Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Padé approximation and generalized moments

We propose studying generalized moment representations of a form in which it suffices to apply a system of orthogonal polynomials in order to procure the biorthogonality conditions in the construction of superdiagonal Padé polynomials using generalized moment representations. The algebraic polynomials in the moment representation are to be sought as the linear forms of biorthogonal polynomials. We obtain the relations between the coefficients of these linear forms and the generalized moments, and we also establish conditions for the existence and uniqueness of generalized moment representations of polynomial form. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 110–115.     

Optimally Space Localized Polynomials with Applications in Signal Processing

For the filtering of peaks in periodic signals, we specify polynomial filters that are optimally localized in space. The space localization of functions f having an expansion in terms of orthogonal polynomials is thereby measured by a generalized mean value ε(f). Solving an optimization problem including the functional ε(f), we determine those polynomials out of a polynomial space that are optimally localized. We give explicit formulas for these optimally space localized polynomials and determine in the case of the Jacobi polynomials the relation of the functional ε(f) to the position variance of a well-known uncertainty principle. Further, we will consider the Hermite polynomials as an example on how to get optimally space localized polynomials in a non-compact setting. Finally, we investigate how the obtained optimal polynomials can be applied as filters in signal processing.     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.     

Orthogonal polynomials in two variables

Summary Some properties of orthogonal (and generalized orthogonal) polynomial sets in two variables are obtained, in particular a characterization of such sets based on generating functions. Then those linear homogeneous partial differential eqnations of the form L[w]+λw=0, having a set of polynomials as solution, are characterized; and a detailed study is made of all such equations of second order whose polynomial solutions form an orthogonal (or generalized orthogonal) set. Supported byN.S.F. Grant GP-5311.     

Plane wave approximation of homogeneous Helmholtz solutions

In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω ² u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations.     

Characterization of Classes of Polynomial Functions

In this paper, some classes of local polynomial functions on abelian groups are characterized by the properties of their variety. For this characterization, we introduce a numerical quantity depending on the variety of the local polynomial only. Moreover, we show that the known characterization of polynomials among generalized polynomials can be simplified: a generalized polynomial is a polynomial if and only if its variety contains finitely many linearly independent additive functions.     

On the complexity of generalized polynomials of <Emphasis Type="Italic">k</Emphasis>-valued functions

Specification of k-valued functions with generalized polynomials (for simple k) is considered. A generalized polynomial is a mod k polynomial in which each variable may also occur with one or several Post negations. The upper and lower estimates of the complexity of generalized polynomials are found for k-valued functions.     

To solving multiparameter problems of algebra. 9. The Ψ<Emphasis Type="Italic">F-q</Emphasis> method for factorizing invariant polynomials and its applications

A new method (the ΨF-q method) for computing the invariant polynomials of a q-parameter (q ≥ 1) polynomial matrix F is suggested. Invariant polynomials are computed in factored form, which permits one to analyze the structure of the regular spectrum of the matrix F, to isolate the divisors of each of the invariant polynomials whose zeros belong to the invariant polynomial in question, to find the divisors whose zeros belong to at least two of the neighboring invariant polynomials, and to determine the heredity levels of points of the spectrum for each of the invariant polynomials. Applications of the ΨF-q method to representing a polynomial matrix F(λ) as a product of matrices whose spectra coincide with the zeros of the corresponding divisors of the characteristic polynomial and, in particular, with the zeros of an arbitrary invariant polynomial or its divisors are considered. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 165–173.     

On rational approximations of functions

Rational analogues of Taylor and Fourier polynomials and polynomials of the Lagrange type are constructed and investigated. These analogues are shown to approximate a function with a remainder term of the same order as in the case of the aforementioned polynomials. Conditions are established under which a polynomial of the Lagrange type and its rational analogue are two-sided approximations of a function on a segment and their derivatives are two-sided approximations of the derivative of the function at collocation nodes. Bibliography: 2 titles. Translated fromObchuslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 32–38.     

