We consider random functions defined by atomic decompositions in a wavelet dictionary. The scale and dilation of the wavelet atoms are not dyadic constraints, but the function is modelled as a sum of wavelet functions at arbitrary positions and scales. The locations of the wavelet atoms and the magnitudes of their coefficients are chosen with respect to a certain marked Poisson process model, a...

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B π,q on a regular domain of R . The result is: if s − d(1/π − 1/p)+ > 0, then the Kolmogorov metric entropy satisfies H( ) ∼ −d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower ...

In this paper it is shown that recent approximation results for scalar conservation laws in one space dimension imply that solutions of these equations with smooth, convex fluxes have more regularity than previously believed. Regularity is measured in spaces determined by quasinorms related to the solution’s approximation properties in L1(R) by discontinuous, piecewise linear functions. Using a...

Motivated by boundary value problems for partial differential equations, classical trace and extension theorems characterize traces of spaces of generalized smoothness (e.g., Sobolev, Besov, etc.) to smooth submanifolds of a Euclidean space. But in many cases one needs similar results for subsets of a more complicated geometric structure (for instance, after the change of variables initial data...

Profile decompositions for “critical” Sobolev-type embeddings are established, allowing one to regain some compactness despite the non-compact nature of the embeddings. Such decompositions have wide applications to the regularity theory of nonlinear partial differential equations, and have typically been established for spaces with Hilbert structure. Following the method of S. Jaffard, we treat...

This paper examines asymptotic equivalence in the sense of Le Cam between density estimation experiments and the accompanying Poisson experiments. The significance of asymptotic equivalence is that all asymptotically optimal statistical procedures can be carried over from one experiment to the other. The equivalence given here is established under a weak assumption on the parameter space F . In...

We continue the study of randomized approximation of embeddings between Sobolev spaces on the basis of function values. The source space is a Sobolev space with nonnegative smoothness order, the target space has negative smoothness order. The optimal order of approximation (in some cases only up to logarithmic factors) is determined. Extensions to Besov and Bessel potential spaces are given and...

The inclusions between the Besov spaces $B^q$, Bloch space $\mathcal{B}$ and standard weighted Bergman $A^p_\alpha$ are completely understood, but norms of corresponding inclusion operators in general unknown. In this work, we compute or estimate asymptotically these inclusions.

We investigate the theoretical performance of Bayes factor estimators in wavelet regression models with independent and identically distributed errors that are not necessarily normally distributed. We compare these estimators in terms of their frequentist optimality in Besov spaces for a wide variety of error and prior distributions. Furthermore, we provide sufficient conditions that determine ...