On a joint generalization of Cauchy's and d'Alembert's functional equations

LetX be a locally compact abelian group and let a compact groupG act onX. We find solutionsf: X inL(X) of the following functional equation:

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

On <Emphasis Type="Italic">Γ</Emphasis>-convergence of quadratic functionals in the plane

In this paper, we prove that if a sequence of homeomorphisms , with bounded planar domains, of Sobolev space has uniformly equibounded distortions in EXP(Ω) and weakly converges to f in then the matrices A(x, f _j ) of the corresponding Laplace-Beltrami operators Γ-converge in the Orlicz–Sobolev space , where Q(t) = t ²log(e + t), to the matrix A(x, f) of the Laplace-Beltrami operator associated to f.      

Rational approximation to Lipschitz and Zygmund classes

In this paper, characterizations for lim _n(R _n (f)/(n ^–1)=0 inH and for lim _n(n ^r+ R _n (f)=0 inW ^r Lip ,r1, are given, while, forZ, a generalization to a related result of Newman is established.Communicated by Ronald A. DeVore.     

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

Weighted norm inequalities for geometric fractional maximal operators

For 0 < let Tf denote one of the operators

The Dyadic Cesàro Operator on R+

The dyadic Cesàro operator C is introduced for functions in the space L ¹ := L ¹(R ₊) by means of the Walsh-Fourier transform defined by

The GHS inequality and the Riemann hypothesis

LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4 _– exp(ixt–V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga.     

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, , of Laurent polynomials (L-polynomials) we consider a linear functional which is positive definite on (0, ) and is defined in terms of a given bisequence, { _k } _– . Two sequences of orthogonal L-polynomials, {Q _n (z) ₀ and , are constructed which span in the order {1,z ^–1,z,z ^–2,z ²,...} and {1,z,z ^–1,z ²,z ^–2,...} respectively. Associated sequences of L-polynomials {P _n (z) ₀ , and are introduced and we define rational functions , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M _n (z, w) are given and correspondence of {M _n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M _n (z, w)}. In particular when { _k } _– is a log-normal bisequence, {M _2n (z, w)} and {M _2n+1 (z, w)} yield such solutions.Research supported in part by the National Science Foundation under Grant DMS-9103141.     

On the metric theory of the nearest integer continued fraction expansion

Supposek _n denotes either (n) or (p _n) (n=1,2,...) where the polynomial maps the natural numbers to themselves andp _k denotes thek ^th rationals prime. Also let denote the sequence of convergents to a real numberx and letc _n(x)) _n=1 be the corresponding sequence of partial quotients for the nearest integer continued fraction expansion. Define the sequence of approximation constants _n(x)) _n=1 by

The strongp-variation of martingales and orthogonal series

Let 1p< and letx=(x _n)n0 be a sequence of scalars. The strongp-variation ofx, denoted byW _p (x), is defined as

A Tauberian Theorem for Ergodic Averages,Spectral Decomposability,and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (,) is a -finite measure space, and 1 < p < . Let T:L^p( L ^p() be a bounded invertible linear operator such that T and T ^–1 are positive. Denote by _n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {T^j+T^–j}_{j =1} . Martín-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of E_n(T)_n=1 ; (b) a uniform A _p p estimate in terms of discrete weights generated by the pointwise action on of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {T^j+T^–j}_j=1 already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform A _p weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages E_n(T)_{n = 1} .     

An inverse scattering problem for the Schrödinger equation with a potential having a periodic asymptotics

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = –d ^z/dx ²+q(x)(–<x<)with a potential q(x), satisfying the condition

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that

Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights

The asymptotic behavior of thenth root of the leading coefficient of orthogonal polynomials on (–,) and the distribution of their zeros is studied for nonsymmetric weights that behave like exp(–2B|x|) whenx>0 and exp(–2Ax ) whenx<0,>AB. These results generalize previous investigations of Rakhmanov and Mhaskar and Saff who handle the symmetric caseA=B.Communicated by Paul Nevai.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.     

The symbols of a Frechet algebra of pseudodifferential operators on a polycylinder

H. O. Cordes investigated a C^*-algebra C of singular integral operators on a polycylinder =ⁿ×B and defined symbol homomorphisms where is a certain compact space, and C_x is the Laplace comparison algebra of the compact space B (cf. [2]). We give a characterization of the Frechet algebra CC obtained as the closure of a certain algebra of dos in the topology generated by all H^S norms. We define B:L(H^S)L(H^S)(S) by analogy with [4, 7] and prove that B maps C into C continuously. As a corollary we get , generalizing surjectivity results from [4] and [7]. It seems that no characterization of (C) is known, but it is clear that where is the Frechet algebra studied in [1] and [7].     

A strong approximation for logarithmic averages of partial sums of random variables

LetS _n be the partial sums of -mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S _n / _n ) converges almost surely to _– f(x)d(x). We also obtain strong approximation forH(n)= _k=1 ⁿ k ^–1 f(S _k /k)=logn _– f(x)d(x) which will imply the asymptotic normality ofH(n)/log^1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for -mixing random variables.     

Fourier series of functions whose Hankel transform is supported on [0, 1]

LetJ denote the Bessel function of order . For >–1, the system x^–/2–1/2J_+2n+1(x^1/2, n=0, 1, 2,..., is orthogonal onL ²((0, ),x dx). In this paper we study the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on [0, 1].Communicated by Mourad Ismail.