We prove Lp-boundedness of oscillating multipliers on certain wide classes rank one locally symmetric spaces.

We investigate Besov spaces and their connection with dyadic spline approximation in Lp(Cl), 0 < p < oo. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines.

For Lp spaces on Td, Rd and Sd−1 sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on Td, Rd and Sd−1) for which M(u1/q) is convex for some q, 1 < q ≤ 2, where M(u) is the Orlicz function. Sharp converse inequalities for such spaces are deduced.

We prove that the only rearrangement invariant (r.i.) spaces for which there exists a crossnorm verifying that the tensor product of these spaces preserves the “natural” r.i. space structure, in the sense that it makes the multiplication operator B a topological isomorphism, are the Lp spaces.

We define and characterize the harmonic Besov space Bp, 1 ≤ p ≤ ∞, on the unit ball B in Rn. We prove that the Besov spaces Bp, 1 ≤ p ≤ ∞, are natural quotient spaces of certain Lp spaces. The dual of Bp, 1 ≤ p < ∞, can be identified with Bq , 1/p + 1/q = 1, and the dual of the little harmonic Bloch space B0 is B1.

We study nonlinear approximation in Lp(R) (0 < p < ∞, d > 1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of Rd. To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spac...

A very general uncertainty principle is given for operators on Banach spaces. Many consequences are derived, including uncertainty principles for Bessel sequences in Hilbert spaces and for integral operators between measure spaces. In particular it implies an uncertainty principle for Lp(G), 1 ≤ p ≤ ∞, for a locally compact Abelian group G, concerning simultaneous approximation of f ∈ Lp(G) by ...

Abstract. There are well-known relationships between compressed sensing and the geometry of the finite-dimensional lp spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via l1minimization in terms of the Gelfand widths of certain identity mappings between finitedimensional l1 and l2 spaces, whereas a more recen...

and Applied Analysis 3 Theorem 2.2. Let X be a Hausdorff locally convex space, let R be a family of seminorms on X determining its topology, and let A be a set. Then each x ∈ l A,X 1 ≤ p < ∞ is represented by x Σa∈A Ia ◦ x a , 2.1 where Ia : X → l A,X is defined by Ia t b { t, b a, 0, b / a, b ∈ A. 2.2 Proof. We denote byF the family of all finite subsets of the index setA. We write x Σa∈A Ia◦ ...

For 0 < p < 1 we give examples of Banach spaces isometrically embedding into Lp but not into any Lr with p < r ≤ 1.