Orthogonal Polynomials In Several Variables Research Articles

A Stieltjes Algorithm for Generating Multivariate Orthogonal Polynomials

Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.

Maximal functions and multiplier theorem for Fourier orthogonal series

Under the assumption that orthogonal polynomials of several variables admit an addition formula, we can define a convolution structure and use it to study the Fourier orthogonal expansions on a space of homogeneous type. We define a maximal function via the convolution structure induced by the addition formula and use it to establish a Marcinkiewicz multiplier theorem. For the space of homogeneous type defined by a family of weight functions on conic domains, we show that the maximal function is bounded by the Hardy-Littlewood maximal function so that the multiplier theorem holds on conic domains.

Optimal approximants and orthogonal polynomials in several variables II: Families of polynomials in the unit ball

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function f ( z ) = 1 − 1 2 ( z 1 + z 2 ) f(z)=1-\frac {1}{\sqrt {2}}(z_1+z_2) and a scale of Hilbert function spaces in the unit 2 2 -ball having reproducing kernel ( 1 − ⟨ z , w ⟩ ) − γ (1-\langle z,w\rangle )^{-\gamma } , γ > 0 \gamma >0 . Our arguments are elementary but do not rely on reduction to the one-dimensional case.

Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Optimal approximants and orthogonal polynomials in several variables

AbstractWe discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators.In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any k∈N0 we obtain families of orthogonal polynomials in n variables with values in the N×N-matrices, where N=(n+kk). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n≥2. We also give explicit expressions of the spherical functions that determine the matrix weight for k=1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n=1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n,k) equal to (2,1) and (3,1).

Towards an Algebraic Theory of Orthogonal Polynomials in Several Variables

In this paper, we review on a general theory of orthogonal polynomials in several variables (O.P.S.V) in which we present two different approaches for the three-term recurrence relation. We draw attention to the fact that it is possible to take advantage of the orthogonal projection approach of the three-term recurrence relation towards the development of the algebraic theory of O.P.S.V.

A superintegrable discrete harmonic oscillator based on bivariate Charlier polynomials

A simple discrete model of the two-dimensional isotropic harmonic oscillator is presented. It is superintegrable with su(2) as its symmetry algebra. It is constructed with the help of the algebraic properties of the bivariate Charlier polynomials. This adds to the other discrete superintegrable models of the oscillator based on Krawtchouk and Meixner orthogonal polynomials in several variables.

Connection coefficients for classical orthogonal polynomials of several variables

Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained from the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.

Symmetric powers of symmetric bilinear forms, homogeneous orthogonal polynomials on the sphere and an application to compact Hyperkähler manifolds

The Beauville–Fujiki relation for a compact Hyperkähler manifold [Formula: see text] of dimension [Formula: see text] allows to equip the symmetric power [Formula: see text] with a symmetric bilinear form induced by the Beauville–Bogomolov form. We study some of its properties and compare it to the form given by the Poincaré pairing. The construction generalizes to a definition for an induced symmetric bilinear form on the symmetric power of any free module equipped with a symmetric bilinear form. We point out how the situation is related to the theory of orthogonal polynomials in several variables. Finally, we construct a basis of homogeneous polynomials that are orthogonal when integrated over the unit sphere [Formula: see text], or equivalently, over [Formula: see text] with a Gaussian kernel.

Orthogonal polynomials through the invariant theory of binary forms

We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring. Explicit determinantal formulae and multivariable extension of the Heine integral formula are stated. Moreover, a general family of covariants that includes transvectants is introduced. Such covariants turn out to be the average value of classical basis of symmetric polynomials over a set of roots of suitable orthogonal polynomials.

Multidimensional generalization of the Korous inequality

ABSTRACTThis paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.

