Singular integrals, scale-space and wavelet transforms

Singular integrals, scale-space and wavelet transforms

The Gaussian scale-space is a singular integral convolution operator with scaled Gaussian kernel. For a large class of singular integral convolution operators with differentiable kernels, a general method for constructing mother wavelets for continuous wavelet transforms is developed, and Calderón type inversion formulas, in both integral and semi-discrete forms, are derived for functions in L ...

Singular Integrals on Hilbert Space

exists as a bounded operator on L(H) as ô tends to zero and p tends to infinity. A theorem of this type extends the Calderon-Zygmund theory of singular integral operators on En to infinite dimensions. For if fc(#)||x||-" is a Calderon-Zygmund kernel and if £ is a bounded Borel set which is disjoint from a neighborhood of the origin then v{E) =fEk(x)\\x\\~ dx satisfies v(tE) = v(E) for £>0; if g...

Multiscaled Wavelet Transforms, Ridgelet Transforms, and Radon Transforms on the Space of Matrices

Let Mn,m be the space of real n × m matrices which can be identified with the Euclidean space R. We introduce continuous wavelet transforms on Mn,m with a multivalued scaling parameter represented by a positive definite symmetric matrix. These transforms agree with the polar decomposition on Mn,m and coincide with classical ones in the rank-one case m = 1. We prove an analog of Calderón’s repro...

Singular Integrals

In this paper I will a t tempt to describe the subject as it has developed in the last fifteen years, outlining the methods by which its problems have been approached and discussing its connections with other branches of Analysis. I t will perhaps be best to start by considering certain classical situations which lead naturally to singular integrals and which contain the seeds of some of the me...

Evaluating Singular and Nearly Singular Integrals Numerically