Christoffel-Darboux Formula Research Articles

Universality in Gaussian random normal matrices

We prove that for Gaussian random normal matrices the correlation function has universal behavior. Using the technique of orthogonal polynomials and identities similar to the Christoffel-Darboux formula, we find that in the limit, as the dimension of the matrix tends to infinity, the density of eigenvalues converges to a constant inside of an ellipse and to zero outside. The convergence holds locally uniformly. At the boundary, in scaled coordinates holding the distance between eigenvalues constant, we show that the density is falling off to zero like the complementary error function. The convergence is uniform on the boundary. Further we give an explicit expression for the the correlation function.

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.

An Analytical Method for the Multiplierless 2-D FIR Filter Functions and Hilbert Transform in $z2$ Domain

A novel analytical method for the new class of linear-phase multiplierless 2-D finite-impulse response (FIR) digital filter functions generated by applying the Christoffel-Darboux formula for classical Chebyshev polynomials of the first and the second kind, in a compact explicit representative form, is proposed in this brief. Correct transformation from a continuous 2-D domain into the z domains without residuum and without errors is described. The proposed solution with high selectivity is a filter function in the z1 domain and the Hilbert transformer in the z2 domain. A corresponding example of the new class of linear-phase highly selective 2-D FIR digital filters of low complexity, which is obtained by the proposed approximation technique, is shown in this brief.

Generalised Christoffel–Darboux formula most directly applied in generating fully symmetric doubly resistively terminated LC lossless ladder filters

In this article, the approximation problem of a continuous analogue filter function of even and odd order is solved mathematically most directly applying the proposed generalised Christoffel–Darboux formula for two continuous orthogonal polynomials (Chebyshev first and second kind) on the equal finite segment of orthogonality in a compact explicit representative form. A set of analytic expressions of the proposed formula for the representative examples of odd-orders is given. Additionally, these expressions are applied in generating new excellent class all pole low-pass prototype analogue filter functions of odd orders (7, 9, 11, and 13th order). Based on the generated functions, the fully symmetric doubly resistively terminated LC ladder filter networks are realised by Darlington realisation. The generated filters are analysed, and their characteristics are presented. The effects of finite tolerance of a critical reactance of these ladder filters on the filter characteristics are considered. The filters generated by the proposed formula show excellent properties and possess important advantages in comparison to the corresponding filters generated by other methods.

Ladder operators and differential equations for multiple orthogonal polynomials

In this paper, we obtain the ladder operators and associated compatibility conditions for types I and II multiple orthogonal polynomials. These ladder equations extend known results for orthogonal polynomials and can be used to derive the differential equations satisfied by multiple orthogonal polynomials. Our approach is based on Riemann–Hilbert problems and the Christoffel–Darboux formula for multiple orthogonal polynomials, and the nearest-neighbor recurrence relations. As an illustration, we give several explicit examples involving multiple Hermite and Laguerre polynomials, and multiple orthogonal polynomials with exponential weights and cubic potentials.

Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies

We connect the theory of orthogonal Laurent polynomials on the unit circle and the theory of Toda-like integrable systems using the Gauss–Borel factorization of a Cantero–Moral–Velázquez moment matrix, that we construct in terms of a complex quasi-definite measure supported on the unit circle. The factorization of the moment matrix leads to orthogonal Laurent polynomials on the unit circle and the corresponding second kind functions. We obtain Jacobi operators, 5-term recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas from this point of view in a completely algebraic way. We generalize the Cantero–Moral–Velázquez sequence of Laurent monomials, recursion relations, Christoffel–Darboux kernels, and corresponding Christoffel–Darboux formulas in this extended context. We introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a Toda-type integrable system, which is connected with the well known Toeplitz lattice. We obtain the Lax and Zakharov–Shabat equations using the classical integrability theory tools. We explicitly derive the dynamical system associated with the coefficients of the orthogonal Laurent polynomials and we compare it with the classical Toeplitz lattice dynamical system for the Verblunsky coefficients of Szegő polynomials for a positive measure. Discrete flows are introduced and related to Darboux transformations. Finally, we obtain the representation of the orthogonal Laurent polynomials (and their second kind functions), using the formalism of Miwa shifts in terms of τ-functions and the subsequent bilinear equations.

