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Fractional powers of operators, K-functionals, Ulyanov inequalities

Walter Trebels, Ursula Westphal (2010)

Banach Center Publications

Given an equibounded (₀)-semigroup of linear operators with generator A on a Banach space X, a functional calculus, due to L. Schwartz, is briefly sketched to explain fractional powers of A. Then the (modified) K-functional with respect to ( X , D ( ( - A ) α ) ) , α > 0, is characterized via the associated resolvent R(λ;A). Under the assumption that the resolvent satisfies a Nikolskii type inequality, | | λ R ( λ ; A ) f | | Y ≤ c φ ( 1 / λ ) | | f | | X , for a suitable Banach space Y, an Ulyanov inequality is derived. This will be of interest if one has good control...

Functional characterization of best and good approximation in normed product spaces.

Carlos Benítez, Manuel Fernández (1987)

Extracta Mathematicae

In [6], C. Dierick deals with a small but important collection of norms in the product of a finite number of normed linear spaces and he extends to such products some results on functional characterization of best approximations. In this paper we establish the widest scope in which the mentioned results remain valid.

Greedy approximation and the multivariate Haar system

A. Kamont, V. N. Temlyakov (2004)

Studia Mathematica

We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in L p ( [ 0 , 1 ] ) , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis d = × . . . × in L p ( [ 0 , 1 ] d ) ,...

Implicit difference methods for nonlinear first order partial functional differential systems

Elżbieta Puźniakowska-Gałuch (2010)

Applicationes Mathematicae

Initial problems for nonlinear hyperbolic functional differential systems are considered. Classical solutions are approximated by solutions of suitable quasilinear systems of difference functional equations. The numerical methods used are difference schemes which are implicit with respect to the time variable. Theorems on convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability is based on a comparison technique with nonlinear estimates...

Invariant approximation for fuzzy nonexpansive mappings

Ismat Beg, Mujahid Abbas (2011)

Mathematica Bohemica

We establish results on invariant approximation for fuzzy nonexpansive mappings defined on fuzzy metric spaces. As an application a result on the best approximation as a fixed point in a fuzzy normed space is obtained. We also define the strictly convex fuzzy normed space and obtain a necessary condition for the set of all t -best approximations to contain a fixed point of arbitrary mappings. A result regarding the existence of an invariant point for a pair of commuting mappings on a fuzzy metric...

Kaczmarz algorithm in Hilbert space

Rainis Haller, Ryszard Szwarc (2005)

Studia Mathematica

The aim of the Kaczmarz algorithm is to reconstruct an element in a Hilbert space from data given by inner products of this element with a given sequence of vectors. The main result characterizes sequences of vectors leading to reconstruction of any element in the space. This generalizes some results of Kwapień and Mycielski.

Kaczmarz algorithm with relaxation in Hilbert space

Ryszard Szwarc, Grzegorz Świderski (2013)

Studia Mathematica

We study the relaxed Kaczmarz algorithm in Hilbert space. The connection with the non-relaxed algorithm is examined. In particular we give sufficient conditions when relaxation leads to the convergence of the algorithm independently of the relaxation coefficients.

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