A spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.     

On integrality of Eisenstein liftings

Letf be a holomorphic Siegel modular form of integral weightk for Sp_2r (Z). Forn≥r, let[f] _r ⁿ be the lift off to Sp_2n (Z) via the Klingen type Eisenstein series, which is defined under some conditions onk. We study an integrality property of the Fourier coefficients of[f] _r ⁿ . A common denominator for them is described in terms of a critical value of the standardL-function attached tof, some Bernoulli numbers, and a certain ideal depending only onf. The result specialized to the caser=0 coincides with the Siegel-Böcherer theorem on the Siegel type Eisenstein series.     

An Efficient Algorithm for the Complex Roots Problem

Given a univariate polynomialf(z) of degreenwith complex coefficients, whose norms are less than 2^min magnitude, the root problem is to find all the roots off(z) up to specified precision 2^−μ. Assuming the arithmetic model for computation, we provide an algorithm which has complexityO(nlog⁵nlogB), whereb= χ + μ, and χ = max{n,m}. This improves on the previous best known algorithm of Pan for the problem, which has complexityO(n²log²nlog(m+ μ)). A remarkable property of our algorithm is that it does not require any assumptions about the root separation off, which were either explicitly, or implicitly, required by previous algorithms. Moreover it also has a work-efficient parallel implementation. We also show that both the sequential and parallel implementations of the algorithm work without modification in the Boolean model of arithmetic. In this case, it follows from root perturbation estimates that we need only specify θ = ⌈n(B+ logn+ 3)⌉ bits of the binary representations of the real and imaginary parts of each of the coefficients off. We also show that by appropriate rounding of intermediate values, we can bound the number of bits required to represent all complex numbers occurring as intermediate quantities in the computation. The result is that we can restrict the numbers we use in every basic arithmetic operation to those having real and imaginary parts with at most φ bits, where[formula]and[formula]Thus, in the Boolean model, the overall work complexity of the algorithm is only increased by a multiplicative factor ofM(φ) (whereM(ψ) =O(ψ(log ψ) log log ψ) is the bit complexity for multiplication of integers of length ψ). The key result on which the algorithm is based, is a new theorem of Coppersmith and Neff relating the geometric distribution of the zeros of a polynomial to the distribution of the zeros of its high order derivatives. We also introduce several new techniques (splitting sets and “centered” points) which hinge on it. We also observe that our root finding algorithm can be efficiently parallelized to run in parallel timeO(log⁶nlogB) usingnprocessors.     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

Constrained best approximation in Hilbert space III. Applications ton-convex functions

This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on application to then-convex functions. A subtle separation theorem is utilized to significantly extend the results in [4] and to obtain new results even for the “classical” cone of nonnegative functions. It was shown in [4] that finding best approximations inK to anyf inX can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation off from either the coneC or a certain subconeC _F. We will show how to determine this subconeC _F, give the precise condition characterizing whenC _F=C, and apply and strengthen these general results in the practically important case whenC is the cone ofn-convex functions inL ₂ (a,b),     

On linear summation methods of Fourier-Laplace series. II

Let Σ _n be the unit sphere inR ⁿ for somen≥3 with centre at the origin, L(Σ _n ) the space of all functions integrable on Σ _n . We prove a theorem on the representation of functions by singular integrals at double Lebesgue points, which is analogous to a theorem by D. K. Faddeev in the one-dimensional case. On the basis of this theorem, we give necessary and sufficient conditions for the fulfillment of the relation $\mathop {\lim }\limits_{x \to \infty } U_N (f,x,\Lambda ) = f(x)$ for an arbitrary integrable functionf at its double Lebesgue pointsx, where byU _N (f, x Λ) we denote the linear means of the Fourier-Laplace series off defined by means of the triangular matrix $\Lambda = \left\{ {\lambda _k^{(N)} :N = 0,1,...;k = 0,1...,N + 1;\lambda _k^{(N)} = 1,\lambda _{N + 1}^{(N)} = 0} \right\}$      

The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.     

Approximation by non-negative algebraic polynomials

Theorem 1 gives an estimate for the approximation of a continuous functionf by polynomials resulting from the convolution off with non-negative algebraic polynomialsp _n . Jackson's theorem can be deduced from it by choosing a particularp _n whose second Chebyshev-Fourier coefficient is sufficiently close to –1.Work supported in part by the Atomic Energy Commission under contract U.S. AEC AT (11-1) 1469, and in part by the National Science Foundation under grant NSF-GJ-812.     

Polynomial approximation errors for functions of low-order continuity

Given a functionf defined on [-1, 1] we obtain, in terms of (n+1)st divided differences, expressions for the minimax errorE _n(f) and the errorS _n(f) obtained by truncating the Chebyshev series off aftern+1 terms. The advantage of using divided differences is thatf is required to have no more than a continuous second derivative on [-1, 1].     

