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.     

Integer Powers of Anti-Bidiagonal Hankel Matrices

In this paper we derive a general expression for integer powers of real upper and lower anti-bidiagonal matrices with constant anti-diagonals using Chebyshev polynomials. An explicit formula for the inverse of these matrices is also provided.     

The Multivariate Integer Chebyshev Problem

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in ℂ ^d . We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete result.      

A Class of Extremal Functions and Trigonometric Polynomials

In this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL¹-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials.     

Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded.     

On the extremality of the polynomials used for obtaining the best known upper bounds for the kissing numbers

We prove that the polynomials used for obtaining the best known upper bounds for some kissing numbers (the maximum number of nonoverlapping unit spheres that can touch a unit sphere in n dimensions) are best between the polynomials of the same or lower degree. We give also some extremal polynomials we have obtained using a method proposed in [4]. The upper bounds obtained in this way are slightly better than these from [1]. However the improvements are not in the integer part for dimensionsn 18.     

On Sign Conditions Over Real Multivariate Polynomials

We present a new probabilistic algorithm to find a finite set of points intersecting the closure of each connected component of the realization of every sign condition over a family of real polynomials defining regular hypersurfaces that intersect transversally. This enables us to show a probabilistic procedure to list all feasible sign conditions over the polynomials. In addition, we extend these results to the case of closed sign conditions over an arbitrary family of real multivariate polynomials. The complexity bounds for these procedures improve the known ones.     

Weighted analogues of capacity,transfinite diameter,and Chebyshev constant

For an arbitrary closed subsetE of the complex plane, the notions of logarithmic capacity, transfinite diameter, and Chebyshev constant ofE with respect to an admissible weightw onE are introduced. For thew-modified capacity, an electrostatics problem for logarithmic potentials in the presence of an external field is analyzed. This leads to an extremal measure whose support is the “smallest” compact set where the sup norm of weighted polynomials “live.” The introduction of a weightw has the advantage that the classical quantities mentioned in the title can be considered for unbounded setsE. Some of the theorems presented are generalizations of the authors' previous results for the case whenE?R.     

Diophantine m-tuples for linear polynomials II. Equal degrees

In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21-33].     

Eigenvalue problems to compute almost optimal points for rational interpolation with prescribed poles

Explicit formulas exist for the (n,m) rational function with monic numerator and prescribed poles that has the smallest possible Chebyshev norm. In this paper we derive two different eigenvalue problems to obtain the zeros of this extremal function. The first one is an ordinary tridiagonal eigenvalue problem based on a representation in terms of Chebyshev polynomials. The second is a generalised tridiagonal eigenvalue problem which we derive using a connection with orthogonal rational functions. In the polynomial case (m = 0) both problems reduce to the tridiagonal eigenvalue problem associated with the Chebyshev polynomials of the first kind. Postdoctoral researcher FWO-Flanders.     

Extremal polynomials for obtaining bounds for spherical codes and designs

We investigate two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs, respectively. We consider two basic properties of the solutions of these problems. Namely, we estimate from below the number of double zeros and find zero Gegenbauer coefficients of extremal polynomials. Our results allow us to search effectively for such solutions using a computer. The best polynomials we have obtained give substantial improvements in some cases on the previously known bounds for spherical codes and designs. Some examples are given in Section 6. This research was partially supported by the Bulgarian NSF under Contract I-35/1994.     

Combinatorics & Topology of Stratifications of The Space of Monic Polynomials With Real Coefficients

We study the stratification of the space of monic polynomials with real coefficients according to the number and multiplicities of real zeros. In the first part, for each of these strata we provide a purely combinatorial chain complex calculating (co)homology of its one-point compactification and describe the homotopy type by order complexes of a class of posets of compositions. In the second part, we determine the homotopy type of the one-point compactification of the space of monic polynomials of fixed degree which have only real roots (i.e., hyperbolic polynomials) and at least one root is of multiplicity k. More generally, we describe the homotopy type of the one-point compactification of strata in the boundary of the set of hyperbolic polynomials, that are defined via certain restrictions on root multiplicities, by order complexes of posets of compositions. In general, the methods are combinatorial and the topological problems are mostly reduced to the study of partially ordered sets.     

On some classes of polynomials with nonnegative coefficients and a given factor

For a given real polynomial f without positive roots we study polynomials g of lowest degree such that the product gf has positive (nonnegative, respectively) coefficients. We show that for quadratic f with negative linear coefficient every such g must have positive coefficients and exhibit an easy procedure for the determination of g. If f has only integer coefficients we show that g with integer coefficients can be found. Furthermore, for some classes of polynomials f we give upper (lower, respectively) bounds for the degrees of g.     

Improvement of the formal and numerical estimation of the constant in some Markov-Bernstein inequalities

Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant in Markov-Bernstein inequalities in L ² for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new sufficient condition is given in order that a polynomial has some complex zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximal Hilbert Functions

The set of Hilbert functions of standard graded algebras is considered as a partially ordered set under numerical comparison. For the set of algebras H(d, e₀), of a given dimension d and multiplicity e₀, we describe the requirements its maximal elements must satisfy; under fairly general conditions, the extremal functions arise from Cohen-Macaulay algebras. We also examine the subset H(d, e₀, e₁), of those functions whose first two coefficients of their Hilbert polynomials are assigned. Finally, we show how these results and the use of certain extended multiplicities can be used to prove finiteness theorems for the number of corresponding functions.     

On the eigenvalues of some transfer matrices

We consider the roots of two families of polynomials which can be derived as the characteristic polynomials of some (generalized) transfer matrices. We study the possible multiplicities and the number of real roots. Moreover, the number of roots lying inside the unit disk is determined, and bounds for their modulus and for the modulus of the other roots are given.     

On real and complex-valued bivariate Chebyshev polynomials

In the real uniform approximation of the function x^myⁿ by the space of bivariate polynomials of total degree m + n − 1 on the unit square, the product of monic univariate Chebyshev polynomials yields an optimal error. We exploit the fundamental Noether's theorem of algebraic curves theory to give necessary and sufficient conditions for unicity and to describe the set of optimal errors in case of nonuniqueness. Then, we extend these results to the complex approximation on biellipses. It turns out that the product of Chebyshev polynomials also provides an optimal error and that the same kind of uniqueness conditions prevail in the complex case. Yet, when nonuniqueness occurs, the characterization of the set of optimal errors presents peculiarities, compared to the real problem.     

第一类双变量Chebyshev多项式的最小零偏差性质研究

利用Rivlin和Shapiro提出的符号理论,证明了文献[10]中提出的第一类双变量Chebyshev多项式恰为所谓的Steiner区域上具有特殊首项的最小零偏差多项式,并由此导出了几类具有一定代数精度的数值积分公式.     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Products of polynomials in uniform norms

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment . Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.