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In this paper we consider the INTERPOLATION by certain functions such as trigonometric and rational functions for finite dimensional linear space X. Then we extend this to infinite dimensional linear SPACES.

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, {An}) and sequence SPACES S, if S1ÌS2Ìc0 (with strict inclusions) then the approximation space A(X, S1, {An}) is properly contained into A(X, S2, {An}). We also initiate a study of strict inclusions between INTERPOLATION SPACES, for Petree’s real INTERPOLATION method.

Let (X, d, m) be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors establish an INTERPOLATION result that a sublinear operator which is bounded from the Hardy space H1(m) to L1,¥(m) and from L¥ (m) to the BMO-type space RBMO (m) is also bounded on Lp(m) for all pÎ(1, ¥). This extension is not completely straightforward and improves the existing result.