Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Summary We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IEX²= but yet satisfying the Law of the Iterated Logarithm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l _p or L _p with 2<p< which satisfy the CLT. As an application we show that in l _p (2<p<) the stochastic boundedness of the weighed partial sums does not imply the CLT.Research partially supported by NSF Grant MCS 75-07605 A01     

Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometrically well-bounded provided the sequence {Uⁿ} _n =– is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2](Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means _n(U, t) of the trigonometric series _k0 k^–e^iktU^k, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {_n(U,t): n 1, t [0,2]} <      

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Localization of the spectrum of certain non-self-adjoint operators

Let the self-adjoint operator A and the bounded operator B be specified in Hilbert space We let denote the spectral family of the operator A. If (E – E _N ) B ₂+E–_NB ² 0 npnN , then in the complex plane z=+ there will exist the curve ¦ ¦ =f (), limf () = 0 for ± such that the entire spectrum of the operator A+B lies within the region ¦ ¦ f(). In particular, the condition of the theorem will be satisfied when B is a completely continuous operator.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 415–420, April, 1968.The author expresses his appreciation to R. S. Ismagilov for his discussion of the results.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

Spectral properties of Neumann Laplacian of horns

We study the Neumann Laplacian of unbounded regions in ⁿ with cusps at infinity so that the corresponding Dirichlet Laplacian has compact resolvent. Typical of our results is that of the region {(x, y)²x, y|<1} the Neumann Laplacian has absolutely continuous spectrum [0, ) of uniform multiplicity four and an infinity of eigenvaluesE _o<E ₁... and that for the region {(x, y)²y|1}, it has absolutely continuous spectrum [1/4, ) of uniform multiplicity 2 and an infinity of eigenvaluesE ₀=0<E ₁.... We use the Enss theory with a suitable asymptotic dynamics.The second author's research is partially funded under NSF grand number DMS-8801918     

On the Space of Sequences of p-Bounded Variation and Related Matrix Mappings

The difference sequence spaces (), c(), and c ₀() were studied by Kzmaz. The main purpose of the present paper is to introduce the space bv _p consisting of all sequences whose differences are in the space _p , and to fill up the gap in the existing literature. Moreover, it is proved that the space bv _p is the BK-space including the space _p . We also show that the spaces bv _p and _p are linearly isomorphic for 1 p . Furthermore, the basis and the -, -, and -duals of the space bv _p are determined and some inclusion relations are given. The last section of the paper is devoted to theorems on the characterization of the matrix classes (bv _p : ), (bv : _p ), and (bv _p : ₁), and the characterizations of some other matrix classes are obtained by means of a suitable relation.     

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.     

Multidimensional Ostrowski inequalities, revisited

New very general multidimensional Ostrowski type inequalities are established, some of them prove to be sharp. They involve the · and ·_p norms of the engaged mixed partial of nth order n1. In establishing them, other important multivariate results of Montgomery type identity are developed and presented for the first time.     

A correction theorem and the dyadic space H1, ∞

It is shown that for every l-function f and for every , >0, there exists a function g such that mes {t=g} <, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (^–1)f. In the proof we make use of the characterization of the dyadic space H^1, in terms of atomic decompositions (it is, apparently, new).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 149, pp. 67–75, 1986.     

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations E_m,n(f)p with undetermined n nodes in the metric of the space L_p(0, 1) (1p) is studied in a class of functionsf(x) for which f ^m+1 (x)L_q(0, 1)1(1qt8) or var {f(m) (x); 0, 1}1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n , and asymptotic formulas are obtained for p= and 1q in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.     

D ∞-Differentials and A ∞-Structures in Spectral Sequences

We introduce the notion of D -differential and use this notion for studying the structure of differential A -algebra on multiplicative spectral sequences. We review basic notions, constructions, and results of our previous papers, where the homotopy invariance of D -differentials was proved, D -differentials and differential perturbations of chain complexes were compared, and the connection between D -differentials and differentials of multiplicative spectral sequences was established. As a consequence of this connection, we describe a method of construction of the structure of A -coalgebra on the Milnor coalgebra dual to the Steenrod algebra immediately by the differentials of the Adams spectral sequence. We describe the method of construction of the structure of A -comodule over a Milnor A -coalgebra on homologies of an arbitrary spectrum.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

Extension of Subgradient Techniques for Nonsmooth Optimization in Banach Spaces

In this paper we continue the study of the subgradient method for nonsmooth convex constrained minimization problems in a uniformly convex and uniformly smooth Banach space. We consider the case when the stepsizes satisfy _{k=1 k} =, lim _{k k} =0.     

Stationary Gaussian processes with a finite correlation function

If the correlation function vanishes outside the segment [–R, R], then an upper estimate (uniform with respect to all such processes) is possible for the probability of the fact that on an other segment [–r, r] the process remains between – and . Such an estimate is obtained, decreasing for 0 asexp(–f(r/R ln ²+ ) and, moreover,r/R may be either 0 or +. The proof is based on an estimate of the form P_mQ_n c_mn P_m Q_n for norms of polynomials on a circle in the complex plane.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 279–288, 1990.