Abstract: ACubic Decompositionof Sequencesof Orthogonal Polynomialson the Unit Circle MANUEL ALFARO*, MARI¤A JOSE¤ CANTERO b,y and FRANCISCOMARCELLA¤ Nc,z Departamento de Matema¤ ticas,Universidad de Zaragoza, 50009 Zaragoza, Spain; Departamento de Matema¤ tica Aplicada,Universidad de Zaragoza, 50015 Zaragoza, Spain; Departamento de Matema¤ ticas,Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911Legane¤ s, Madrid, Spain

Abstract: Preface Part I. Polynomials and Numerical Analysis: 1. Polynomials 2. Representations of polynomial ideals 3. Polynomials with coefficients of limited accuracy 4. Approximate numerical computation Part II. Univariate Polynomial Problems: 5. Univariate polynomials 6. Various tasks with empirical univariate polynomials Part III. Multivariate Polynomial Problems: 7. One multivariate polynomial 8. Zero-dimensional systems of multivariate polynomials 9. Systems of empirical multivariate polynomials 10. Numerical basis computation Part IV. Positive-Dimensional Polynomial Systems: 11. Matrix eigenproblems for positive-dimensional systems Index.

Abstract: Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal. We call such a pair a Leonard pair on $V$. We give a correspondence between Leonard pairs and a class of orthogonal polynomials. This class coincides with the terminating branch of the Askey scheme and consists of the $q$-Racah, $q$-Hahn, dual $q$-Hahn, $q$-Krawtchouk, dual $q$-Krawtchouk, quantum $q$-Krawtchouk, affine $q$-Krawtchouk, Racah, Hahn, dual Hahn, Krawtchouk, Bannai/Ito, and orphan polynomials. We describe the above correspondence in detail. We show how, for the listed polynomials, the 3-term recurrence, difference equation, Askey-Wilson duality, and orthogonality can be expressed in a uniform and attractive manner using the corresponding Leonard pair. We give some examples that indicate how Leonard pairs arise in representation theory and algebraic combinatorics. We discuss a mild generalization of a Leonard pair called a tridiagonal pair. At the end we list some open problems. Throughout these notes our argument is elementary and uses only linear algebra. No prior exposure to the topic is assumed.

Abstract: We show that the average characteristic polynomial Pn(z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z 1 n exp(−Tr(V (M) − AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue aj of A, there is a weight and Pn has nj orthogonality conditions with respect to this weight, if nj is the multiplicity of aj. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.

Abstract: We develop a general method that allows us to introduce families of orthogonal matrix polynomials of size N × N satisfying second-order differential equations. The presence of this extra property should make these orthogonal polynomials into useful tools in several areas of mathematics and its applications. Historically, this has certainly been the case for their scalar-valued versions. The subtlety of the noncommutative algebra of matrices can be exploited to yield many different such families, almost dwarfing the scalar situation by comparison. All these families form a nice and rich hierarchy starting from the classical Jacobi, Hermite, and Laguerre families, N=1, and increasing in number and variety as the size N increases. We illustrate the use of our method by giving large classes of generic examples of arbitrary size N.

Abstract: We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern–Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes–Wigert polynomials), and the deformation parameter turns out to be the usual q parameter in Chern–Simons theory. In this way, we give a matrix model computation of the Chern–Simons partition function on S3 and show that there are infinitely many matrix models with this partition function.

Abstract: Power amplifiers are the major source of nonlinearity in communications systems. Such nonlinearity causes spectral regrowth as well as in-band distortion, which leads to adjacent channel interference and increased bit error rate. Polynomials are often used to model the nonlinear power amplifier or its predistortion linearizer. In this paper, we present a novel set of orthogonal polynomials for baseband Gaussian input to replace the conventional polynomials and show how they alleviate the numerical instability problem associated with the conventional polynomials. The orthogonal polynomials also provide an intuitive means of spectral regrowth analysis.

Abstract: Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely differentiable function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi–Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Jacobi and Hermite–Jacobi are developed.

Abstract: We revisit the ladder operators for orthogonal polynomials and re-interpret two supplementary conditions as compatibility conditions of two linear over-determined systems; one involves the variation of the polynomials with respect to the variable z (spectral parameter) and the other a recurrence relation in n (the lattice variable). For the Jacobi weight w(x) = (1 - x) α (1 + x) β , x ∈ [-1, 1], we show how to use the compatibility conditions to explicitly determine the recurrence coefficients of the monic Jacobi polynomials.

Abstract: We lift the constraint of a diagonal representation of the Hamiltonian by searching for square integrable bases that support an infinite tridiagonal matrix representation of the wave operator. The class of solutions obtained as such includes the discrete (for bound states) as well as the continuous (for scattering states) spectrum of the Hamiltonian. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, include problems in one and three dimensions.

Abstract: In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguerre, and Jacobi ensembles of Random Matrix theory. We also compute multivariate hypergeometric functions, and offer both symbolic and numerical evaluations for all these quantities. We prove that all algorithms are well-defined, analyze their complexity, and illustrate their performance in practice. Finally, we also present a few of the possible applications of this library.

