zbMATH — the first resource for mathematics

Wavelets and operators. (English) Zbl 0810.42015
Daubechies, Ingrid (ed.), Different perspectives on wavelets. American Mathematical Society short course on wavelets and applications, held in San Antonio, TX (USA), January 11-12, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Appl. Math. 47, 35-58 (1993).
This is an expository paper concerned with such topics as linear operators on the \(L^ 2\)-space, singular integral operators, frames in a Hilbert space (a substitute for approximation purposes of an orthonormal system), pseudo-differential operators, wavelets and their use in constructing approximations for linear (bounded or not) operators. Many other related concepts are involved in the presentation. The focus is on the so-called Calderón-Zygmund operators which are defined by \[ (Tf)(x)= \text{p.v.} \int K(x,y)f(y)dy, \] with \(K(x,y)\) satisfying the following conditions: \[ (a)\quad K(y,x)= -K(x,y);\qquad(b)\quad | K(x,y)|\leq C_ 0| x- y|^{-n}; \] (c) there exists an exponent \(\gamma\in (0,1)\) and a constant \(C\) such that for \(| x'- x|\leq | x- y|/2\), \[ | K(x',y)- K(x,y)|\leq C| x'- x|^ \gamma | x-y|^{-n-\gamma}. \] Using wavelets one constructs an approximation for the operator \(T\), say \(T_ m\), for which the following estimate is indicated: \(\| T- T_ m\|\leq Cm^{- \gamma}\sqrt{\log m}\), for each \(m\geq 1\), with \(C= C(n,\gamma)\).
For the entire collection see [Zbl 0782.00059].

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
35S05 Pseudodifferential operators as generalizations of partial differential operators
44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)