Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations

Optimal Sobolev Embeddings on R

The aim of this paper is to study Sobolev-type embeddings and their optimality. We work in the frame of rearrangement-invariant norms and unbounded domains. We establish the equivalence of a Sobolev embedding to the boundedness of a certain Hardy operator on some cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the ...

Optimal Domain Spaces in Orlicz-sobolev Embeddings

We deal with Orlicz-Sobolev embeddings in open subsets of R. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given Orlicz space. Moreover, the optimal Orlicz-Sobolev space is exhibited whenever it exists. Parallel questions are addressed for Orlicz-Sobolev embeddings into Orlicz spaces ...

Optimal Sobolev embeddings and Function Spaces

Further characterizations of Sobolev spaces

Let (Fn)n∈N be a sequence of non-decreasing functions from [0,+∞) into [0,+∞). Under some suitable hypotheses on (Fn)n∈N, we prove that if g ∈ Lp(RN ), 1 < p < +∞, satisfies lim inf n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy < +∞, then g ∈ W1,p(RN ) and moreover lim n→∞ ∫ RN ∫ RN Fn(|g(x)− g(y)|) |x − y|N+p dx dy = KN,p ∫ RN |∇g(x)| dx, whereKN,p is a positive constant depending only onN a...

Quantum approximation II. Sobolev embeddings

A basic problem of approximation theory, the approximation of functions from the Sobolev space W r p ([0, 1] ) in the norm of Lq([0, 1] ), is considered from the point of view of quantum computation. We determine the quantum query complexity of this problem (up to logarithmic factors). It turns out that in certain regions of the domain of parameters p, q, r, d quantum computation can reach a sp...