Examples of Wavelets for Local Fields

It is well known that the Haar and Shannon wavelets in L2(R) are at opposite extremes, in the sense that the Haar wavelet is localized in time but not in frequency, whereas the Shannon wavelet is localized in freqency but not in time. We present a rich setting where the Haar and Shannon wavelets coincide and are localized both in time and in frequency. More generally, if R is replaced by a group G with certain properties, J. Benedetto and the author have proposed a theory of wavelets onG, including the construction of wavelet sets [2]. Examples of such groups G include the padic rational group G = Qp, which is simply the completion of Q with respect to a certain natural metric topology, and the Cantor dyadic group F2((t)) of formal Laurent series with coefficients 0 or 1. In this expository paper, we consider some specific examples of the wavelet theory on such groups G. In particular, we show that Shannon wavelets on G are the same as Haar wavelets on G. We also give several examples of specific groups (such as Qp and Fp((t)), for any prime number p) and of various wavelets on those groups. All of our wavelets are localized in frequency; the Haar/Shannon wavelets are localized both in time and in frequency. One of the principal goals of wavelet theory has been the construction of useful orthonormal bases for L(R). The group R has been an appropriate setting both because of its use in applications and because of its special property of containing lattices such as Z which induce discrete groups of translation operators on L(R). Of such wavelets, the easiest to describe are the Haar wavelet and the Shannon wavelet in L(R). The Haar is a compactly supported step function in time but has noncompact support and slow decay in frequency. On the other hand, the Shannon is considered to be at the opposite extreme, being a compactly supported step function in frequency but with noncompact support and slow decay in time. This distinction between Haar and Shannon wavelets appears to caused by certain properties of the topological group R. J. Benedetto and the author [2] have presented a theory of wavelets on L(G), for groups G with different properties to be described in Section 2 below. In this paper, we shall present some of that theory and some examples. Our main result, Theorem 4.3, is the surprising fact that Haar and Shannon wavelets not only exist in L(G), but the two are the same. Moreover, 2000 Mathematics Subject Classification. 11S85, 42C40.