Orthogonal polynomials on the unit circle and chain sequences

Szego{double acute} has shown that real orthogonal polynomials on the unit circle can be mapped to orthogonal polynomials on the interval [-1,1] by the transformation 2x=z+z-1. In the 80's and 90's Delsarte and Genin showed that real orthogonal polynomials on the unit circle can be mapped to symmetric orthogonal polynomials on the interval [-1,1] using the transformation 2x=z1/2+z-1/2. We extend the results of Delsarte and Genin to all orthogonal polynomials on the unit circle. The transformation maps to functions on [-1,1] that can be seen as extensions of symmetric orthogonal polynomials on [-1,1] satisfying a three-term recurrence formula with real coefficients {cn} and {dn}, where {dn} is also a positive chain sequence. Via the results established, we obtain a characterization for a point w(|w|=1) to be a pure point of the measure involved. We also give a characterization for orthogonal polynomials on the unit circle in terms of the two sequences {cn} and {dn}. © 2013 Elsevier Inc.

Interlacing of zeros of orthogonal polynomials under modification of the measure

We investigate the mutual location of the zeros of two families of orthogonal polynomials. One of the families is orthogonal with respect to the measure dμ (x), supported on the interval (a, b) and the other with respect to the measure |x -c|τ|x -d|γdμ (x), where c and d are outside (a, b) We prove that the zeros of these polynomials, if they are of equal or consecutive degrees, interlace when either 0 < τ, γ ≤ 1 or γ = 0 and 0 < τ ≤ 2. This result is inspired by an open question of Richard Askey and it generalizes recent results on some families of orthogonal polynomials. Moreover, we obtain further statements on interlacing of zeros of specific orthogonal polynomials, such as the Askey-Wilson ones. © 2013 Elsevier Inc.

Zeros of classical orthogonal polynomials of a discrete variable

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

A Favard type theorem for orthogonal polynomials on the unit circle from a three term recurrence formula

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Stieltjes functions and discrete classical orthogonal polynomials

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Random regression models to estimate genetic parameters for milk production of Guzerat cows using orthogonal Legendre polynomials

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Sieved para-orthogonal polynomials on the unit circle

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Model updating of the non-linear vibrating structures through Volterra series and proper orthogonal decomposition

Classical procedures for model updating in non-linear mechanical systems based on vibration data can fail because the common linear metrics are not sensitive for non-linear behavior caused by gaps, backlash, bolts, joints, materials, etc. Several strategies were proposed in the literature in order to allow a correct representative model of non-linear structures. The present paper evaluates the performance of two approaches based on different objective functions. The first one is a time domain methodology based on the proper orthogonal decomposition constructed from the output time histories. The second approach uses objective functions with multiples convolutions described by the first and second order discrete-time Volterra kernels. In order to discuss the results, a benchmark of a clamped-clamped beam with an pre-applied static load is simulated and updated using proper orthogonal decomposition and Volterra Series. The comparisons and discussions of the results show the practical applicability and drawbacks of both approaches.

Para-orthogonal polynomials from constant Verblunsky coefficients

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Computational tools for comparing asymmetric GARCH models via Bayes factors

In this paper we use Markov chain Monte Carlo (MCMC) methods in order to estimate and compare GARCH models from a Bayesian perspective. We allow for possibly heavy tailed and asymmetric distributions in the error term. We use a general method proposed in the literature to introduce skewness into a continuous unimodal and symmetric distribution. For each model we compute an approximation to the marginal likelihood, based on the MCMC output. From these approximations we compute Bayes factors and posterior model probabilities. (C) 2012 IMACS. Published by Elsevier B.V. All rights reserved.

Reconstruction of spatiotemporal capture data by means of orthogonal functions: the case of skipjack tuna (Katsuwonus pelamis) in the central-east Atlantic

[EN] The information provided by the International Commission for the Conservation of Atlantic Tunas (ICCAT) on captures of skipjack tuna (Katsuwonus pelamis) in the central-east Atlantic has a number of limitations, such as gaps in the statistics for certain fleets and the level of spatiotemporal detail at which catches are reported. As a result, the quality of these data and their effectiveness for providing management advice is limited. In order to reconstruct missing spatiotemporal data of catches, the present study uses Data INterpolating Empirical Orthogonal Functions (DINEOF), a technique for missing data reconstruction, applied here for the first time to fisheries data. DINEOF is based on an Empirical Orthogonal Functions decomposition performed with a Lanczos method. DINEOF was tested with different amounts of missing data, intentionally removing values from 3.4% to 95.2% of data loss, and then compared with the same data set with no missing data. These validation analyses show that DINEOF is a reliable methodological approach of data reconstruction for the purposes of fishery management advice, even when the amount of missing data is very high.

