Ratio Tauberian Theorems for Positive Functions and Sequences in Banach Lattices

We prove two ratio Tauberian theorems and deduce two generalized Tauberian theorems for functions and sequences with values in positive cones of Banach lattices. Two counter-examples are given to show that the hypotheses in the ratio Tauberian theorems are essential.     

Ratio limit theorems and Tauberian theorems for vector-valued functions and sequences

We prove ratio limit theorems for (C,γ)-means (γ?0) and Abel means of functions and sequences in Banach spaces, and ratio Tauberian theorems for (C,γ)-means (γ?1) and Abel means of functions and sequences in Banach lattices.     

Fixed point theorems for one-parameter asymptotically nonexpansive semigroups in general Banach spaces

In this paper, we first prove two fixed points theorems for one-parameter asymptotically nonexpansive semigroups in general Banach spaces. Using these results, we prove a strong convergence theorem of Mann's type sequences for the asymptotically nonexpansive semigroups. This is a generalization of the result of Suzuki and Takahashi for one-parameter nonexpansive semigroups in general Banach spaces.     

Tauberian theorems for weighted means of double sequences

We discuss the relations between weighted mean methods and ordinary convergence for double sequences. In particular, we study Tauberian theorems also for methods not being products of the related one-dimensional summability methods. For the special case of theC _1,1-method, the results contain a classical Tauberian theorem by Knopp [9] as special case and generalize theorems given by Móricz [16] thereby showing that one of his Tauberian conditions can be weakened.     

Comparison theorems for eigenvalues

Summary Comparison theorems are proved relating the smallest positive eigenvalues λ and λ* of two operator equations of type Au=λBu and A*v=λ*B*v, respectively. Sufficient conditions on the operators are given which guarantee that λ⩽λ*, with special reference to the case that A and A* are elliptic differential operators. One novelty of the theory is that B is not required to be positive. Two general techniques are described: 1) A generalization of the classical minimum principle for eigenvalues, which is appropriate for selfadjoint elliptic operators A of arbitrary even order; and 2) A differential identity related to Picone's identity, appropriate for nonselfadjoint second order elliptic operators and strongly elliptic quasilinear systems. Entrata in Redazione il 24 maggio 1972.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

Harmonic Analysis in the p-Adic Lizorkin Spaces: Fractional Operators, Pseudo-Differential Equations, p-Adic Wavelets, Tauberian Theorems

In this article the p-adic Lizorkin spaces of test functions and distributions are introduced. Multi-dimensional Vladimirov’s and Taibleson’s fractional operators, and a class of p-adic pseudo-differential operators are studied on these spaces. Since the p-adic Lizorkin spaces are invariant under these operators, they can play a key role in considerations related to fractional operator problems. Solutions of pseudo-differential equations are also constructed. Some problems of spectral analysis of pseudo-differential operators are studied. p-Adic multidimensional Tauberian theorems connected with these pseudo-differential operators for the Lizorkin distributions are proved.     

Statistical gap Tauberian theorems in metric spaces

By using the concept of statistical convergence we present statistical Tauberian theorems of gap type for the Cesàro, Euler-Borel family and the Hausdorff families applicable in arbitrary metric spaces. In contrast to the classical gap Tauberian theorems, we show that such theorems exist in the statistical sense for the convolution methods which include the Taylor and the Borel matrix methods. We further provide statistical analogs of the gap Tauberian theorems for the Hausdorff methods and provide an explanation as to how the Tauberian rates over the gaps may differ from those of the classical Tauberian theorems.     

Bernstein functions and rates in mean ergodic theorems for operator semigroups

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form g(A, where ?A is the generator of the semigroup and g is a Bernstein function. In addition, we obtain some new results on Bernstein functions which are of independent interest.     

Browder’s type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces

In this paper, we prove Browder’s type convergence theorems for one-parameter strongly continuous semigroups of nonexpansive mappings in Banach spaces. The author is supported in part by Grants-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology.     

Tauberian and Abelian theorems for correlation function of a homogeneous isotropic random field

We prove theorems of Tauberian and Abelian types for nonintegrable correlation functions of homogeneous isotropic random fields and use them to study asymptotic distributions of local functionals of Gaussian fields.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1652–1664, December, 1991.     

Strong ergodic theorems for commutative semigroups of operators

We prove strong mean convergence theorems and the existence of ergodic projection and retraction for commutative semigroups of operators which is Eberlein-weakly almost periodic.

     

Abstract Korovkin-type theorems in modular spaces and applications

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones.     

Tauberian theorems and mean values of multiplicative functions

One gives an application of Tauberian theorems to the problem of connection between the mean values of multiplicative functions for the cases when the argument runs through all natural numbers and all prime numbers, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 145–157, 1979.The author dedicates this paper to A. I. Vinogradov on the occasion of his 50th birthday.     

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Asymptotic resolvent and theorems on generation of semigroups of operators in locally convex spaces

The concept of the asymptotic resolvent of an operator is introduced and used to establish theorems on the generation of semigroups of distributions and operator semigroups of class (ℒ _loc ^p , ℬ) in a locally convex space. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 433–442, September, 1977.     

Fixed points of asymptotically regular semigroups in Banach spaces

In this paper we study in Banach spaces the existence of fixed points of (nonlinear) asymptotically regular semigroups. We establish for these semigroups some fixed point theorems in spaces with weak uniform normal structure, in a Hilbert space, inL ^p spaces, in Hardy spacesH ^p and in Sobolev spacesW ^r.p for 1<p<∞ andr≥0, in spaces with Lifshitz’s constant greater than one. These results are the generalizations of [8, 10, 16].     

Principles of functional analysis in scales of spaces: Hahn-Banach theorem, Banach theorem on homomorphisms, and theorems on open mappings and closed graphs

The Hahn-Banach theorem, the Banach theorem on homomorphisms, and theorems on open mappings and closed graphs are transferred to functionals and operators acting in inductive scales of spaces. The applications to operators acting in inductive limits and dual spaces are considered. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 153–173, 2005.     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].