Boundedness of orthogonal polynomials in several variables

Boundedness of orthogonal polynomials in several variables

Book Reviews

Plus ça change, plus c'est la même chose (to borrow a famous line). I hope you enjoy Joe Grcar's featured review of the book by Claude Brezinski and Dominique Tournès about the French army officer André-Louis Cholesky. Grcar includes quite a bit of background information on Cholesky's life and times and throws in a few opinions of his own. Although the challenges that creative people faced in Cholesky's time were somewhat different from what they are today, they were also somehow the same. As usual, we have a great variety of reviews in this issue. We have two books in the realm of numerical linear algebra. \AA ke Björck's Numerical Methods in Matrix Computations is intended to cover most of the main topics. Froilán Dopico praises it as encyclopedic but highly readable. Much more narrowly focused is the book Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, by Málek and Strakoš. In fact it lies on the borderline between numerical and functional analysis. Andy Wathen says it's “a thought-provoking read for experts and a piece of real mathematical scholarship for beginners to chew on łdots”. Don't miss Krešo Josić's review of Statistics Done Wrong: The Woefully Complete Guide, by Alex Reinhart. This reminds us of Darrell Huff's famous How to Lie with Statistics from sixty years ago. Apparently our modern-day computing power makes it even easier to do statistics wrong (to lie much faster?), so it's time for a new book. The focus is on frequentist statistics, so perhaps there is even room for one more book in this genre: Bayesian Statistics Done Wrong, or perhaps How to Lie with Bayesian Statistics. And there is much more: discrete mathematical methods in biology, orthogonal polynomials of several variables, the mathematics of insurance, and a second course in calculus.

Markov–Bernstein and Landau–Kolmogorov type inequalities in several variables for the Hermite and closely connected measures

This paper is devoted to Markov–Bernstein and Landau–Kolmogorov type inequalities in several variables in L2 norm. The measures which are used, are the Hermite measure and some closely connected measures in such a way that the partial derivatives of the orthogonal polynomials are also orthogonal. These orthogonal polynomials in several variables are built by tensor product of the orthogonal polynomials in one variable. These inequalities are obtained by using a variational method and they involve the square norms of a polynomial p and at most those of the partial derivatives of order 2 of p.It is also proved that the sequences of polynomials in several variables, orthogonal with respect to the Hermite and Laguerre–Sonin measures, are closed in a certain Sobolev space regarding its natural inner product and the inner product linked to the bilinear functional which is used in the variational method. This result shows us that the given Landau–Kolmogorov type inequalities are satisfied not only for polynomials, but also for functions belonging to this Sobolev space.

A higher order Sobolev-type inner product for orthogonal polynomials in several variables

We consider polynomials in several variables orthogonal with respect to a Sobolev-type inner product, obtained from adding a higher order gradient evaluated in a fixed point to a standard inner product. An expression for these polynomials in terms of the orthogonal family associated with the standard inner product is obtained. A particular case using polynomials in the unit ball is analyzed, and some asymptotic results are derived.

On linearly related orthogonal polynomials in several variables

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.

Well-conditioned Orthonormal Hierarchical $\mathcal{L}_{2}$ Bases on ℝ n Simplicial Elements

We construct well-conditioned orthonormal hierarchical bases for simplicial $\mathcal{L}_{2}$ finite elements. The construction is made possible via classical orthogonal polynomials of several variables. The basis functions are orthonormal over the reference simplicial elements in two and three dimensions. The mass matrices M are identity while the conditioning of the stiffness matrices S grows as $\mathcal{O}(p^{3})$ with respect to the order p. The diagonally normalized stiffness matrices are well conditioned. The diagonally normalized composite matrices ?M+S are also well conditioned for a wide range of ?. For the mass, stiffness and composite matrices, the bases in this study have much better conditioning than existing high-order hierarchical bases.

Orthogonal polynomials in several variables for measures with mass points

Let dν be a measure in ℝ d obtained from adding a set of mass points to another measure dμ. Orthogonal polynomials in several variables associated with dν can be explicitly expressed in terms of orthogonal polynomials associated with dμ, so are the reproducing kernels associated with these polynomials. The explicit formulas that are obtained are further specialized in the case of Jacobi measure on the simplex, with mass points added on the vertices, which are then used to study the asymptotics kernel functions for dν.