Hermite interpolation with trigonometric polynomials

Interpolation methods of Hermite type in translation invariant spaces of trigonometric polynomials for any position of interpolation points and any number of derivatives are constructed. For the case of an odd number of interpolation conditions-periodic trigonometric polynomials of minimum order are chosen as interpolation functions while for the case of an even number of interpolation conditions-antiperiodic trigonometric polynomials of minimum order are appropriate.     

The boundary of higher rank numerical ranges

The rank-k-numerical range of an n×n matrix A is defined as

     

Noncommutative biorthogonal polynomials

We define and study biorthogonal sequences of polynomials over noncommutative rings, generalizing previous treatments of biorthogonal polynomials over commutative rings and of orthogonal polynomials over noncommutative rings. We extend known recurrence relations for specific cases of biorthogonal polynomials and prove a general version of Favard?s theorem.     

Almost Hermitian trigonometric interpolation on three equidistant nodes

Summary We find necessary and sufficient conditions for the regularity (or unique solvability) of a trigonometric interpolation problem on three equidistant nodes in [0, 2) where the data are almost Hermitian. We shall show that in many cases the conditions only depend upon the number of even and odd derivatives prescribed in the problem. In general such a simplification seems to be difficult.Dedicated to the memory of Alexander M. Ostrowski on occasion of the 100th anniversary of his birth     

On a monotonic trigonometric sum

By establishing a cosine analogue of a result of Askey and Steinig on a monotonic sine sum, this paper sharpens and unifies several results associated with Young's inequality for the partial sums of k ^–1 cosk.     

Szeg? transformations and rational spectral transformations for associated polynomials

In this contribution we analyze rational spectral transformations related to associated polynomials with respect to probability measures supported on the interval [−1, 1]. The connection with rational spectral transformations of measures supported on the unit circle using the Szeg? transformation is presented.     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.     

Orthogonal polynomials and measures on the unit circle. The Geronimus transformations

In this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle. This perturbation is the inverse of the Christoffel transformation and is called the Geronimus transformation. We study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated Hessenberg matrices. Finally, we show an example of this kind of transformation.     

Arithmetic-Geometric Mean determinantal identity

In this paper, we give a generalization of a determinantal identity posed by Charles R. Johnson about minors of a Toeplitz matrix satisfying a specific matrix identity. These minors are those appear in the Dodgson’s condensation formula.     

Restricted colored permutations and Chebyshev polynomials

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schröder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation.     

Elementary transformations of pfaffian representations of plane curves

Let C be a smooth curve in P² given by an equation F=0 of degree d. In this paper we consider elementary transformations of linear pfaffian representations of C. Elementary transformations can be interpreted as actions on a rank 2 vector bundle on C with canonical determinant and no sections, which corresponds to the cokernel of a pfaffian representation. Every two pfaffian representations of C can be bridged by a finite sequence of elementary transformations. Pfaffian representations and elementary transformations are constructed explicitly. For a smooth quartic, applications to Aronhold bundles and theta characteristics are given.     

Perturbation bounds for polynomials

Using two different elementary approaches we derive a global and a local perturbation theorem on polynomial zeros that significantly improve the results of Ostrowski (Acta Math 72:99–257, 1940), Elsner et al. (Linear Algebra Appl 142:195–209, 1990). A comparison of different perturbation bounds shows that our results are better in many cases than the similar local result of Beauzamy (Can Math Bull 42(1):3–12, 1999). Using the matrix theoretical approach we also improve the backward stability result of Edelman and Murakami (Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, SIAM, Philapdelphia, 1994; Math Comput 64:210–763, 1995).     

Counting points on varieties over finite fields related to a conjecture of Kontsevich

We describe a characteristic-free algorithm for reducing an algebraic variety defined by the vanishing of a set of integer polynomials. In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field. The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph. We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures.Partially supported by NSF Grant DMS-9700787 and RIMS, Kyoto University.     

Symmetric determinantal representation of polynomials

We give an elementary proof, only using linear algebra, of a result due to Helton, Mccullough and Vinnikov, which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil. We give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2.     

The expressive power of voting polynomials

We consider the problem of approximating a Boolean functionf∶{0,1} ⁿ →{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP ^A is properly contained in PSPACE ^A . We are also able to prove the old AC⁰ exponential-size lower bounds in a new way. This allows us to prove the new result that an AC⁰ circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.     

Toward a language theoretic proof of the four color theorem

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of R_n , It is shown that if A,BεR_n , det(A) det(B:) ≠ 0, then d_k (AB) ≡ 0 mod d_k (A) d_k (B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k (AB) = d_k (A) d_k (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Nikol'skij-type and maximal inequalities for generalized trigonometric polynomials

We consider some Nikol'skij-type inequalities, thus inequalities between different metrics of a function, for almost periodic trigonometric polynomials. Some basic methods of probability theory are applied to prove the existence of the distribution function for an almost periodic function in the sense of Besicovitch. Finally, the Maximal function of Hardy and Littlewood is considered and maximal inequalities on Besicovitch spaces are proved. Received: 23 July 1998 / Revised version: 8 March 1999     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

Linear spaces of matrices of constant rank and instanton bundles

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P³${\mathbb{P}}^3$ and derived category tools. The method allows one to prove the existence of new examples of size 10×10$10\times 10$ and 14×14$14\times 14$ via instanton bundles of charge 2 and 4 respectively, and it provides an explanation for what used to be the only known example (Westwick 1996 [25]). We also give an algorithm to construct explicitly a matrix of size 14 of this type.