Fully measurable small Lebesgue spaces

Decomposition of Lebesgue Spaces

Spaces of Measurable Transformations

By a space we shall mean a measurable space, i.e. an abstract set together with a <r-ring of subsets, called measurable sets, whose union is the whole space. The structure of a space will be the <r-ring of its measurable subsets. A measurable transformation from one space to another is a mapping such that the inverse image of every measurable set is measurable. Let X and F be spaces, F a set of...

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...

Generalized Lebesgue Spaces and Application to Statistics

Statistics requires consideration of the “ideal estimates” defined through the posterior mean of fractional powers of finite measures. In this paper we study L1= , the linear space spanned by th power of finite measures, 2 (0; 1). It is shown that L1= generalizes the Lebesgue function space L1= ( ), and shares most of its important properties: It is a uniformly convex (hence reflexive) Banach s...