Normalized Frobenius condition number of the orthogonal projections of the identity

[EN]This paper deals with the orthogonal projection (in the Frobenius sense) AN of the identity matrix I onto the matrix subspace AS (A ? Rn×n, S being an arbitrary subspace of Rn×n). Lower and upper bounds on the normalized Frobenius condition number of matrix AN are given. Furthermore, for every matrix subspace S ? Rn×n, a new index bF (A, S), which generalizes the normalized Frobenius condition number of matrix A, is defined and analyzed...

Geometrical and Spectral Properties of the Orthogonal Projections of the Identity

[EN]We analyze the best approximation

Chemische und enzymatische Synthese von orthogonal stabil geschützten Kohlenhydrat-Scaffolds für kombinatorische Anwendungen

Auf der Suche nach potenten pharmakologischen Wirkstoffen hat die Kombinatorische Chemie in der letzten Dekade eine große Bedeutung erlangt, um innerhalb kurzer Zeit ein breites Spektrum von Verbindungen für biologische Tests zu erzeugen. Kohlenhydrate stellen als Scaffolds interessante Kandidaten für die kombinatorische Synthese dar, da sie mehrere Derivatisierungspositionen in stereochemisch definierter Art und Weise zur Verfügung stellen. So ist die räumlich eindeutige Präsentation angebundener pharmakophorer Gruppen möglich, wie es für den Einsatz als Peptidmimetika wünschenswert ist. Zur gezielten Derivatisierung einzelner Hydroxylfunktionen ist der Einsatz eines orthogonalen Schutz-gruppenmusters erforderlich, das gegenüber den im Lauf der kombinatorischen Synthese herrschenden Reaktionsbedingungen hinreichend stabil ist. Weiterhin ist ein geeignetes Ankersystem zu finden, um eine Festphasensynthese und damit eine Automatisierung zu ermöglichen.Zur Minimierung der im Fall von Hexosen wie Galactose benötigten fünf zueinander orthogonal stabilen Schutzgruppen wurde bei der vorliegenden Arbeit von Galactal ausgegangen, bei dem nur noch drei Hydroxylfunktionen zu differenzieren sind. Das Galactose-Gerüst kann anschließend wiederhergestellt werden. Die Differenzierung wurde über den Einsatz von Hydrolasen durch regioselektive Acylierungs- und Deacylierungs-reaktionen erreicht, wobei auch immobilisierte Enzyme Verwendung fanden. Dabei konnte ein orthogonales Schutzgruppenmuster sequentiell aufgebaut werden, das auch die nötigen Stabilitäten gegenüber sonstigen, teilweise geeignet modifizierten Reaktionsbedingungen aufweist. Für die Anbindung an eine Festphase wurde ein metathetisch spaltbarer Anker entwickelt, der über die anomere Position unter Wiederherstellung des Galactose-Gerüsts angebunden wurde. Auch ein oxidativ spaltbares und ein photolabiles Ankersystem wurden erprobt.

Orthogonal gamma-based expansions for volatility option prices under jump-diffusion dynamics

In my work I derive closed-form pricing formulas for volatility based options by suitably approximating the volatility process risk-neutral density function. I exploit and adapt the idea, which stands behind popular techniques already employed in the context of equity options such as Edgeworth and Gram-Charlier expansions, of approximating the underlying process as a sum of some particular polynomials weighted by a kernel, which is typically a Gaussian distribution. I propose instead a Gamma kernel to adapt the methodology to the context of volatility options. VIX vanilla options closed-form pricing formulas are derived and their accuracy is tested for the Heston model (1993) as well as for the jump-diffusion SVJJ model proposed by Duffie et al. (2000).