Tauberian Theorems for the Wavelet Transform

Abstract

Abstract. We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on R \ {0} to R; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series.

Wavelet expansions and asymptotic behavior of distributions

“…We may refer to this question as a quasiasymptotic extension problem. Such a problem has been recently studied in detail by the second author and collaborators [38,34]. It is also connected with the results from [33,Sec.…”

“…Observe that slowly varying functions are asymptotically invariant under rescaling at small scale (resp. large scale), and therefore, wavelet analysis is a very convenient tool for studying this class of functions [20][21][22][23][24][25]38].…”

“…This notion has found many applications in areas such as quantum field theory [43,39,40], PDE theory [39], and in the asymptotic study of various integral transforms [5,19,36,39]. Wavelet methods in asymptotic analysis have attracted much attention, both in the classical [7][8][9]18] and distributional [20][21][22][23][24][25]29,38] contexts. In this paper we relate the (quasi)asymptotic behavior of a distribution with the ones of the wavelet coefficients.…”

Regularizations at the origin of distributions having prescribed asymptotic properties

“…On the other hand, the relationship between regularizations and asymptotic properties of distributions is also of importance from the point of view of pure mathematics, for instance, in areas such as singular integral equations [8], the study of boundary properties of holomorphic functions [5], or in Tauberian theory for integral transforms [4,12,13,22,23]. In fact, as shown in recent studies [4,11,22], the asymptotic analysis of various integral transforms may be completely reduced to the study of asymptotic properties of regularizations of distributions; this is the case for the Laplace and wavelet transforms.…”

“…We emphasize that such a problem is essentially a Tauberian one and may be restated in terms of Mellin convolution type integral transforms: quasiasymptotic behavior is nothing but knowledge of asymptotic information over (Mellin) convolution transforms for all kernels in a Schwartz space of test functions. Recently, this problem has been investigated in [5,18,19,20,22]; we shall give extensions of those results, and in particular we provide more detailed asymptotic information for critical degrees than that from [5]. We shall consider distributions with values in a Banach space.…”