Linear Spectral Transformation Research Articles

Recovering orthogonality from quasi‐type kernel polynomials using specific spectral transformations

In this work, the concept of quasi‐type Kernel polynomials with respect to a moment functional is introduced. Difference equation satisfied by these polynomials along with the criterion for orthogonality conditions are discussed. The process of recovering orthogonality for the linear combination of a quasi‐type kernel polynomial with another orthogonal polynomial, which is identified by involving linear spectral transformation, is provided. This process involves an expression of ratio of iterated kernel polynomials. This leads to considering the limiting case of ratio of kernel polynomials involving continued fractions. Special cases of such ratios in terms of certain continued fractions are exhibited. Moreover, examples are discussed for obtaining the orthogonality of quasi‐type kernel polynomials of order 1.

Spectral transformations and second kind polynomials associated with a hermitian linear functional

The aim of this paper is to analyze the relation between the Christoffel transformations in the framework of hermitian linear functionals and the associated polynomials of the second kind through of their associated Carathéodory formal power series. In particular, some results are given in the way of characterization of linear spectral transformations using Christoffel and Geronimus transformations. This gives a new example of rational spectral transformation that has not yet been studied in the literature.

Multivariate Toda hierarchies and biorthogonal polynomials

A new multivariate Toda hierarchy of nonlinear partial differential equations adapted to multivariate biorthogonal polynomials is discussed. This integrable hierarchy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov–Shabat equations, KP type equations, appropriate reductions, Darboux or linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.

Zeros of para–orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para---orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle dµ and para---orthogonal polynomials associated with a modification of dµ by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.

Perturbations on the antidiagonals of Hankel matrices

Given a strongly regular Hankel matrix, and its associated sequence of moments which defines a quasi-definite moment linear functional, we study the perturbation of a fixed moment, i.e., a perturbation of one antidiagonal of the Hankel matrix. We define a linear functional whose action results in such a perturbation and establish necessary and sufficient conditions in order to preserve the quasi-definite character. A relation between the corresponding sequences of orthogonal polynomials is obtained, as well as the asymptotic behavior of their zeros. We also study the invariance of the Laguerre-Hahn class of linear functionals under such perturbation, and determine its relation with the so-called canonical linear spectral transformations.

A new linear spectral transformation associated with derivatives of Dirac linear functionals

In this contribution, we analyze the regularity conditions of a perturbation on a quasi-definite linear functional by the addition of Dirac delta functionals supported on N points of the unit circle or on its complement. We also deal with a new example of linear spectral transformation. We introduce a perturbation of a quasi-definite linear functional by the addition of the first derivative of the Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. Necessary and sufficient conditions for the quasi-definiteness of the new linear functional are obtained. Outer relative asymptotics for the new sequence of monic orthogonal polynomials in terms of the original ones are obtained. Finally, we prove that this linear spectral transform can be decomposed as an iteration of Christoffel and Geronimus linear transformations.

Linear Spectral Transformations and Laurent Polynomials

In this manuscript we analyze some linear spectral transformations of a Hermitian linear functional using the multiplication by some class of Laurent polynomials. We focus our attention in the behavior of the Verblunsky parameters of the perturbed linear functional. Some illustrative examples are pointed out.

A Closed-Form Solution of Linear Spectral Transformation for Robust Speech Recognition

The maximum likelihood linear spectral transformation (ML-LST) using a numerical iteration method has been previously proposed for robust speech recognition. The numerical iteration method is not appropriate for real-time applications due to its computational complexity. In order to reduce the computational cost, the objective function of the ML-LST is approximated and a closed-form solution is proposed in this paper. It is shown experimentally that the proposed closed-form solution for the ML-LST can provide rapid speaker and environment adaptation for robust speech recognition.

Verblunsky Parameters and Linear Spectral Transformations

In this paper we analyze the behavior of Verblunsky parameters for hermitian linear functionals deduced from canonical linear spectral transformations of a quasi-definite hermitian linear functional. Some illustrative examples are studied.

Rapid adaptation using linear spectral transformation for embedded speech recognisers

Embedded speech recognisers are typically used in unknown mobile environments where the acoustic conditions frequently change. Since a large amount of adaptation data is not usually available for such environments, the adaptation methods for the acoustic models of these recognisers must improve the recognition performance with only a small amount of adaptation data. In this Letter, we show that maximum likelihood linear spectral transformation provides the advantage of rapid adaptation using a very limited amount of adaptation data for the embedded acoustic models.

Linear Spectral Transformation for Robust Speech Recognition Using Maximum Mutual Information

This paper presents a transformation-based rapid adaptation technique for robust speech recognition using a linear spectral transformation (LST) and a maximum mutual information (MMI) criterion. Previously, a maximum likelihood linear spectral transformation (ML-LST) algorithm was proposed for fast adaptation in unknown environments. Since the MMI estimation method does not require evenly distributed training data and increases the a posteriori probability of the word sequences of the training data, we combine the linear spectral transformation method and the MMI estimation technique in order to achieve extremely rapid adaptation using only one word of adaptation data. The proposed algorithm, called MMI-LST, was implemented using the extended Baum-Welch algorithm and phonetic lattices, and evaluated on the TIMIT and FFMTIMIT corpora. It provides a relative reduction in the speech recognition error rate of 11.1% using only 0.25 s of adaptation data.

Spectral transformations, self-similar reductions and orthogonal polynomials

We study spectral transformations in the theory of orthogonal polynomials which are similar to Darboux transformations for the Schr?dinger equation. Linear transformations of the Stieltjes function with coefficients that are rational in the argument are constructed as iterations of the Christoffel and Geronimus transformations. We describe a characteristic property of semi-classical orthogonal polynomials (SCOP) on the uniform and the exponential lattice; namely, that all these polynomials can be obtained through simple quasi-periodic and q-periodic (self-similar) closures of the chain of linear spectral transformations. In the self-similar setting, a characterization of the Laguerre - Hahn polynomials on linear and q-linear lattices is obtained by considering rational transformations of the Stieltjes function generated by transitions to the associated polynomials.

On the properties of linear spectral transformations for 2-dimensional digital filters

A set of eight linear spectral transformations which can be used in the design of two-dimensional digital filters is studied from a group-theoretic point of view. Several properties of the transformations, some of them known and some of them new, are deduced and are then applied in the implementation of 2-D digital filters. It is shown that trade-offs exist which can be used to reduce either the amount of memory required for the programming or the amount of data manipulation.

A group of linear spectral transformation for two-dimensional digital filters

There are eight linear transformations of the spectral plane which map the frequency axes onto themselves. These transformations can be used to change the pass and stop regions of a two-dimensional digital filter. A stable realization is assured by transforming the data rather than the system transfer function.