Microlocal analysis of a seismic linearized inverse problem. Abstract: The seismic inverse problem is to determine the wavespeed c(x) in the interior of a medium from measurements at the boundary. In this paper we analyze the linearized inverse problem in general acoustic media. The problem is to nd a left inverse of the linearized forward map F, or, equivalently, to nd the inverse of the norma...

In the present work the space  $L_{p;r} $ which is continuously embedded into $L_{p} $  is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is...

In this paper we apply a quadrature method based on the tensor product trapezoidal rule to the solution of a singular integral equation over the two-dimensional torus. We prove that this method is stable if and only if a certain numerical symbol does not vanish. For a special kernel function, we present a plot of numerically computed symbol values and, for symmetric kernels (Mikhlin-Giraud kern...

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the BeurlingAhlfors singular integral operator, acting on LpX(R ) with p ∈ (1,∞). Moreover, replacing equality by a linear equivalence, this is fo...

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1 |t−s|μ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On t...