Indices defined by interpolation scales and applications (original) (raw)

Inclusion indices of function spaces and applications

Mathematical Proceedings of the Cambridge Philosophical Society, 2004

We investigate inclusion indices for general function spaces, not necessarily symmetric. Using them, we estimate the grade of proximity between two spaces E → F when we have certain information on the inclusion. The results are based on ideas from interpolation theory.

On strict inclusion relations between approximation and interpolation spaces

Banach Journal of Mathematical Analysis, 2011

Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by Butzer and Scherer in 1968 and, independently, by Y. Brudnyi and N. Kruglyak in 1978, and popularized by Pietsch in his seminal paper of 1981. Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces. In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes (X, {A n }) and sequence spaces S, if S 1 ⊂ S 2 ⊂ c 0 (with strict inclusions) then the approximation space A(X, S 1 , {A n }) is properly contained into A(X, S 2 , {A n }). We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.

Numerical index of vector-valued function spaces

Studia Mathematica, 2000

We show that the numerical index of a c 0-, l 1-, or l ∞-sum of Banach spaces is the infimum numerical index of the summands. Moreover, we prove that the spaces C(K, X) and L 1 (µ, X) (K any compact Hausdorff space, µ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.

On interpolation and integration in finite-dimensional spaces of bounded functions

Communications in Applied Mathematics and Computational Science, 2006

We observe that, under very mild conditions, an n-dimensional space of functions (with a finite n) admits numerically stable n-point interpolation and integration formulae. The proof relies entirely on linear algebra, and is virtually independent of the domain and of the functions to be interpolated.

Interpolation theory and measures related to operator ideals

The Quarterly Journal of Mathematics, 1999

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Interpolation of Operators for a Spaces

2000

Lorentz and Shimogaki [2] have characterized those pairs of Lorentz A spaces which satisfy the interpolation property with respect to two other pairs of A spaces. Their proof is long and technical and does not easily admit to generalization. In this paper we present a short proof of this result whose spirit may be traced to Lemma 4.3 of [4] or perhaps more accurately to the theorem of Marcinkiewicz [5, p. 112]. The proof involves only elementary properties of these spaces and does allow for generalization to interpolation for n pairs and for M spaces, but these topics will be reported on elsewhere. The Banach space A^ [1, p. 65] is the space of all Lebesgue measurable functions ƒ on the interval (0, /) for which the norm is finite, where </> is an integrable, positive, decreasing function on (0, /) and/* (the decreasing rearrangement of |/|) is the almost-everywhere unique, positive, decreasing function which is equimeasurable with \f\. A pair of spaces (A^, A v) is called an interpolation pair for the two pairs (A^, A Vl) and (A^2, A V2) if each linear operator which is bounded from A^ to A v (both /== 1, 2) has a unique extension to a bounded operator from A^ to A v. THEOREM (LORENTZ-SHIMOGAKI). A necessary and sufficient condition that (A^, A w) be an interpolation pair for (A^, A Vi) and (A^2, A V2) is that there exist a constant A independent of s and t so that (*) ^(0/0(5) ^ A max(TO/^(a)) t=1.2 holds, where O 00=ƒ S {r) dr,-" , VaC'Wo Y a (r) dr.

On certain Banach spaces in connection with interpolation theory

Journal of Computational and Applied Mathematics, 1997

By using a norm generated by the error series of a sequence of interpolation polynomials, we obtain in this paper ~ertain Banach spaces. A relation between these spaces and the space (Co, S) with norm generated by the error series of the best polynomial approximations (minimax series) is established.