobolev inner product, that is, an inner product involving derivatives.     

A note on a theorem of Berkes and Philipp for dependent sequences

We improve an almost sure invariance principle for f-mixing sequences of real random variables with finite (2 + δ)th moment (0 < δ 2) due to Berkes and Philipp (1979). The exponent of the slow logarithmic rate of mixing in that theorem is relaxed from 160/δ to (1 + ε)(1 + 2/δ) and the undetermined quantities a_N are replaced by Nσ² for some σ > 0.     

The Divergence of Lagrange Interpolation for |x| at Equidistant Nodes

In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

Some inequalities for algebraic polynomials with the Laguerre weight

There is a series of publications which have considered inequalities of Markov–Bernstein–Nikolskii type for algebraic polynomials with the Jacobi weight (see [N.K. Bari, A generalization of the Bernstein and Markov inequalities, Izv. Akad. Nauk SSSR Math. Ser. 18 (2) (1954) 159–176; B.D. Bojanov, An extension of the Markov inequality, J. Approx. Theory 35 (1982) 181–190; P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995; I.K. Daugavet, S.Z. Rafalson, Some inequalities of Markov–Nikolskii type for algebraic polynomials, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 1 (1972) 15–25; A. Guessab, G.V. Milovanovic, Weighted L²-analogues of Bernstein's inequality and classical orthogonal polynomials, J. Math. Anal. Appl. 182 (1994) 244–249; I.I. Ibragimov, Some inequalities for algebraic polynomials, in: V.I. Smirnov (Ed.), Fizmatgiz, 1961, Research on Modern Problems of Constructive Functions Theory; G.K. Lebed, Inequalities for polynomials and their derivatives, Dokl. Akad. Nauk SSSR 117 (4) (1957) 570–572; G.I. Natanson, To one theorem of Lozinski, Dokl. Akad. Nauk SSSR 117 (1) (1957) 32–35; M.K. Potapov, Some inequalities for polynomials and their derivatives, Vestnik Moskov. Univ. Ser. Mat. Mekh. 2 (1960); E. Schmidt, Über die nebst ihren Ableitungen orthogonalen Polynomsysteme und das zugehörige Extremum, Math. Ann. 119 (1944) 165–209; P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (25) (1960) 373–378]). In this paper we find an inequality of the same type for algebraic polynomials on (0,∞) with the Laguerre weight function e^-xx^α (α>-1).     

Ordinal machines and admissible recursion theory

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory.     

The rate of convergence of q-Bernstein polynomials for

In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {B_n,q(f)} for 0<q<1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions , and the orders do not depend on α, unlike the classical case.     

Classification of graphs by regularity

We give a classification of graphs by two parameters s and t such that a graph is regular iff t ≥ 2, edge-regular iff t ≥ 3, and distance regular of diameter δ iff s = δ, t ≥ 2δ − 2. We investigate the algebra of polynomials in the adjacency matrix and relate to every graph a family of orthogonal polynomials. This generalizes various results on distance regular graphs.     

L approximation with trigonometric n-nomials

We compare the degree of approximation to L²(−π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, − ∞ < k < ∞.     

Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

In this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Im z|<2π. Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain.     

Note on multidimensional empirical processes for φ-mixing random vectors

In 1974, Sen proved weak convergence of the empirical processes (in the J₁-topology on D^p[0, 1]) for a stationary φ-mixing sequence of stochastic p( 1)-vectors. In this note, we show that Sen's theorem on weak convergence of the multidimensional empirical process for a stationary φ-mixing sequence of stochastic vectors remains true under a less restrictive condition on the mixing constants {φ_n}, i.e., φ_n = O(n^−1−δ) for some δ > 0.     

Blumenthal''s Theorem for Laurent Orthogonal Polynomials

We investigate polynomials satisfying a three-term recurrence relation of the form B_n(x)=(x−β_n)B_n−1(x)−α_nxB_n−2(x), with positive recurrence coefficients α_n+1,β_n (n=1,2,…). We show that the zeros are eigenvalues of a structured Hessenberg matrix and give the left and right eigenvectors of this matrix, from which we deduce Laurent orthogonality and the Gaussian quadrature formula. We analyse in more detail the case where α_n→α and β_n→β and show that the zeros of B_n are dense on an interval and that the support of the Laurent orthogonality measure is equal to this interval and a set which is at most denumerable with accumulation points (if any) at the endpoints of the interval. This result is the Laurent version of Blumenthal's theorem for orthogonal polynomials.     

Liouville-type theorems for some complex hypoelliptic operators

It is shown that the natural analogue of Liouville's theorem holds for the well-known hypoelliptic operators _α (for ¦α¦ < n) introduced and studied by Folland and Stein on the Heisenberg group H_n. Since these operators are non-real for α ≠ 0, the usual methods of potential theory fail and are replaced by an explicit use of the fundamental solution.     

Random walks generated by area preserving maps with zero Lyapounov exponents

We study the asymptotic limit distributions of Birkhoff sums S_n of a sequence of random variables of dynamical systems with zero entropy and Lebesgue spectrum type. A dynamical system of this family is a skew product over a translation by an angle α. The sequence has long memory effects. It comes that when α/π is irrational the asymptotic behavior of the moments of the normalized sums S_n/f_n depends on the properties of the continuous fraction expansion of α. In particular, the moments of order k, , are finite and bounded with respect to n when α/π has bounded continuous fraction expansion. The consequences of these properties on the validity or not of the central limit theorem are discussed.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

Representations for the first associated q-classical orthogonal polynomials

Let {p_k(x; q)} be any system of the q-classical orthogonal polynomials, and let be the corresponding weight function, satisfying the q-difference equation D_q(σ)=τ, where σ and τ are polynomials of degree at most 2 and exactly 1, respectively. Further, let {p_k⁽¹⁾(x;q)} be associated polynomials of the polynomials {p_k(x; q)}. Explicit forms of the coefficients b_n,k and c_n,k in the expansions

are given in terms of basic hypergeometric functions. Here _k(x) equals x^k if σ⁺(0)=0, or (x;q)_k if σ⁺(1)=0, where σ⁺(x)σ(x)+(q−1)xτ(x). The most important representatives of those two classes are the families of little q-Jacobi and big q-Jacobi polynomials, respectively.Writing the second-order nonhomogeneous q-difference equation satisfied by p_n−1⁽¹⁾(x;q) in a special form, recurrence relations (in k) for b_n,k and c_n,k are obtained in terms of σ and τ.