Gregory Beylkin

On the representation of operators in bases of compactly supported wavelets

This paper describes exact and explicit representations of the differential operators,

Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform

{{d^n } / {dx^n }}

A new slant on seismic imaging: Migration and integral geometry

,

Discrete radon transform

n = 1,2, \cdots

Wavelet-like bases for the fast solutions of second-kind integral equations

, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring

Algorithms for Numerical Analysis in High Dimensions

O(N\log N)

Linearized inverse scattering problems in acoustics and elasticity

operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length

Numerical operator calculus in higher dimensions

N = 2^n

Multiresolution quantum chemistry: Basic theory and initial applications

is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from

Multiresolution representations using the autocorrelation functions of compactly supported wavelets

O(N)