Compactness Results for a Class of Limiting Interpolation Methods

We study the behavior of compact operators when we interpolate them by real methods defined through slowly varying functions and rearrangement invariant spaces. We apply these results to prove compactness of certain integral operators acting between grand Lebesgue spaces and between small Lebesgue spaces.

     

Reiteration theorems with extreme values of parameters

We consider real interpolation methods defined by means of slowly varying functions b and symmetric spaces E, for which we present extreme reiteration theorems. As an application we identify, for all possible values of θ∈[0,1], the interpolation spaces (L ₁,LlogL) _θ,b,E and (L _exp,L _∞) _θ,b,E .     

q-concavity and related properties on symmetric sequence spaces

We introduce a new property between the q-concavity and the lower q-estimate of a Banach lattice and we get a general method to construct maximal symmetric sequence spaces that satisfies this new property but fails to be q-concave. In particular this gives examples of spaces with the Orlicz property but without cotype 2.     

Limit cases of reiteration theorems

We consider real interpolation methods defined by means of slowly varying functions and rearrangement invariant spaces, for which we present a collection of reiteration theorems for interpolation and extrapolation spaces. As an application we obtain interpolation formulas for Lorentz‐Karamata type spaces, for Zygmund spaces , and for the grand and small Lebesgue spaces.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Improvement of the Discrete Cosine Transform calculation by means of a recursive method

This paper presents a recursive method for function evaluation. The proposal argues for the use of a more complete primitive, namely a weighted sum, which converts the calculation of the function values into a recursive operation defined by a two input table. The weighted sum can be tuned for different values of the weighting parameters that hold the features of the concrete evaluated function. The calculation of the Discrete Cosine Transform (DCT) is improved by this method which provides a decrease of the operation count as well as an acceptable error bound. These results have repercussions on the compression standard JPEG, for the purposes of storage and transmission in web-based applications