Let Lm,p(Rn) be the homogeneous Sobolev space for p?(n,?), ? a Borel regular measure on Rn, and Lm,p(Rn)+Lp(d?) of measurable functions with finite seminorm ?f?Lm,p(Rn)+Lp(d?):=inff1+f2=f?{?f1?Lm,p(Rn)p+?Rn|f2|pd?}1/p. We construct linear operator T:Lm,p(Rn)+Lp(d?)?Lm,p(Rn), that nearly optimally decomposes every function in sum space: ?Tf?Lm,p(Rn)p+?Rn|Tf?f|pd??C?f?Lm,p(Rn)+Lp(d?)p C dependent...

Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of ...

In an earlier paper [SIAM J. Matrix Anal. Appl. vol. 30 (2008), 925–938] we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving singular linear systems whose coefficient matrix is Hermitian and positive semidefinite. In this paper we show in which cases these conditions are also necessary, and show that they are not necesary in othe...

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if p : Ω × Ω → (1,∞) and q : ∂Ω→ (1,∞) are continuous functions such that (n− 1)p(x, x) n− sp(x, x) > q(x) in ∂Ω ∩ {x ∈ Ω: n− sp(x, x) > 0}, then the inequality ‖f‖Lq(·)(∂Ω) ≤ C { ‖f‖Lp̄(·)(Ω) + [f ]s,p(·,·) } holds. Here p̄(x) = p(x, x) and [f ]s,p(·,·) denotes the fractional semi...

We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, specifically by using the BV seminorm. Although our procedure applies in quite general situations it was obtained by geometric...

By a quantum metric space we mean a C∗-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example ...

Let p. be a symmetric p-stable measure, 0 < p < 1, on a locally convex separable linear metric space E and let q be a lower semicontinuous seminorm on E. It is known that F(t) = u{x : q(x) < t) is absolutely continuous with respect to the Lebesgue measure. We prove an explicit formula for the density F'(t) and give an asymptotic estimate of it at infinity. 1. Let X be a symmetric Gaussian rando...

In this paper, a new stable nonconforming mixed finite element scheme is proposed for the stationary conduction-convection problem, in which a new low order Crouzeix-Raviart type nonconforming rectangular element is taken as approximation space for the velocity, the piecewise constant element for the pressure and the bilinear element for the temperature, respectively. The convergence analysis i...

LetM be a compact spin manifold with a smooth action of the ntorus. Connes and Landi constructed θ-deformations Mθ of M , parameterized by n×n real skew-symmetric matrices θ. TheMθ’s together with the canonical Dirac operator (D,H) on M are an isospectral deformation of M . The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show th...