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This work is devoted to image restoration (denoising and deblurring) by variational models. As in our prior work [Inverse Probl. Imaging, 3 (2009), pp. 43--68], the image $\tilde f$ to be restored is assumed to be the sum of a cartoon component $u$ (a function of bounded variation) and a texture component $v$ (an oscillatory function in a Sobolev space with negative degree of differentiability). In order to separate noise from texture in a blurred noisy textured image, we need to collect some information that helps distinguish noise, especially Gaussian noise, from texture. We know that homogeneous Sobolev spaces of negative differentiability help capture oscillations in images very well; however, these spaces do not directly provide clear distinction between texture and noise, which is also highly oscillatory, especially when the blurring effect is noticeable. Here, we propose a new method for distinguishing noise from texture by considering a family of Sobolev norms corresponding to noise and texture. I...

because of the Carleson theorem that (2) holds if and only if every interpolation problem f(zv) = wJn i/ = 1,2,... , for {wv} G f°°, has a solution / G H°°. Although the interpolating Blaschke products comprise a small subset of the set of all Blaschke products, they play a central role in the theory ofH°°. See the last three chapters of [3]. The theorem in this paper helps explain why interpolating Blaschke products are so important in that theory.

of D, then d(Tp,Tq) = d(p,q) for all TeJi\ thus maps in Ji are hyperbolic isometrics. The Schwarz-Pick theorem asserts that if/:Z)->Z) is analytic then / decreases distances, . fl .t . „ d{Ap\M)^d{p,q), (1.1) or lnnnitesimally, i-W/OI 0. Following C. McMullen, we write M(c) for the set of analytic / : / ) -> / ) such that whenever B is a hyperbolic ball in D, diam (/(£)) ^ diam(5)-c, where diam denotes diameter in the hyperbolic metric. For example, n M(C) = jt, c>0 while f(z) = z eM(c) provided c is large. This paper gives three characterizations of the set M(c). The first characterization concerns nearly isometric behavior along certain geodesies, and the second is in terms of angular derivatives at boundary points. Each/eM(c) is a Blaschke product, and the third characterization is by the distribution of the zeros. We thank Curt McMullen for bringing M(c) to our attention and for the results of the next section.

Let ψ be an orientation-reversing homeomorphism of the unit circle onto itself. We consider approximation in certain Sobolev norms by functions of the form f(z) + g(ψ), where / and g are polynomials. The methods involve conf ormal welding and Hardy space theory. We construct a Jordan arc of positive continuous analytic capacity such that the harmonic measures for the two complementary domains are mutually absolutely continuous. Let C(S) denote the space of continuous, complex-valued functions on the unit circle S. For feC(S), let P(f) denote the space of all polynomials in / with complex coefficients. Let z denote the identity function. Browder and Wermer proved the following theorem [3, p. 551]. Let ψ be a direction-reve rsing homeomorphism of S onto S. Then the vector space sum P(z) + P(ψ) is uniformly dense in C(S). In the first part of this paper we prove a result that partially extends this theorem to the C1 norm. We say that a direction-reve rsing homeomorphism ψ:S—*S is an involution if ψoψ = z. Let C^S) denote the space of continuously-differentiable, complex-valued functions on S, with the norm

Those open sets U of S2 for which A(U) is pointwise boundedly dense in H∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in CR(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H∞(U).

For bilipschitz images of Cantor sets in Rd we estimate the Lipschitz har- monic capacity and prove that this capacity is invariant under bilipschitz homeomor- phisms. A crucial step of the proof is an estimate of the L 2 norms of the Riesz tranforms on L 2 (G,p) where p is the natural probability measure on the Cantor set E and G E has p(G) > 0.