共查询到20条相似文献,搜索用时 15 毫秒
1.
María Álvarez de Morales Juan J. Moreno–Balcázar Teresa E. Pérez Miguel A. Piñar 《Acta Appl Math》2000,61(1-3):257-266
In this work, we study algebraic and analytic properties for the polynomials { Q
n
}
n 0, which are orthogonal with respect to the inner product
where , R such that – 2 > 0. 相似文献
2.
Aleksandar Pejčev 《Numerical Algorithms》2014,66(3):569-590
The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein?Szeg? weights, $${\int\limits_{-1}^{1}}f(t)w(t)\, dt=G_{n}[f]+R_{n}(f),\quad G_{n}[f]=\sum\limits_{\nu=1}^{n}\lambda_{\nu} f(\tau_{\nu}) \quad(n\in\textbf{N}),$$ where f is an analytic function inside an elliptical contour \(\mathcal{E}_{\rho}\) with foci at \(\mp 1\) and sum of semi-axes \(\rho > 1\) , and w is a nonnegative and integrable weight function of Bernstein?Szeg? type. The derivation of effective bounds on \(|R_{n}(f)|\) is possible if good estimates of \(\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|\) are available, especially if one knows the location of the extremal point \(\eta\in\mathcal{E}_{\rho}\) at which \(|K_{n}|\) attains its maximum. In such a case, instead of looking for upper bounds on \(\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|\) , one can simply try to calculate \(|K_{n}(\eta,w)|\) . In the case under consideration, i.e. when $$w(t)= \frac{(1-t^{2})^{-1/2}}{\beta(\beta-2\alpha)\,t^{2} +2\delta(\beta-\alpha)\,t+\alpha^{2}+\delta^{2}},\quad t\in(-1,1),$$ for some \(\alpha,\beta,\delta\) , which satisfy \(0<\alpha<\beta,\ \beta\ne 2\alpha,\vert\delta\vert<\beta-\alpha\) , the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on \(|R_{n}(f)|\) . The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands. 相似文献
3.
Jacob S. Christiansen 《Advances in Mathematics》2012,229(2):1180-1204
In this paper, we generalize Szeg?'s theorem for orthogonal polynomials on the real line to infinite gap sets of Parreau–Widom type. This notion includes Cantor sets of positive measure. The Szeg? condition involves the equilibrium measure which in turn is absolutely continuous. Our approach builds on a canonical factorization of the M-function and the covering space formalism of Sodin–Yuditskii. 相似文献
4.
Szegő polynomials are orthogonal with respect to an inner product on the unit circle. Numerical methods for weighted least-squares
approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of Szegő polynomials. This
paper discusses the conditioning of several mappings involving Szegő polynomials and, thereby, sheds light on the sensitivity
of some approximation problems involving trigonometric polynomials.
This Research supported in part by NSF grant DMS-0107858. 相似文献
5.
6.
Sorin G. Gal 《Mediterranean Journal of Mathematics》2008,5(3):253-272
In this paper, firstly we prove the Voronovskaja’s convergence theorem for complex Bernstein polynomials in compact disks
in , centered at origin, with quantitative estimates of this convergence. Secondly, we study the approximation properties of
the iterates of complex Bernstein polynomials and we prove that they preserve in the unit disk (beginning with an index) the
univalence, starlikeness, convexity and spirallikeness.
Received: May 5, 2007 Revised: September 14, 2007 and November 11, 2007 Accepted: November 26, 2007 相似文献
7.
Arnaldo Nogueira 《Journal d'Analyse Mathématique》2001,85(1):1-41
A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on
the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm;
then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms
such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard
continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the
multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms.
Supported by a grant from the CNP
q
-Brazil, 301456/80, and FINEP/CNP
q
/MCT 41.96.0923.00 (PRONEX). 相似文献
8.
Milivoje Lukic 《Constructive Approximation》2013,38(1):161-169
We disprove a conjecture of Simon for higher-order Szeg? theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture. 相似文献
9.
We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition
is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions
for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms
of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant.
In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions. 相似文献
10.
Maxim Derevyagin Olga Holtz Sergey Khrushchev Mikhail Tyaglov 《Journal of Approximation Theory》2012,164(9):1238-1261
We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg?’s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager. 相似文献
11.
According to the Erd?s–Szekeres theorem, for every n, a sufficiently large set of points in general position in the plane contains n in convex position. In this note we investigate the line version of this result, that is, we want to find n lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains n in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd?s–Szekeres theorem. 相似文献
12.
Szeg?’s First Limit Theorem provides the limiting statistical distribution of the eigenvalues of large Toeplitz matrices. Szeg?’s Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large Toeplitz matrices. In this paper we survey results extending the First and Second Limit Theorems to Kac–Murdock–Szeg? (KMS) matrices. These are matrices whose entries along the diagonals are not necessarily constants, but modeled by functions. We clarify and extend some existing results, and explain some apparently contradictory results in the literature. 相似文献
13.
In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8]. 相似文献
14.
In this paper, we prove a sharp Bernstein-type inequality for algebraic polynomials on circular arcs. 相似文献
15.
U. Fidalgo Prieto A. López García G. López Lagomasino V. N. Sorokin 《Constructive Approximation》2010,32(2):255-306
We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality
conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and
type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal
solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described
by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane. 相似文献
16.
17.
In this paper, we give an explicit formula for the Szegö kernel for (0, q) forms on the Heisenberg group Hn+1. 相似文献
18.
19.
A quantitative version of Pólya–Szeg? inequality is proven for log-concave functions in the case of Steiner and Schwarz rearrangements. 相似文献
20.
Albrecht Böttcher 《Integral Equations and Operator Theory》2014,78(3):407-414
We consider the determinants of compressions of Toeplitz operators to finite-dimensional model spaces and establish analogues of the Borodin–Okounkov formula and the strong Szeg? limit theorem in this setting. 相似文献