WAVELET TRANSFORM OF PERIODIC GENERALIZED-FUNCTIONS

The aim of this paper is to define the wavelet transform for spaces of periodic functions, then extend this definition to spaces of generali zed functions larger than the space of periodic Schwartz distributions , such as spaces of periodic Beurling ultradistributions and hyperfunc tions on the unit circle. It is shown that the wavelet transforms of s uch generalized functions are nice and smooth functions defined on an infinite cylinder, provided that the analyzing wavelet is also nice an d smooth. For example, it is shown that the growth rate of the derivat ives of the wavelet transform is almost as good as that of the analyzi ng wavelet. More precisely, if the mother wavelet g satisfies Sup(x is -an-element-of R)\x(g)g(q)(x)\ less-than-or-equal-to CA(k)B(q)k(kbeta) q(qalpha) (k, q = 0, 1, 2, ...), then the wavelet transform W(g)(f) of a periodic Beurling ultradistribution f satisfies sup(r,theta) is-an- element-of Y epsilon\r(k) partial derivative(theta)p partial derivativ e(r)q)W(g)(f)(r, theta)\ less-than-or-equal-to DA(k)k(alphak)B(p)C(q)p (palpha)q(q)(alpha + beta); k, p, q greater-than-or-equal-to 0, where Y(epsilon) = {(r, theta): r greater-than-or-equal-to epsilon > 0, thet a is-an-element-of T}. (C) 1994 Academic Press, Inc.