Wavelet Transform Of Periodic Generalized-Functions

    Authors

    A. I. Zayed

    Comments

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    Abbreviated Journal Title

    J. Math. Anal. Appl.

    Keywords

    ATOMIC DECOMPOSITIONS; SAMPLING THEORY; BANACH-SPACES; HARDY-SPACES; SIGNAL; PROPAGATION; Mathematics, Applied; Mathematics

    Abstract

    The aim of this paper is to define the wavelet transform for spaces of periodic functions, then extend this definition to spaces of generalized functions larger than the space of periodic Schwartz distributions, such as spaces of periodic Beurling ultradistributions and hyperfunctions on the unit circle. It is shown that the wavelet transforms of such generalized functions are nice and smooth functions defined on an infinite cylinder, provided that the analyzing wavelet is also nice and smooth. For example, it is shown that the growth rate of the derivatives of the wavelet transform is almost as good as that of the analyzing 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 derivative(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, theta is-an-element-of T}. (C) 1994 Academic Press, Inc.

    Journal Title

    Journal of Mathematical Analysis and Applications

    Volume

    183

    Issue/Number

    2

    Publication Date

    1-1-1995

    Document Type

    Article

    Language

    English

    First Page

    391

    Last Page

    412

    WOS Identifier

    WOS:A1994NN53500006

    ISSN

    0022-247X

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