Summability for Nonunital Spectral Triples

This paper examines the issue of summability for spectral triples for the class of nonunital algebras introduced in [23]. For the case of (p, )-summability, we prove that the Dixmier trace can be used to define a (semifinite) trace on the algebra of the spectral triple. We show this trace is well-behaved, and provide a criteria for measurability of an operator in terms of zeta functions. We also show that all our hypotheses are satisfied by spectral triples arising from geodesically complete Riemannian manifolds. In addition, we indicate how the Local Index Theorem of Connes-Moscovici extends to our nonunital setting.     

Rigid factors of ergodic transformations

We prove a theorem concerning cartesian products of ergodic not necessarily measuring preserving transformations, using the notion of rigid factors for such transformations.     

Summability for solutions to some quasilinear elliptic systems

We prove local regularity in Lebesgue spaces for weak solutions \(u\) of quasilinear elliptic systems whose off-diagonal coefficients are small when \(|u|\) is large: the faster off-diagonal coefficients decay, the higher integrability of \(u\) becomes.     

Summability of Laguerre expansions

В РАБОтЕ пОлУЧЕНы УсИ лЕНИь РЕжУльтАтОВ МА РкЕттА О сУММИРУЕМОстИ МОДИФ ИцИРОВАННых РАжлОжЕНИИ лАгЕРРА. У стАНОВлЕНО, ЧтОα=1/6 Ест ь кРИтИЧЕскИИ ИНДЕкс Д ль сУММИРУЕМОстИ пО ЧЕжАРО. ДОкАжАНО, Чт О пРИα=1/6 сРЕДНИЕ ЧЕжАР О схОДьтсь пОЧтИ ВсУДУ. пОлУЧЕН тАкжЕ АНАлОг тЕОРЕМы ФЕИЕРА-лЕБЕгА.     

Summability and vector amarts

We show that an operator is absolutely summing if and only if it maps amarts into uniform amarts, from which we can deduce a theorem of A. Bellow and another of Edgar-Sucheston. We also show that the absolute value of a Banach lattice valued potential is a potential if and only if the lattice is an A-M space from which we deduce that the L¹-bounded amarts form a Ricsz space if and only if the space is finite dimensional.     

Summability of Hermite Series

Let f∈L ^p (R), 1≤p≤t8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy $\left| {c_j } \right| \cdot f = o\left( 1 \right)\;\quad as\;j \to \infty $ If r=5/4, then the Hermite series Σc _j h _j conerges to f almost everywhere. If r=9/4-1/p, the Σ c _j h _j converges to f in L ^p (R).     

Summability in topological spaces

The main purpose of the paper is to introduce the notion of summability in abstract Hausdorff topological spaces. We give a characterization of such summability methods when the space allows a countable base. We also provide several Tauberian theorems in topological structures. Some open problems are discussed.     

Summability of Hermite Series

Let fL ^p (R), 1pt8, and c _j be the inner product of f and the Hermite function h _j . Assume that c _j 's satisfy If r=5/4, then the Hermite series c _j h _j conerges to f almost everywhere. If r=9/4-1/p, the c _j h _j converges to f in L ^p (R).     

On a New Summability Result for Operator Matrices

For matrices of quasi-homogeneous operators we establish a summability result which is a useful complement of a recent summability result.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Pointwise Summability of Gabor Expansions

A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L ₁ or, more generally, when f∈W(L ₁,ℓ _∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.     

Summability results for operator matrices on topological vector spaces

Basic summability results are established for matrices of linear and some nonlinear mappings between topological vector spaces.