Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).     

Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifoldsM embedded in ℝ ^k which satisfy some mild curvature conditions. The inequality |q·x| <Ψ(|q|) whereΨ(r) → 0 asr → ∞ has finitely or infinitely many solutionsqεℤ ^k for almost all (in induced measure) points x onM according as the sum Σ _r ^{= 1/∞} Ψ(r)r ^k−2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point inM satisfying the inequality infinitely often whenψ(r) =r ^−t . τ >k − 1.     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

Singular values, quasiconformal maps and the Schottky upper bound

Some properties and asymptotically sharp bounds are obtained for singdar values of Ramanujan’s generalized modular equation. from which infinite-product representations of the Hersch-Pfluger ϕdimtortion function ϕ _K (r) and the Agard η-distortion function η _K (t) follow. By these results, the explicit quasiconformal Schwan lemma is improved, several properties are obtained for the Schottky upper bound, and a conjecture on the linear distortion function λ (K) is proved to be true.     

Higher order Lipschitz classes of functions and absolutely convergent Fourier series

We introduce the higher order Lipschitz classes Λ _r (α) and λ _r (α) of periodic functions by means of the rth order difference operator, where r = 1, 2, ..., and 0 < α ≦ r. We study the smoothness property of a function f with absolutely convergent Fourier series and give best possible sufficient conditions in terms of its Fourier coefficients in order that f belongs to one of the above classes. This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

Characterising the Accuracy of Multivariable Refinable Functions via the Symbol Function

Suppose that f(x) = (f ₁(x),...,f _r (x)) ^T , x∈R ^d is a vector-valued function satisfying the refinement equation f(x) = ∑_Λ c _κ f(2x−κ) with finite set Λ of Z ^d and some r×r matricex c _κ. The requirements for f to have accuracy p are given in terms of the symbol function m(ξ). Supported by NSFC     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

On the discounted global CLT for some weakly dependent random variables

In this paper, we consider L ₁ upper bounds in the global central limit theorem for the sequence of r.v.’s (not necessarily stationary) satisfying the ψ-mixing condition. In a particular case, under the finiteness of the third absolute moments of summands A _i and that of the series ∑ _r⩾1 r ² φ(r), we obtain bounds of order O(n ^−1/2) for Δ _n1:= ∫ _−∞ ^∞ |ℙ{A ₁ + ⋯ + A _n < x} − Φ(x)|dx, where is the standard normal distribution function, and ψ is the function participating in the definition of the ψ-mixing condition. Moreover, we apply the obtained results to get the convergence rate in the so-called discounted global CLT for a sequence of r.v.’s, satisfying the ψ-mixing condition. The bounds obtained provide convergence rates in the discounted global CLT of the same order as in the case of i.i.d. summands with a finite third absolute moment, i.e., of order O((1 − υ)^1/2), where υ is a discount factor, 0 < υ < 1. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 584–597, October–December, 2006.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Sixteen Eisenstein series

S. Ramanujan gave fourteen families of series in his Second Notebook in Chap. 17, Entries 13–17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. The aim of this paper is to develop a powerful tool (four versatile functions f ₀,f ₁,f ₂, and f ₃) to collect all of Ramanujan’s examples together.      

Strong Cesàro summability of double Fourier integrals

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on . Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

On Taylor-like integral representations of analytic functions

Let f be an analytic function in a complex domain D and (without loss of generality) assume 0∈D. Then the paper’s aim is to derive a Taylor-like integral expression for f, i.e. an integral representation analogous to the corresponding power series, say, ∑ _k=0^∞ a _k t ^k /k!. We start from the simplest case f(t)=e ^t , which leads to the identity

Small fractional parts of additive forms

Letf(x)=θ₁ x ₁ ^k +...+θ _s x _s ^k be an additive form with real coefficients, and ∥α∥ = min {|α-u|:uεℤ} denote the distance fromα to the nearest integer. We show that ifθ ₁,…,θ _s , are algebraic ands = 4k then there are integersx ₁,…,x _s , satisfying l ≤x ₁,≤ N and ∥f(x)∥ ≤ N ^E , withE = − 1 + 2/e. Whens = λk, 1 ≤λ ≤ 2k, the exponentE may be replaced byλE/4, and if we drop the condition thatθ ₁,…,θ _s , be algebraic then the result holds for almost all values of θεℝ ^s . Whenk ≥ 6 is small a better exponent is obtained using Heath-Brown’s version of Weyl’s estimate.     

A homotopy operator for Spencer’s sequence in the C ∞-case

The main result of the formal theory of overdetermined systems of differential equations says that any regular system Au = f with smooth coefficients on an open set U ⊂ ℝ ⁿ admits a solution in smooth sections of the bundle of formal power series provided that f satisfies a compatibility condition in U. Our contribution consists in detailed study of the dependence of formal solutions on the point of the base U of the bundle. We also parameterize these solutions by their Cauchy data. In doing so, we prove that, under absence of topological obstructions, there is a formal solution which smoothly depends on the point of the base. This leads to a concept of a finitely generated system (do not mix up it with holonomic or finite -type systems) for which we then prove a C ^∞-Poincaré lemma. The text was submitted by the authors in English.     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

A superadditive property of Hadamard’s gamma function

Hadamard’s gamma function is defined by