Complexity of oscillatory integration for univariate Sobolev spaces

Complexity of oscillatory integration for univariate Sobolev spaces

We analyze univariate oscillatory integrals for the standard Sobolev spaces Hs of periodic and non-periodic functions with an arbitrary integer s ≥ 1. We find matching lower and upper bounds on the minimal worst case error of algorithms that use n function or derivative values. We also find sharp bounds on the information complexity which is the minimal n for which the absolute or normalized er...

Boundedness of Oscillatory Singular Integrals on Weighted Sobolev Spaces

In this paper, an oscillatory singular integral operator T deened by T f (x) = Z IR e ixP(y) f (x ? y) y dy is showed to be bounded on a weighted Sobolev space H

Sobolev Spaces

Global automorphic Sobolev spaces

The goal is legitimization of term-wise differentiation of L spectral expansions, so that computations producing a classical outcome are correct. We are fond of L expansions because they are what Plancherel gives. Typically, L expansions are not continuous, much less differentiable, so the issue cannot be proving classical differentiability, which does not hold. To say that L spectral expansion...

Lifting in Sobolev Spaces

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...