Polynomials Of Several Real Variables Research Articles

Christoffel–Darboux Kernels in Several Real Variables

The Christoffel–Darboux kernels for orthogonal polynomials in several real variables are investigated within the context of the three-term relation reformulated for this purpose. As suggestive examples of orthogonality, we propose to discuss the two simple algebraic cases: the unit circle and the Bernoulli lemniscate.

Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T1≤T≤T2 for the linear operator T, only in terms of quadratic forms, is deduced. Here, T1, T,and T2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples.

Growth and approximation of solutions to a class of certain linear partial differential equations in ℝN

Abstract In this paper we consider the equation ∇2 φ + A(r 2)X · ∇φ + C(r 2)φ = 0 for X ∈ ℝN whose coefficients are entire functions of the variable r = |X|. Corresponding to a specified axially symmetric solution φ and set C n of (n + 1) circles, an axially symmetric solution Λn*(x, η;C n) and Λn(x, η;C n) are found that interpolates to φ(x, η) on the C n and converges uniformly to φ(x, η) on certain axially symmetric domains. The main results are the characterization of growth parameters order and type in terms of axially symmetric harmonic polynomial approximation errors and Lagrange polynomial interpolation errors using the method developed in [MARDEN, M.: Axisymmetric harmonic interpolation polynomials in ℝN, Trans. Amer. Math. Soc. 196 (1974), 385–402] and [MARDEN, M.: Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137–154].

Orthogonal Homogeneous Polynomials

An addition formula, Pythagorean identity, and generating function are obtained for orthogonal homogeneous polynomials of several real variables. Application is made to the study of series of such polynomials. Results include an analog of the Funk-Hecke theorem.

On Nonconvexity of Graphs of Polynomials of Several Real Variables

We consider transversal (orthogonal) perturbations of finite-dimensional convex sets and estimate the 'degree of nonconvexity' of resulting sets, i.e. we estimate the nonconvexity of graphs of continuous functions. We prove that a suitable estimate of nonconvexity of graphs over all lines induces a 'nice' estimate of the nonconvexity of graphs of the entire function. Here, the term 'nice' means that in the well-known Michael selection theorem it is possible to replace convex sets of a multivalued mapping by such nonconvex sets. As a corollary, we obtain positive results for poly- nomials of degree two under some restrictions on coefficients. Our previous results concerned the polynomials of degree one and Lipschitz functions. We show that for a family of polynomials of degree three such estimate of convexity in general does not exist. Moreover, for degree 9 we show that the nonconvexity of the unique polynomialP.x;y/Dx 9 Cx 3 y realizes the worst possible case. Mathematics Subject Classifications ( 1991): Primary: 54C60, 54C65; Secondary: 52A21, 54C35.

Deformation quantization and Nambu Mechanics

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

Multivariate polynomial interpolation under projectivities II: Neville-Aitken formulas

This is the second part of a note on interpolation by real polynomials of several real variables. For certain regular knot systems (geometric or regular meshes, tensor product grids), Neville-Aitken algorithms are derived explicitly. By application of a projectivity they can be extended in a simple way to arbitrary (k+1)-pencil lattices as recently introduced by Lee and Phillips. A numerical example is given.

Multivariate polynomial interpolation under projectivities part I: lagrange and newton interpolation formulas

In this note interpolation by real polynomials of several real variables is treated. Existence and unicity of the interpolant for knot systems being the perspective images of certain regular knot systems is discussed. Moreover, for such systems a Newton interpolation formula is derived allowing a recursive computation of the interpolant via multivariate divided differences. A numerical example is given.

On multivariate interpolation by generalized polynomials on subsets of grids

This note may be regarded as a complement to a paper of H. Werner [17] who has carried over Newton's classical interpolation formula to Hermite interpolation by algebraic polynomials of several real variables on certain subsets of grids. Here generalized polynomials of several real or complex variables are treated. Recursive procedures are presented showing that interpolation by generalized multivariate polynomials is performed nearly as simply as interpolation by algebraic polynomials. Having in general the same approximation power, generalized polynomials may be better adapted to special situations. In particular, the results of this note can be used for constructing nonpolynomial finite elements since in that case the interpolation points usually are rather regular subsystems of grids. Though the frame is more general than in [17] some of our proofs are simpler. As an alternative method to evaluate multivariate generalized interpolation polynomials for rectangular grids a Neville-Aitken algorithm is presented.

A set of inequalities

It is my purpose to establish a set of inequalities between symmetric homogeneous polynomials in several real variables. That may sound a bit grand, but the simplest example of such an inequality is the familiarA more complicated inequality of the same general type is the inequalityAgain increasing the degree of complexity we have the pair of inequalities

Structural rigidity II: Almost infinitesimally rigid bar frameworks

Structural rigidity of bar frameworks is studied in the case that the corresponding rigidity matrix has rank one less than the number of columns. This represents the case just next to infinitesimal rigidity and is likely to occur at the exceptional loci of generically inf-rigid frameworks. Applying a general rigidity criterion from the preceding paper we obtain necessary and sufficient rigidity conditions in which (i) vectors are assigned to binary trees, (ii) fixed linear combinations (whose coefficients are polynomials of several real variables) are formed and (iii) it is tested whether these combinations may simultaneously belong to a certain vector space. Although this criterion is too complex for theorem proving it may be easily applied in concrete situations.

Extension of Bernstein's Theorem

where D,P(z) denotes the partial derivative of P with respect to z, at z for i = 1, ... ., m. A function H(z, w) defined for z, w E Cm with values in C is called a hermitian symmetric form if H(z, w) is linear in z for every fixed w and H(w, z) = H(z, w). In this paper, we extend the following complex version of Bernstein's theorem [5, p. 57] from a polynomial on the unit disk to a polynomial on the unit ball in several complex variables. Similar extensions of Bernstein's theorem and Markoff's theorem for polynomials of several real variables were obtained by Kellogg [4].

Value Distribution of Harmonic Polynomials in Several Real Variables

Value Distribution of Harmonic Polynomials in Several Real Variables

Value distribution of harmonic polynomials in several real variables.

Using Bergman’s integral operator method, the author studies an arbitrary axisymmetric harmonic polynomial H ( x , ρ ) H(x,\rho ) in R 3 {R^3} and R N {R^N} in relation to its associate polynomial h ( ζ ) h(\zeta ) in C C . His results pertain to the value distributions and critical circles of H ( x , ρ ) H(x,\rho ) in certain cones; bounds on the gradient of an H ( x , ρ ) H(x,\rho ) assumed bounded in sphere x 2 + ρ 2 ≦ 1 {x^2} + {\rho ^2} \leqq 1 ; axisymmetric harmonic vectors. Corresponding results are also obtained for axisymmetric harmonic functions F ( x , ρ ) F(x,\rho ) with rational associate f ( ζ ) f(\zeta ) .