Generalized characteristic polynomials of graph bundles

In this paper, we find computational formulae for generalized characteristic polynomials of graph bundles. We show that the number of spanning trees in a graph is the partial derivative (at (0,1)) of the generalized characteristic polynomial of the graph. Since the reciprocal of the Bartholdi zeta function of a graph can be derived from the generalized characteristic polynomial of a graph, consequently, the Bartholdi zeta function of a graph bundle can be computed by using our computational formulae.     

Quasi-permutation polynomials

A quasi-permutation polynomial is a polynomial which is a bijection from one subset of a finite field onto another with the same number of elements. This is a natural generalization of the familiar permutation polynomials. Basic properties of quasi-permutation polynomials are derived. General criteria for a quasi-permutation polynomial extending the well-known Hermite’s criterion for permutation polynomials as well as a number of other criteria depending on the permuted domain and range are established. Different types of quasi-permutation polynomials and the problem of counting quasi-permutation polynomials of fixed degree are investigated.     

Computing ECT-B-splines recursively

ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices. Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences [24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined by different local ECT-systems on knot intervals of a finite partition of a compact interval [a,b] connected at inner knots all of multiplicities zero by full connection matrices A ^[i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines or the B-spline curves. *Supported in part by INTAS 03-51-6637.     

The probability function of a generalized poisson distribution and special polynomials

The probability function of a generalized Poisson distribution is written in terms of modified Hermite polynomials of 2nd kind in many variables. Their relationship is established with special polynomials previously derived for this purpose. The properties of the polynomials and the probability function are studied. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996. pp. 160–169.     

Differential operators associated with zonal polynomials. I

Summary Associated with each zonal polynomial,C _k(S), of a symmetric matrixS, we define a differential operator ∂_k, having the basic property that ∂_kC_λδ_kλ, δ being Kronecker's delta, whenever κ and λ are partitions of the non-negative integerk. Using these operators, we solve the problems of determining the coefficients in the expansion of (i) the product of two zonal polynomials as a series of zonal polynomials, and (ii) the zonal polynomial of the direct sum,S⊕T, of two symmetric matricesS andT, in terms of the zonal polynomials ofS andT. We also consider the problem of expanding an arbitrary homogeneous symmetric polynomial,P(S) in a series of zonal polynomials. Further, these operators are used to derive identities expressing the doubly generalised binomial coefficients ( _P ^λ ),P(S) being a monomial in the power sums of the latent roots ofS, in terms of the coefficients of the zonal polynomials, and from these, various results are obtained.     

Polynomial numerical hulls of matrix polynomials,II

In this article, we study some algebraic and geometrical properties of polynomial numerical hulls of matrix polynomials and joint polynomial numerical hulls of a finite family of matrices (possibly the coefficients of a matrix polynomial). Also, we study polynomial numerical hulls of basic A-factor block circulant matrices. These are block companion matrices of particular simple monic matrix polynomials. By studying the polynomial numerical hulls of the Kronecker product of two matrices, we characterize the polynomial numerical hulls of unitary basic A-factor block circulant matrices.     

Analysis of the distribution of roots of a polynomial using a generalized Routh scheme

We study the problem of determining the numerical characteristics of the distribution of the set of roots of a polynomial. The complete solution of this problem is achieved by applying a generalized Routh scheme, proposed previously for computing the Cauchy indices and the greatest common divisor of polynomials. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 107–118.     

The ideal of separants in the ring of differential polynomials

We obtained the criterion of existence of a quasi-linear polynomial in a differential ideal in the ordinary ring of differential polynomials over a field of characteristic zero. We generalized the “going up” and “going down” theorems onto the case of Ritt algebras. In particular, new finiteness criteria for differential standard bases and estimates that characterize calculation complexity were obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 215–227, 2007.     

Greatest common divisor of generalized polynomials and polynomial matrices

The comrade matrix was introduced recently as the analogue of the companion matrix when a polynomial is expressed in terms of a basis set of orthogonal polynomials. It is now shown how previous results on determining the greatest common divisor of two or more polynomials can be extended to the case of generalized polynomials using the comrade form. Furthermore, a block comrade matrix is defined, and this is used to extend to the generalized case another previous result on the regular greatest common divisor of two polynomial matrices.