Eventual Positivity of Hermitian Polynomials and Integral Operators

Quillen proved that if a Hermitian bihomogeneous polynomial is strictly positive on the unit sphere, then repeated multiplication of the standard sesquilinear form to this polynomial eventually results in a sum of Hermitian squares. Catlin-D'Angelo and Varolin deduced this positivstellensatz of Quillen from the eventual positive-definiteness of an associated integral operator. Their arguments involve asymptotic expansions of the Bergman kernel. The goal of this article is to give an elementary proof of the positive-definiteness of this integral operator.     

Positivity of third order linear recurrence sequences

It is shown that the Positivity Problem for a sequence satisfying a third order linear recurrence with integer coefficients, i.e., the problem whether each element of this sequence is nonnegative, is decidable.     

Positivity of second order linear recurrent sequences

We give a decision method for the Positivity Problem for second order recurrent sequences: it is decidable whether or not a recurrent sequence defined by u_n=au_n-1+bu_n-2 has only nonnegative terms.     

Positivity for polyharmonic problems on domains close to a disk

We study the problem of positivity preserving of the Green operator for the polyharmonic operator (?Δ) ^m under homogeneous Dirichlet boundary conditions on domains Ω of ?R ². Here we will treat only Ω, which are ε-close to a disk B in C ^m,γ-sense, meaning, there exists a C ^m,γ-mapping g : \( \bar{B}\longrightarrow \bar{\Omega}\) such that g?(B) = ?Ω and \(||g -- Id||_{C^{m,\gamma}}(\bar{B})\!\leq\!\varepsilon\). We show that ε-closeness in C ^{m, γ}-sense is enough in order to ensure positivity preserving. For the clamped plate equation (i.e. m = 2), this means that it is a Hölder norm of the curvature of ? Ω, which governs the positivity behavior. This improves the previous work by Grunau and Sweers, where closeness to the disk in C ^2m -sensewas required (in C ⁴-sense for thethe clamped plate).     

Positivity of solutions to boundary value problems for infinite functional differential systems

In this paper the problems of unique solvability of boundary value problems for infinite functional differential systems and positivity of their solutions are reduced to analysis of scalar functional equations in the space of measurable essentially bounded functions y:[0,+∞)→(−∞,+∞). Efficient tests of the solvability and the positivity are obtained.     

André-Quillen cohomology and rational homotopy of function spaces

We develop a simple theory of André-Quillen cohomology for commutative differential graded algebras over a field of characteristic zero. We then relate it to the homotopy groups of function spaces and spaces of homotopy self-equivalences of rational nilpotent CW-complexes. This puts certain results of Sullivan in a more conceptual framework.     

On a combinatorial theorem related to a theorem of G. Szegö

In 1954, M. Kac discovered a probabilistic interpretation of a theorem of G. Szegö of Toeplitz matrices and demonstrated that this theorem can be provedin an elementary way by using a combinatorial identify of G. A. Hunt. In this paper Hunt's combinatorial identity is derived from a more general combinatorial result.     

Positivity and boundedness of trigonometric sums

We give a systematic account of results which assure positivity and boundedness of partial sums of cosine or sine series. New proofs of recent results are sketched.     

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S²ⁿ⁻¹ in ⁿ. We prove that Szegö map is a unitary operator from a subspace of (p, q −1)-forms on the sphere defined by the operators and the normal vector field onto the space of L²-harmonic (p, q)-forms on the unit ball. Our results generalize earlier result of Folland.     

The growth of valuations on rational function fields in two variables

Given a valuation on the function field , we examine the set of images of nonzero elements of the underlying polynomial ring under this valuation. For an arbitrary field , a Noetherian power series is a map that has Noetherian (i.e., reverse well-ordered) support. Each Noetherian power series induces a natural valuation on . Although the value groups corresponding to such valuations are well-understood, the restrictions of the valuations to underlying polynomial rings have yet to be characterized. Let denote the images under the valuation of all nonzero polynomials of at most degree in the variable . We construct a bound for the growth of with respect to for arbitrary valuations, and then specialize to valuations that arise from Noetherian power series. We provide a sufficient condition for this bound to be tight.

     

Pushing the envelope of the test functions in the Szegö and Avram–Parter theorems

The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far.     

A Björck–Pereyra-type algorithm for Szegö–Vandermonde matrices based on properties of unitary Hessenberg matrices

In this paper we carry over the Björck-Pereyra algorithm for solving Vandermonde linear systems to what we suggest to call Szegö-Vandermonde systems V_Φ(x), i.e., polynomial-Vandermonde systems where the corresponding polynomial system Φ is the Szegö polynomials. The properties of the corresponding unitary Hessenberg matrix allow us to derive a fast O(n²) computational procedure. We present numerical experiments that indicate that for ill-conditioned matrices the new algorithm yields better forward accuracy than Gaussian elimination.     

ODEs and Wiman-Valiron theory in the unit disc

An asymptotic equality of Wiman-Valiron type is proved for the derivatives of analytic functions in the unit disc and applied to ODEs with analytic coefficients.     

Pointwise universal trigonometric series

A series is called a pointwise universal trigonometric series if for any , there exists a strictly increasing sequence of positive integers such that converges to f(z) pointwise on . We find growth conditions on coefficients allowing and forbidding the existence of a pointwise universal trigonometric series. For instance, if as |n|→∞ for some ε>0, then the series S_a cannot be pointwise universal. On the other hand, there exists a pointwise universal trigonometric series S_a with as |n|→∞.     

On the exact distribution and mean value function of a geometric process with exponential interarrival times

The geometric process is considered when the distribution of the first interarrival time is assumed to be exponential. An analytical expression for the one dimensional probability distribution of this process is obtained as a solution to a system of recursive differential equations. A power series expansion is derived for the geometric renewal function by using an integral equation and evaluated in a computational perspective. Further, an extension is provided for the power series expansion of the geometric renewal function in the case of the Weibull distribution.     

Orthonormal rational function vectors

Summary In this paper, we develop a matrix framework to solve the problem of finding orthonormal rational function vectors with prescribed poles with respect to a certain discrete inner product that is defined by a set of data points and corresponding weight vectors w_i,j. Our algorithm for solving the problem is recursive, and it is of complexity If all data points are real or lie on the unit circle, then the complexity is reduced by an order of magnitude.     

Evaluation of a rational function

Evaluation of a given rational function can be carried out in various ways, such as by Horner's method, directly through a continued fraction representation, or by a subsequent process that uses three-term recurrences. Four such algorithms are summarized, and their relative performances when they are implemented using a parallel architecture (a four-node transputer system) is investigated.     

The Fekete–Szegö problem for k-uniformly convex functions and for a class defined by the Owa–Srivastava operator

By making use of a fractional differential operator due to Owa and Srivastava, a subclass of analytic functions related to k-uniformly convex functions, is introduced. For this class and in particular for the class of k-uniformly convex functions the Fekete–Szegö problem is completely solved.     

有理Fourier级数在变差条件下的收敛性研究

本文把Fourier级数的一些经典结论推广到有理Fourier级数的情况下. 首先给出了有理Fourier级数和共轭有理Fourier级数在有界变差条件下的收敛速度估计. 利用此结论, 得到了类似于Fourier级数的Dirichlet-Jordan定理和W. H. Young定理. 最后, 证明了这两个定理在调和有界变差条件下也成立.