Wavelet-based image denoising algorithm depends upon the energy compaction property of wavelet transforms. However, for many real-world images, we cannot expect good energy compaction in a single wavelet domain, because most real-world images consist of components of a variety of smoothness. We can relieve this problem by using multiple wavelet bases to match different characteristics of images...

Let (M,ρ, μ) be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces B p,q (M) and Triebel–Lizorkin-type spaces F s,τ p,q (M) associated to a non-negative self-adjoint operator L whose heat kernels satisfy some Gaussian upper bound estimate, Hölder continuity, and the stochastic completeness property. Characterizations of these spaces via P...

The fact that symbols in the modulation space M1,1 generate pseudo-differential operators of the trace class was first mentioned by Feichtinger and the proof was given by Gröchenig [12]. In this paper, we show that the same is true if we replace M1,1 by more general α-modulation spaces which include modulation spaces (α = 0) and Besov spaces (α = 1) as special cases. The result with α = 0 corre...

In this paper, we show that the solution of the supercrtical surface quasi-geostrophic (SQG) equation, with initial data in a critical Besov space, belongs to a subanalytic Gevrey class. In order to prove this, a suitable estimate on the nonlinear term, in the form of a commutator, is required. We express the commutator as a bilinear multiplier operator and obtain single-scale estimates for its...

In this work we investigate a technique for accelerating convergence of adjoint–based optimization of PDE systems based on a nonlinear change of variables in the control space. This change of variables is accomplished in the differentiate–then–discretize approach by constructing the descent directions in a control space not equipped with the Hilbert structure. We show how such descent direction...

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...

Seeger replaced the Sobolev space L2s with s > n/2 appeared in the Hörmander multiplier theorem by the Besov space B n/2 . In this paper, we replace L2s with s > n/2 by the modulation space M 2,1 s with s > 0.

The problem of density estimation on R is considered. Adopting the maxiset point of view, we focus on performance of adaptive procedures. Any rule which consists in neglecting the wavelet empirical coefficients smaller than a sequence of thresholds vn will be called an elitist rule. We prove that for such a procedure the maximal space for the rate v n , with 0 < α < 1, is always contained in th...

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted L classes. We establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solv...