Shape of impulse response characteristics of linear-phase nonrecursive 2D FIR filter function

An analytical method for the new class of linear-phase multiplierless 2D FIR filter functions generated by applying the Christoffel-Darboux formula for classical Chebyshev polynomials of the first and the second kind, proposed in [6] was used for designing of linear-phase multiplierless 2D FIR filter described in this paper. Correct transformation from continuous two-dimensional domain into the z domains without residuum and without errors is described. The proposed solution high selectivity is a filter function in the z1 domain, and the Hilbert transformer in the z2 domain. The impulse response coefficients of proposed 2D FIR filter functions are presented in this paper, and corresponding examples of impulse response are illustrated. The paper also presents detailed analysis of the size of pass-band and stop-band of proposed multiplierless linear-phase 2D FIR filter function. Normalized surface area of the filter function pass-band is 3.45789156 10-5 for given maximal attenuation of 0.28 dB. Normalized surface area of the filter function stop-band is 80.395% for the given minimal attenuation of 100 dB.

Analysis of finite tolerance effect of critical pole on characteristics of selective RC active special Gegenbauer filters

A precise analytical method for finding the explicit expression for the characteristic function of special Gegenbauer filters applicable to the design of RC active filters is suggested in this paper. The adverse parasite effects of limited finite gain-bandwidth product of operational amplifiers are decreased by using filters with the low pass-band attenuation. The new class of continual filter functions generated by analytical method by extremal Christoffel-Darboux formula for orthogonal Gegenbauer polynomials has two parameters. One is the filter order, n, and the second one is real free parameter, v, which provides a wide range of the amplitude responses. In this paper, a detailed analysis of attenuation and insertion loss in the bandwidth and around the stop-band cut-off frequency, wcs, are carried out using 3D plots and using examples of the effect of finite tolerance of quality factor module, Q, of critical conjugate-complex poles of considered RC active filter functions.

Christoffel-Darboux Formula Most Directly Applied in Generating Economical Linear Phase Low-Pass Digital FIR Filter Functions

The approximation problem of a filter function of even and odd order is solved mathematically in this paper most directly applying the proposed Christoffel-Darboux formula for two continual orthogonal polynomials on the equal finite segment. As a result, a linear phase digital finite impulse response (FIR) filter function is generated in compact explicit form. In addition, a new difference equation and structure of linear phase digital FIR filter are proposed. Two examples of the extremely economic FIR filter (with four adders and without multipliers) designed by the proposed technique are presented. The proposed solutions are extremely efficiency in regard to energy consumption. DOI: http://dx.doi.org/10.5755/j01.eee.18.8.2639

New class of filter functions generated directly by the modified Christoffel–Darboux formula for classical orthonormal Jacobi polynomials

ABSTRACTThe new solution of determining the prototype analogue filter function as the response that satisfies the specifications of all pole low‐pass continual time filter functions of odd and even order is presented in this article.The approximation problem of filter function was solved mathematically, applying the summed Christoffel–Darboux formula for the orthonormal polynomials. The starting point in solving the approximation problem is the direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthonormal polynomials in the finite interval [−1, + 1] in full respect to the weighting function with two free real parameters.Solution for the monotonic and non‐monotonic filter functions is obtained in an explicit form, which is shown to enable generation of the Jacobi filter functions in a simple way by choosing the values of the free real parameters.Moreover, the proposed solution with the same criterion of approximation is used to generate for particular specifications the appropriate best known classical approximation functions of free parameters: the Gegenbauer, Legendre, and Chebyshev filter functions of the first and second kind as well. The example of the proposed all‐pole low‐pass lossless LC filter in the frequency domain for frequency responses of the 11th‐order filter is illustrated.This work is concerned with the approximation of an all‐pole low‐pass filter to design a circuit assuring an optimum realization of the required specifications. The article proposes new class filter functions with an approach to ideal filter magnitude characteristic. Copyright © 2012 John Wiley & Sons, Ltd. image

Linear Phase Two-Dimensional FIR Digital Filter Functions Generated by applying Christoffel-Darboux Formula for Orthonormal Polynomials

Global Christoffel-Darboux formula for four orthonormal polynomials on two equal finite segments for generating linear phase two-dimensional finite impulse response (FIR) digital filter functions in a compact explicit representative form is proposed in this paper. The formula can be most directly applied for solving mathematically the approximation problem of a filter function of even and odd order. An example of a new class extremely economic linear phase two-dimensional FIR digital filter without multipliers obtained by the proposed approximation technique is presented. The generated linear phase two-dimensional FIR filter functions have two symmetries, that is, the following relations are valid: H(z1,z2)=H(z2,z1) and H(z1,z2)=H(-z1,-z2) and . Ill. 3, bibl. 15 (in English; abstracts in English and Lithuanian).DOI: http://dx.doi.org/10.5755/j01.eee.120.4.1449

New class of filter functions generated most directly by the Christoffel–Darboux formula for classical orthonormal Jacobi polynomials

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel–Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions.

