1. Provocative example 2. Natural function spaces on the circle S = R/2πZ 3. Topology on C∞(S1) 4. Pointwise convergence of Fourier series 5. Distributions: generalized functions 6. Invariant integration, periodicization 7. Hilbert space theory of Fourier series 8. Levi-Sobolev inequality, Levi-Sobolev imbedding 9. C∞(S1) = limC(S) = limH(S) 10. Distributions, generalized functions, again 11. T...

We study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tr...

In this paper we analyze the approximation by standard piecewise linear finite elements of a non homogeneous Neumann problem in a cuspidal domain. Since the domain is not Lipschitz, many of the results on Sobolev spaces which are fundamental in the usual error analysis do not apply. Therefore, we need to work with weighted Sobolev spaces and to develop some new theorems on traces and extensions...

By proving a weighted contraction estimate in uniformly local Sobolev spaces for the flow of gravity water waves, we show that this nonlocal system is in fact pseudo-local in the following sense: locally in time, the dynamic far away from a given bounded region has a small effect on that region (again, in a sense that we will make precise in the article). Our estimate on the flow also implies a...

Let Ω be an arbitrary bounded domain of Rn. We study the right invertibility of the divergence on Ω in weighted Lebesgue and Sobolev spaces on Ω, and relate this invertibility to a geometric characterization of Ω and to weighted Poincaré inequalities on Ω. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when Ω is Lipschitz or, more ge...

We show that the nonlinear wave equation u + ut = 0 is globally well-posed in radially symmetric Sobolev spaces Hk rad(R 3) × Hk−1 rad (R 3) for all integers k > 2. This partially extends the well-posedness in Hk(R3) × Hk−1(R3) for all k ∈ [1, 2], established by Lions and Strauss [12]. As a consequence we obtain the global existence of C∞ solutions with radial C∞ 0 data. The regularity problem ...

This paper investigates statistical performances of Support Vector Machines (SVM) and considers the problem of adaptation to the margin parameter and to complexity. In particular we provide a classifier with no tuning parameter. It is a combination of SVM classifiers. Our contribution is two-fold: (1) we propose learning rates for SVM using Sobolev spaces and build a numerically realizable aggr...

We consider an oblique derivative problem for non-divergence parabolic equations with discontinuous in t coefficients in a half-space. We obtain weighted coercive estimates of solutions in anisotropic Sobolev spaces. We also give an application of this result to linear parabolic equations in a bounded domain. In particular, if the boundary is of class C1,δ, δ ∈ (0, 1], then we present a coerciv...

1 2 The paper deals with the basic integral equation of random field estimation theory by the criterion of minimum of variance of the error estimate. This integral equation is of the first kind. The corresponding integral operator over a bounded domain Ω in Rn is weakly singular. This operator is an isomorphism between appropriate Sobolev spaces. This is proved by a reduction of the integral eq...

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains O ⊂ R with both theoretical and numerical purpose. We use N.V. Krylov’s framework of stochastic parabolic weighted Sobolev spaces H p,θ(O, T ). The summability parameters p and q in space and time may differ. Existence and uniqueness of solutions in these sp...