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1.
A. Branquinho 《Journal of Mathematical Analysis and Applications》2009,356(1):242-256
In this paper we characterize sequences of orthogonal polynomials on the unit circle whose corresponding Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of second order matrix differential equations. In the semi-classical case, a characterization in terms of second order linear differential equations with polynomial coefficients is deduced. 相似文献
2.
Let there be given a probability measure μ on the unit circle of the complex plane and consider the inner product induced by μ. In this paper we consider the problem of orthogonalizing a sequence of monomials {zrk}k, for a certain order of the , by means of the Gram–Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {ψk}k. We show that the matrix representation with respect to {ψk}k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a ‘snake-shaped’ matrix factorization. Here the ‘snake shape’ of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {zrk}k are orthogonalized, while the ‘segments’ of the snake are canonically determined in terms of the Schur parameters for μ. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism. 相似文献
3.
Ruymán Cruz-Barroso Leyla Daruis Pablo González-Vera Olav Njåstad 《Journal of Computational and Applied Mathematics》2007
In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied. 相似文献
4.
Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved. 相似文献
5.
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 , σs is singular, andFor 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. 相似文献
6.
7.
Ricardo Abreu-Blaya Jean-Marie Vilaire 《Journal of Mathematical Analysis and Applications》2010,361(2):579-187
We are interested in finding solvability conditions for the Riemann boundary value problems for hyperanalytic functions in a simply connected bounded open subset of the complex plane whose boundary is merely required to be a d-summable closed curve. 相似文献
8.
In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V3, 3). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
9.
In this paper, we discuss some properties of the Cauchy type integral operator defined over the half space of . As applications, we study a type of Riemann boundary value problem for solutions to polynomially generalized Cauchy–Riemann equations including with and as special cases over the half space of . Making use of Fischer‐type decomposition and the Clifford calculus for solutions to these equations, we will obtain explicit expressions of solutions to the kind of boundary value problems over the half space of . Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
10.
Thomas Dehn 《Journal of Computational and Applied Mathematics》1994,50(1-3):207-219
We consider the asymptotic behavior of the ratios qn+1(z)/qn(z) of polynomials orthonormal with respect to some positive measure μ. Let the recurrence coefficients n and βn converge to 0 and
, respectively. Then, qn+1(z)/qn(z) Φ(z),for n→∞ locally uniformly for
, where Φ maps
conformally onto the exterior of the unit disc (Nevai (1979)). We provide a new and direct proof for this and some related results due to Nevai, and apply it to convergence acceleration of diagonal Padé approximants. 相似文献
11.
In this paper, we study the Rm (m > 0) Riemann boundary value problems for regular functions, harmonic functions and bi-harmonic functions with values in a universal clifford algebra C(Vn,n). By using Plemelj formula, we get the solutions of Rm (m > 0) Riemann boundary value problems for regular functions. Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions, we obtain the solutions of Rm (m > 0) Riemann boundary value problems for harmonic functions and bi-harmonic functions. 相似文献
12.
Fagui Liu 《Journal of Mathematical Analysis and Applications》2009,351(2):595-602
The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R2 in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given. 相似文献
13.
Rakhmanov's theorem for orthogonal polynomials on the unit circle gives a sufficient condition on the orthogonality measure for orthogonal polynomials on the unit circle, in order that the reflection coefficients (the recurrence coefficients in the Szegő recurrence relation) converge to zero. In this paper we give the analog for orthogonal matrix polynomials on the unit circle. 相似文献
14.
D. Barrios Rolanía G. Lpez Lagomasino E. B. Saff 《Journal of Approximation Theory》2003,124(2):263-281
Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szeg
functions. 相似文献
15.
Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval
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Xiaochen Li Xiping Liu Mei Jia Yan Li Sha Zhang 《Mathematical Methods in the Applied Sciences》2017,40(6):1892-1904
In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval with the integral boundary conditions By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
16.
A new numerical quadrature formula on the unit circle 总被引:1,自引:0,他引:1
In this paper we study a quadrature formula for Bernstein–Szegő measures on the unit circle with a fixed number of nodes and
unlimited exactness. Taking into account that the Bernstein–Szegő measures are very suitable for approximating an important
class of measures we also present a quadrature formula for this type of measures such that the error can be controlled with
a well-bounded formula.
This work was supported by Ministerio de Educación y Ciencia under grants number MTM2005-01320 (E. B. and A. C.) and MTM2006-13000-C03-02
(F. M.). 相似文献
17.
Andreas Lasarow 《Mathematische Nachrichten》2008,281(1):75-108
We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
18.
Timothy Nguyen 《Journal of Mathematical Analysis and Applications》2007,327(2):977-990
Consider ergodic orthogonal polynomials on the unit circle whose Verblunsky coefficients are given by αn(ω)=λV(Tnω), where T is an expanding map of the circle and V is a C1 function. Following the formalism of [Jean Bourgain, Wilhelm Schlag, Anderson localization for Schrödinger operators on Z with strongly mixing potentials, Comm. Math. Phys. 215 (2000) 143-175; Victor Chulaevsky, Thomas Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Comm. Math. Phys. 168 (1995) 455-466], we show that the Lyapunov exponent γ(z) obeys a nice asymptotic expression for λ>0 small and z∈∂D?{±1}. In particular, this yields sufficient conditions for the Lyapunov exponent to be positive. Moreover, we also prove large deviation estimates and Hölder continuity for the Lyapunov exponent. 相似文献
19.
Omar Belhamiti 《Communications in Nonlinear Science & Numerical Simulation》2012,17(2):555-565
In this paper, we propose a new approach based on conjunction of the orthogonal collocation on finite elements method with decoupling and quasi-linearization technique to approximate solutions of a set of nonlinear split boundary value problems. The numerical stability, the convergence and the accuracy of the results are checked by this algorithm. The approach developed in this study is illustrated by some numerical examples. These examples are solved using a special software package which implements the proposed algorithms. 相似文献
20.
We obtain a new upper estimate for the Green’s function associated with a higher order fractional boundary value problem. As an application of this result, criteria for the existence of positive solutions of the problem are then established. 相似文献