A bivariate Markov inequality for Chebyshev polynomials of the second kind

This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov’s theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel–Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.

Orthogonal Functions Based on Chebyshev Polynomials

It is known that Chebyshev polynomials are an orthogonal set associated with a certain weight function. In this paper, we present an approach for the contruction of a special wavelet function as well as a special scaling function. Main tool of the special wavelet is a first kind Chebyshev polynomial. Based on Chebyshev polynomials and their zero, we define our scaling function and wavelets, and by using Christoffel-Darboux formula for Chebyshev polynomials, we prove that these functions are orthogonal. Finally, we provide several examples of scaling function and wavelets for illustration. Keywords: Chebyshev polynomial; Christoffel-Darboux formula; Wavelets; and Scaling function. 2010 Mathematics Subject Classification 76D04.

Nearest neighbor recurrence relations for multiple orthogonal polynomials

We show that multiple orthogonal polynomials for r measures ( μ 1 , … , μ r ) satisfy a system of linear recurrence relations only involving nearest neighbor multi-indices n → ± e → j , where e → j are the standard unit vectors. The recurrence coefficients are not arbitrary but satisfy a system of partial difference equations with boundary values given by the recurrence coefficients of the orthogonal polynomials with each of the measures μ j . We show how the Christoffel–Darboux formula for multiple orthogonal polynomials can be obtained easily using this information. We give explicit examples involving multiple Hermite, Charlier, Laguerre, and Jacobi polynomials.

Multiple orthogonal polynomials of mixed type: Gauss–Borel factorization and the multi-component 2D Toda hierarchy

Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss–Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel–Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss–Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov–Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ-function representation of the multiple orthogonality is given.

New class of filter functions generated most directly by Christoffel-Darboux formula for Gegenbauer orthogonal polynomials

A new original formulation of all pole low-pass filter functions is proposed in this article. The starting point in solving the approximation problem is a direct application of the Christoffel-Darboux formula for the set of orthogonal polynomials, including Gegenbauer orthogonal polynomials in the finite interval [−1, +1] with the application of a weighting function with a single free parameter. A general solution for the filter functions is obtained in a compact explicit form, which is shown to enable generation of the Gegenbauer filter functions in a simple way by choosing the value of the free parameter. Moreover, the proposed solution with the same criterion of approximation could be used to generate Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even (10th) and odd (11th) order are illustrated. The approximation is shown to yield a good compromise solution with respect to the filter frequency characteristics (magnitude as well as phase characteristics). The influence of tolerance of the filter critical component (inductor) on the proposed magnitude and group delay characteristics of a resistively terminated LC lossless ladder filter is analysed as well. The proposed filter functions are superior in terms of the excellent magnitude characteristic, which approximates an ideal filter almost perfectly over the entire pass-band range and exhibits the summed sensitivity function better than that of a Butterworth filter. In the article, we present the filter function solution that exhibits optimum amplitude as well as optimum group delay characteristics that are of crucial importance for implementation of digital processing as well as RF analogue parts of communication networks. Derivation of the other band range filter functions, which could be realised either by continuous or digital filters, is also generally possible with the procedure proposed in this article.

Notes On Carlitz's q-Operators

In this paper, Carlitz’s $q$-operator and the auxiliary ones are applied to prove $q$-Christoffel-Darboux formulas and some Carlitz type generating functions. In addition, the technique of exponential operator decomposition to deduce $q$-Mehler’s formulas for Rogers-Szego and Hahn polynomials are shown.

Polynomials with no zeros on the bidisk

We prove a detailed sums of squares formula for two variable polynomials with no zeros on the bidisk $\mathbb{D}^2$ extending previous versions of such a formula due to Cole-Wermer and Geronimo-Woerdeman. The formula is related to the Christoffel-Darboux formula for orthogonal polynomials on the unit circle, but the extension to two variables involves issues of uniqueness in the formula and the study of ideals of two variable orthogonal polynomials with respect to a positive Borel measure on the torus which may have infinite mass. We present applications to two variable Fej\'er-Riesz factorizations, analytic extension theorems for a class of bordered curves called distinguished varieties, and Pick interpolation on the bidisk.

A General Christoffel-Darboux Type Formula

In this note we provide a general expression for multivariable stable polynomials that reduces to Christoffel-Darboux type formulas. As a result we obtain new proofs for existing Christoffel-Darboux formulas based on Schur complements. In some cases we are able to refine existing results.