We prove the $L^2{(}^2)$ boundedness of the oscillatory singular integrals $P_{0}f(x,y)={\int }_{D_x}{e}^{i(M_2\left(x\right)y^{\text{'}}+M_1\left(x\right)x^{\text{'}})}over{x}^{\text{'}}{y}^{\text{'}}f(x-x^{\text{'}},y-y^{\text{'}})d{x}^{\text{'}}d{y}^{\text{'}}$ for arbitrary real-valued $L^{\infty }$ functions $M_1\left(x\right),M_2\left(x\right)$ and for rather general domains $D_x{\subseteq }^2$ whose dependence upon x satisfies no regularity assumptions.

The affine systems generated by Ψ ⊂ L2(Rn) are the systemsAA(Ψ) = {DjA Tk Ψ : j ∈ Z, k ∈ Zn},where Tk are the translations, and DA the dilations with respect to an invertible matrix A. As shown in [5], there is a simple characterization for those affine systems that are a Parseval frame for L2(Rn). In this paper, we correct an error in the proof of the characterization result from [5], by redefining the class of not-necessarily expanding dilation matrices for which this characterization result holds....

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, IsraelAttempts at extending spline subdivision schemes to operate on compact sets are reviewed. The aim is to develop a procedure for approximating a set-valued function with compact images from a finite set of its samples. This is motivated by the problem of reconstructing a 3D object from...

We study the bases and frames of reproducing kernels in the model subspaces $K_{\Theta }^2=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{{\lambda }_n}\left(z\right)=(1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right))/(z-{\overline{\lambda }}_n)$ under “small” perturbations of the points ${\lambda }_n$. We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces $K_{\Theta }^2$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

The notion of “strong boundary values” was introduced by the authors in the local theory of hyperfunction boundary values (boundary values of functions with unrestricted growth, not necessarily solutions of a PDE). In this paper two points are clarified, at least in the global setting (compact boundaries): independence with respect to the defining function that defines the boundary, and the spaces of test functions to be used. The proofs rely crucially on simple results in spectral asymptotics.