arXiv subjects

A. Pena

Publications and source records attributed to A. Pena.

9 recordsLinked to original sources

arXiv2010

A new approach to the asymptotics for Sobolev orthogonal polynomials

In this paper we deal with polynomials orthogonal with respect to an inner product involving derivatives, that is, a Sobolev inner product. Indeed, we consider Sobolev type polynomials which are orthogonal with respect to $$(f,g)=\int fg dμ+\sum_{i=0}^r M_i f^{(i)}(0) g^{(i)}(0), \quad M_i \ge 0,$$ where $μ$ is a certain probability measure with unbounded support. For these polynomials, we obtain the relative asymptotics with respect to orthogonal polynomials related to $μ$, Mehler--Heine type asymptotics and their consequences about the asymptotic behaviour of the zeros. To establish these results we use a new approach different from the methods used in the literature up to now. The development of this technique is highly motivated by the fact that the methods used when $μ$ is bounded do not work.

math.CA↗

arXiv2010

The Planck/LFI Radiometer Electronics Box Assembly

The Radiometer Electronics Box Assembly (REBA) is the control and data processing on board computer of the Low Frequency Instrument (LFI) of the Planck mission (ESA). The REBA was designed and built incorporating state of the art processors, communication interfaces and real time operating system software in order to meet the scientific performance of the LFI. We present a technical summary of the REBA, including a physical, functional, electrical, mechanical and thermal description. Aspects of the design and development, the assembly, the integration and the verification of the equipment are provided. A brief description of the LFI on board software is given including the Low-Level Software and the main functionalities and architecture of the Application Software. The compressor module, which has been developed as an independent product, later integrated in the application, is also described in this paper. Two identical engineering models EM and AVM, the engineering qualification model EQM, the flight model FM and flight spare have been manufactured and tested. Low-level and Application software have been developed. Verification activities demonstrated that the REBA hardware and software fulfil all the specifications and perform as required for flight operation.

astro-ph.IM↗

arXiv2009

Asymptotics for a generalization of Hermite polynomials

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler--Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.

math.CA↗

arXiv2009

Orthogonal polynomials associated with an inverse quadratic spectral transform

Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad n\ge1,$$ with $t_n \not= 0$ for $n\ge2$. We obtain a new characterization of the orthogonality of the sequence $\{Q_n \}_{n\ge0}$ with respect to a linear functional $v$, in terms of the coefficients of a quadratic polynomial $h$ such that $h(x)v= u$. We also study some cases in which the parameters $s_n$ and $t_n$ can be computed more easily, and give several examples. Finally, the interpretation of such a perturbation in terms of the Jacobi matrices associated with $\{P_n \}_{n\ge0}$ and $\{Q_n \}_{n\ge0}$ is presented.

math.CA↗

arXiv2009

Nonuniversal dynamic conductance fluctuations in disordered systems

Sample-to-sample fluctuations of the time-dependent conductance of a system with static disorder have been studied by means of diagrammatic theory and microwave pulsed transmission measurements. The fluctuations of time-dependent conductance are not universal, i.e., depend on sample parameters, in contrast to the universal conductance fluctuations in the steady-state regime. The variance of normalized conductance, determined by the infinite-range intensity correlation C_3(t), is found to increase as a third power of delay time from an exciting pulse, t. C_3(t) grows larger than the long-range intensity correlation C_2(t) after a time t_q ~ ^{1/2} t_D (t_D being the diffusion time, being the average dimensionless conductance).

cond-mat.mes-hall↗

arXiv2007

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given $\{P_n \}$ a sequence of monic orthogonal polynomials, we analyze their linear combinations $\{Q_n \}$with constant coefficients and fixed length $k+1$. Necessary and sufficient conditions are given for the orthogonality of the monic sequence $\{Q_n \}$ as well as an interesting interpretation in terms of the Jacobi matrices associated with $\{P_n \}$ and $\{Q_n \}$. Moreover, in the case $k=2$, we characterize the families $\{P_n \}$ such that the corresponding polynomials $\{Q_n \}$ are also orthogonal.

math.CA↗

arXiv2007

Sobolev orthogonal polynomials: balance and asymptotics

Let $μ_0$ and $μ_1$ be measures supported on an unbounded interval and $S_{n,λ_n}$ the extremal varying Sobolev polynomial which minimizes \begin{equation*} < P, P >_{λ_n}=\int P^2 dμ_0 + λ_n \int P'^2 dμ_1, \quad λ_n >0 \end{equation*} \noindent in the class of all monic polynomials of degree $n$. The goal of this paper is twofold. On one hand, we discuss how to balance both terms of this inner product, that is, how to choose a sequence $(λ_n)$ such that both measures $μ_0$ and $μ_1$ play a role in the asymptotics of $(S_{n, λ_n}).$ On the other, we apply such ideas to the case when both $μ_0$ and $μ_1$ are Freud weights. Asymptotics for the corresponding $S_{n, λ_n}$ are computed, illustrating the accuracy of the choice of $λ_n .$

math.CA↗

arXiv1995

An extension of Milman's reverse Brunn-Minkowski inequality

The classical Brunn-Minkowski inequality states that for $A_1,A_2\subset\R^n$ compact, $$ |A_1+A_2|^{1/n}\ge |A_1|^{1/n}+|A_2|^{1/n}\eqno(1) $$ where $|\cdot|$ denotes the Lebesgue measure on $\R^n$. In 1986 V. Milman {\bf [Mil 1]} discovered that if $B_1$ and $B_2$ are balls there is always a relative position of $B_1$ and $B_2$ for which a perturbed inverse of $(1)$ holds. More precisely:\lq\lq{\sl There exists a constant $C>0$ such that for all $n\in\N$ and any balls $B_1,B_2\subset\R^n$ we can find a linear transformation $u\colon\R^n\to\R^n$ with $|{\rm det}(u)|=1$ and $$|u(B_1)+B_2|^{1/n}\le C(|B_1|^{1/n}+|B_2|^{1/n})"$$} The aim of this paper is to extend this Milman's result to a larger class of sets.

math.FA↗

arXiv1992

The theorems of Caratheodory and Gluskin for $0<p<1$

In this note we investigate some aspects of the local structure of finite dimensional $p$-Banach spaces. The well known theorem of Gluskin gives a sharp lower bound of the diameter of the Minkowski compactum. In [Gl] it is proved that diam$({\cal M}_n^1)\geq cn$ for some absolute constant $c$. Our purpose is to study this problem in the $p$-convex setting. In [Pe], T. Peck gave an upper bound of the diameter of ${\cal M}_n^p$, the class of all $n$-dimensional $p$-normed spaces, namely, diam$({\cal M}_n^p)\leq n^{2/p-1}$. We will show that such bound is optimum.

math.FA↗