On multivariate Baskakov operator

In this paper, we study multivariate Baskakov operator Bn,d(f,x). We first show that the operator can retain some properties of the original function f, such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator Bn,d(f,x) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K-functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.     

A generalization of the Buchstab equation

Letf be a multiplicative function and letΨ _f (x, y) denote the incomplete multiplicative sum Σ _n≤x,P(n)≤y f(n), whereP(n) denotes the greatest prime factor ofn. A Buchstab- and a Hildebrand equation forΨ _f (x, y) are derived.     

Kombinatorische Beweise eines Satzes von Birman und Hilden und eines Satzes von Magnus mit der Theorie der automorphen Mengen und der zugehörigen Zopfgruppenoperationen

We deal with the well-known operation ofARTIN?S Braid GroupB _n on the free groupF _n by automorphisms, and give a proof for a theorem ofBIRMAN/HILDEN (here Satz B) by showing, that the images of the generators ofF _n underB _n are of a special form (Satz C). The theory ofBRIESKORN?S Automorphic Sets comes in here. With similar methods we give a proof of a theorem of Magnus saying thatB _n operates on a certain polynomial ring effectively by automorphisms (here Satz 9.2).     

Limit laws of entrance times for homeomorphisms of the circle

Given a homeomorphismf of the circle with irrational rotation number and a descending chain of renormalization intervalsj _n off, we consider for each interval the point process obtained by marking the times for the orbit of a point in the circle to enterJ _n. Assuming the point is randomly chosen by the unique invariant probability measure off, we obtain necessary and sufficient conditions which guarantee convergence in law of the corresponding point process and we describe all the limiting processes. These conditions are given in terms of the convergent subsequences of the orbit of the rotation number off under the Gauss transformation and under a certain realization of its natural extension. We also consider the case when the point is randomly chosen according to Lebesgue measure,f being a diffeomorphism which isC ¹-conjugate to a rotation, and we show that the same necessary and sufficient conditions guarantee convergence in this case.     

The rate of approximation of Gaussian radial basis neural networks in continuous function space

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks(GRBFNs)in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n~d neurons to any continuous function f defined on a compact subset K(R~d)can be controlled by ω(f, n~(-1/2)), where ω(f, t)is the modulus of continuity of the function f .     

A criterion for the uniform distribution of sequences in compact metric spaces

LetX be a compact metric space, le μ be a non-negative normalized Borel measure onX and letf be a measurable bounded real-valued function defined onX such thatf is μ-almost everywhere continuous and different from zero. It is proved that a sequence (x _n ),n=1,2, … of points inX is μ-uniformly distributed if and only if for every Borel setE?X with μ(Bd(E))=0 we have \(\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f(x_n )} 1_E (x_n ) = \int\limits_E {f(x)d\mu (x)} ,\) where 1 _E denotes the characteristic function ofE andbdE the boundary ofE. Furthermore some quantitative aspects and generalizations of this theorem are discussed.     

On the non-amenability of the reflective quotient

Let O(f, ?) be the integral orthogonal group of an integral quadratic form f of signature (n, 1). Let R(f, ?) be the subgroup of O(f, ?) generated by all hyperbolic reflections. Vinberg [Vi3] proved that if n ≥ 30 then the reflective quotient O(f, ?)/R(f, ?) is infinite. In this note we generalize Vinberg’s theorem and prove that if n ≥ 92 then O(f, ?)/R(f, ?) contains a non-abelian free group (and thus it is not amenable).     

Caractérisation spectrale des algèbres de Banach de dimension finie

Let f be an analytic function mapping a domain in $\text{C}$ into a complex Banach algebra. Using potential theory and a new result on almost continuity of the spectrum, which extends the theorem of Newburgh, we prove that either the set of λ such that the spectrum of f(λ) is finite is of outer capacity zero, or there exists an integer n such that the spectrum of f(λ) has at most n elements for every λ. From this we get extensions of a theorem given, in the complex case, by Kaplansky in 1954 and Hirschfeld and Johnson in 1972. More precisely we show that, if the spectrum is finite for every element of an open set of a real algebra or of the set of Hermitian elements of an algebra with an involution, then the quotient of this algebra by its radical is finite-dimensional.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Another Rigidity Theorem for Affine Immersions

The theorem of Beez-Killing in Euclidean differential geometry states as follows [KN, p.46]. Let f: M ⁿ → Rⁿ⁺¹ be an isometric immersion of an n-dimensional Riemannian manifold into a Euclidean (n + l)-space. If the rank of the second fundamental form of f is greater than 2 at every point, then any isometric immersion of M ⁿ into R ^{n +} 1 is congruent to f. A generalization of this classical theorem to affine differential geometry has been given in [O] (see Theorem 1.5). We shall give in this paper another version of rigidity theorem for affine immersions.     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.