arXiv subjects
Publications and source records attributed to A. Peña.
We study the inverse problem in the theory of (standard) orthogonal polynomials involving two polynomials families $(P_n)_n$ and $(Q_n)_n$ which are connected by a linear algebraic structure such as $$P_n(x)+\sum_{i=1}^N r_{i,n}P_{n-i}(x)=Q_n(x)+\sum_{i=1}^M s_{i,n}Q_{n-i}(x)$$ for all $n=0,1, \dots$ where $N$ and $M$ are arbitrary nonnegative integer numbers.
In this paper the discrete Sobolev inner product $$< p,q > =\int p(x) q(x) \,dμ+ \sum_{i=0}^r M_i \, p^{(i)}(c) \, q^{(i)}(c)$$ is considered, where $μ$ is a finite positive Borel measure supported on an infinite subset of the real line, $c\in\mathbb{R}$ and $\, M_i \ge 0, \, i = 0, 1, ..., r.$ Connection formulas for the orthonormal polynomials associated with $< ., . >$ are obtained. As a consequence, for a wide class of measures $μ$, we give the Mehler-Heine asymptotics in the case of the point $c$ is a hard edge of the support of $μ$. In particular, the case of a symmetric measure $μ$ is analyzed. Finally, some examples are presented.
The propagation of light through samples with random inhomogeneities can be described by way of transmission eigenchannels, which connect incoming and outgoing external propagating modes. Although the detailed structure of a disordered sample can generally not be fully specified, these transmission eigenchannels can nonetheless be successfully controlled and utilized for focusing and imaging light through random media. Here we demonstrate that in deeply localized quasi-1D systems, the single dominant transmission eigenchannel is formed by an individual Anderson localized mode or by a "necklace state". In this single-channel regime, the disordered sample can be treated as an effective 1D system with a renormalized localization length, coupled through all the external modes to its surroundings. Using statistical criteria of the single-channel regime and pulsed excitations of the disordered samples allows us to identify long-lived localized modes and short-lived necklace states at long and short time delays, respectively.
Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$ are constant matrices of proper size and $\mathbb{Q}_0 = \mathbb{P}_0$. The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.
Let $(P_n)_n$ and $(Q_n)_n$ be two sequences of monic polynomials linked by a type structure relation such as $$ Q_{n}(x)+r_nQ_{n-1}(x)=P_{n}(x)+s_nP_{n-1}(x)+t_nP_{n-2}(x)\;, $$ where $(r_n)_n$, $(s_n)_n$ and $(t_n)_n$ are sequences of complex numbers. First, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences $(P_n)_n$ and $(Q_n)_n$ are orthogonal with respect to regular moment linear functionals ${\bf u}$ and ${\bf v}$, respectively. Second, assuming that the above relation is non-degenerate and $(P_n)_n$ is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence $(Q_n)_n$ in terms of the coefficients of the polynomials $Φ$ and $Ψ$ which appear in the rational transformation (in the distributional sense) $Φ{\bf u}=Ψ{\bf v}\; .$ Some illustrative examples of the developed theory are presented.
C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted $L^{p}$ spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre--Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B. Xh. Fejzullahu and F. Marcellán.