Decomposition of Lebesgue Spaces

Decomposition of Lebesgue Spaces

On isomorphism of two bases in Morrey-Lebesgue type spaces

Double system of exponents with complex-valued coefficients is considered. Under some conditions on the coefficients, we prove that if this system forms a basis for the Morrey-Lebesgue type space on $left[-pi , pi right]$, then it is isomorphic to the classical system of exponents in this space.

Affine Synthesis onto Lebesgue and Hardy Spaces

The affine synthesis operator Sc = P j>0 P k∈Zd cj,kψj,k is shown to map the mixed-norm sequence space `(`) surjectively onto L(R) under mild conditions on the synthesizer ψ ∈ L(R), 1 ≤ p < ∞, with R Rd ψ dx = 1. Here ψj,k(x) = |det aj |ψ(ajx−k), and the dilation matrices aj expand, for example aj = 2I . Affine synthesis further maps a discrete mixed Hardy space `(h) onto H(R). Therefore the H-...

The Sampling Theorem in Variable Lebesgue Spaces

hold. The facts above are well-known as the classical Shannon sampling theorem initially proved by Ogura [10]. Ashino and Mandai [1] generalized the sampling theorem in Lebesgue spaces L0(R) for 1 < p0 < ∞. Their generalized sampling theorem is the following. Theorem 1.1 ([1]). Let r > 0 and 1 < p0 < ∞. Then for all f ∈ L 0(R) with supp f̂ ⊂ [−rπ, rπ], we have the norm inequality C p r ‖f‖Lp0(Rn...

The Lebesgue decomposition theorem for arbitrary contents

The decomposition theorem named after Lebesgue asserts that certain set functions have canonical representations as sums of particular set functions called the absolutely continuous and the singular ones with respect to some fixed set function. The traditional versions are for the bounded measures with respect to some fixed measure on a σ algebra, in final form due to Hahn 1921, and for the bou...