Taylor type formula with Fréchet polynomials

We extend Taylor’s formula for functions defined on uniquely 2-divisible Abelian semigroups with values in Banach spaces. As corollaries, we give sufficient conditions for the representation of such functions as sums of generalized power series, and also a stability criterion for Fréchet’s polynomial equation which extends the known results.     

Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

Productively Fréchet Spaces

We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.     

Fréchet Embeddings of Negative Type Metrics

We show that every n-point metric of negative type (in particular, every n-point subset of L ₁) admits a Fréchet embedding into Euclidean space with distortion , a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space. S. Arora supported by David and Lucile Packard Fellowship and NSF grant CCR-0205594. J.R. Lee supported by NSF grant CCR-0121555, NSF 0514993, NSF 0528414 and an NSF Graduate Research Fellowship.     

Smooth Fréchet globalizations of Harish-Chandra modules

We give an alternative proof of the Casselman-Wallach globalization theorem. The approach is based on lower bounds for matrix coefficients on a reductive group.     

Characterizations of nuclearity in Fréchet spaces

We extend the result of A. Bellow (Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799) on the characterization of finite-dimensional Banach spaces, to a characterization of nuclearity for Fréchet spaces. Those spaces are nuclear iff every Pettis-bounded and Pettis-uniformly integrable amart is mean convergent. Several other characterizations are given.     

Block sequences in nuclear Fréchet spaces with basis

It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Fréchet space is isomorphic to = R^N, N = {1,2,...} if it has a basis such that any block sequence with blocks of length 2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.Translated from Matematicheskie Zametki, Vol. 17, No. 6, pp. 899–908, June, 1975.The author wishes to thank B. S. Mityagin for calling his attention to this problem and for his valuable suggestions.     

Coorbit Spaces with Voice in a Fréchet Space

We set up a new general coorbit space theory for reproducing representations of a locally compact second countable group G that are not necessarily irreducible nor integrable. Our basic assumption is that the kernel associated with the voice transform belongs to a Fréchet space \(\mathcal T\) of functions on G, which generalizes the classical choice \(\mathcal T=L_w^1(G)\). Our basic example is \( \mathcal T=\bigcap _{p\in (1,+\infty )} L^p(G)\), or a weighted versions of it. By means of this choice it is possible to treat, for instance, Paley-Wiener spaces and coorbit spaces related to Shannon wavelets and Schrödingerlets.     

On the Fréchet derivative of matrix functions

In 1956 Rinehart [4] discussed the derivatives of matrix functions by considering differences ƒ(A + E) − ƒ(A) for matrices E commuting with A. In that case the derivative turned out to be ƒ′(A). In this paper the case of noncommutative A and E is treated. This leads to the Fréchet derivative of the matrix function ƒ. An explicit integral representation is obtained. Using an approach that is similar to the one in [5], a finite sum in polynomials of A is obtained. The coefficients may be computed recursively. This is useful in computing the Fréchet derivative which is needed for Newton's method to solve nonlinear matrix equations.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Resolutions of the identity in Fréchet spaces

It is a classical result that every Bade -complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Fréchet spaces. Namely, every Bade -complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset of.     

On Topological Classification of Non-Archimedean Fréchet Spaces

We prove that any infinite-dimensional non-archimedean Fréchet space E is homeomorphic to where D is a discrete space with card(D) = dens(E). It follows that infinite-dimensional non-archimedean Fréchet spaces E and F are homeomorphic if and only if dens(E) = dens(F). In particular, any infinite-dimensional non-archimedean Fréchet space of countable type over a field is homeomorphic to the non-archimedean Fréchet space .