Abstract: We present the main ideas and techniques of the proof that the duality-covariant four-dimensional noncommutative \phi^4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation by flow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces.

Abstract: In this paper we give basic concepts of the Mathematica package \Or- thogonalPolynomials". The package \OrthogonalPolynomials" emerged from the need to implement basic algorithms from the theory of orthogonal polynomials to the Mathematica platform which ofiers, in our opinion the highest comput- ing possibilities. Package performs construction of orthogonal polynomials and quadrature formulas. Also, the package has implemented almost all the classes of orthogonal polynomials studied up to date. For the detailed exposition of the material presented in this paper we refer to (3), here we present only basic characteristics of the package.

Abstract: We give a new proof of the operator version of the Fejer-Riesz Theorem using only ideas from elementary operator theory. As an outcome, an algorithm for computing the outer polynomials that appear in the Fejer-Riesz factorization is obtained. The extremal case, where the outer factorization is also *-outer, is examined in greater detail. The connection to Agler’s model theory for families of operators is considered, and a set of families lying between the numerical radius contractions and ordinary contractions is introduced. The methods are also applied to the factorization of multivariate operator-valued trigonometric polynomials, where it is shown that the factorable polynomials are dense, and in particular, strictly positive polynomials are factorable. These results are used to give results about factorization of operator valued polynomials over $$ \mathbb{R}^m, m \geq 1 $$ , in terms of rational functions with fixed denominators.

Abstract: We introduce here a generalization of the modified Bernstein polynomials for Jacobi weights using the $q$-Bernstein basis proposed by G.M. Phillips to generalize classical Bernstein Polynomials. The function is evaluated at points which are in geometric progression in $]0,1[$. Numerous properties of the modified Bernstein Polynomials are extended to their $q$-analogues: simultaneous approximation, pointwise convergence even for unbounded functions, shape-preserving property, Voronovskaya theorem, self-adjointness. Some properties of the eigenvectors, which are $q$-extensions of Jacobi polynomials, are given.

Abstract: In this paper, we obtain certain discrete orthogonal polynomials expressed in terms of the (d + 1,2(d + 1))-hypergeometric functions, from the eigenmatrices of character algebras.

Abstract: General classes of two variables Appell polynomials are introduced by exploiting properties of an iterated isomorphism, related to the so-called Laguerre-type exponentials. Further extensions to the multi-index and multivariable cases are mentioned.

Abstract: We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.

Abstract: We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Levy processes.

Abstract: We give a proof of the Universality Conjecture for orthogonal and symplectic ensembles of random matrices in the scaling limit for a class of weights w(x)=exp(-V(x)) where V is a polynomial, V(x)=kappa_{2m}x^{2m}+..., kappa_{2m}>0. For such weights the associated equilibrium measure is supported on a single interval. The precise statement of our results is given in Theorem 1.1 below. For a proof of the Universality Conjecture for unitary ensembles, for the same class of weights, see [DKMVZ2]. Our starting point is Widom's representation [W] of the orthogonal and symplectic correlation kernels in terms of the kernel arising in the unitary case plus a correction term which is constructed out of derivatives and integrals of orthonormal polynomials (OP's) {p_j(x)}, j=0,1,..., with respect to the weight w(x). The calculations in [W] in turn depend on the earlier work of Tracy and Widom [TW2]. It turns out (see [W] and also Theorems 2.1 and 2.2 below) that only the OP's in the range j=N+O(1), N->infinity, contribute to the correction term. In controlling this correction term, and hence proving Universality for both the orthogonal and symplectic cases, the uniform Plancherel--Rotach type asymptotics for the OP's found in [DKMVZ2] play an important role, but there are significant new analytical difficulties that must be overcome which are not present in the unitary case. We note that we do not use skew orthogonal polynomials.

Abstract: A new class of quasi exactly solvable potentials with a variable mass in the Schroedinger equation is presented. We have derived a general expression for the potentials also including Natanzon confluent potentials. The general solution of the Schroedinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials.

Abstract: We introduce polynomials B n i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonniere, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

Abstract: Based on the two-dimensional (2-D) least squares method, this paper presents a novel numerical method to calculate the magnetic characteristics for switched reluctance motor drives. In this method, the 2-D orthogonal polynomials are used to model the magnetic characteristics. The coefficients in these polynomials are determined by the 2-D least squares method. These coefficients can be computed off line and can also be trained on line. The computed results agree well with the experimental results. In addition, the effect of the order number of the polynomials on the computation errors is discussed. The proposed method is very helpful in torque prediction, simulation studies and development of sensorless control of switched reluctance motor drives.

Abstract: We announce numerous new results in the theory of orthogonal polynomials on the unit circle, most of which involve the connection between a measure on the unit circle in the complex plane and the coefficients in the recursion relations for the polynomials known as Verblunsky coefficients. Included are several applications of the recently discovered matrix realization of Cantero, Moral, and Velazquez. In analogy with the spectral theory of Jacobi matrices, several classes of exotic Verblunsky coefficients are studied. A version of Rahkmanov's theorem is proven with a single gap with eigenvalues allowed in the gap. Analogs of Borg's theorem and the Birman-Schwinger principle are found.