Fourier Analysis on Number Fields

Ramakrishnan, Dinakar; Valenza, Robert J.

doi:10.1007/978-1-4757-3085-2

Cited by 157 publications

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“…The notion of supernatural number (which may be seen as an extension of the notion of natural number) is well known and used in various areas (see, for instance, [13] and [14]). Due to its importance in this paper, we recall it in some detail in the first subsection.…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

On the relative solvability of certain inverse monoids

Cordeiro

Delgado

2010

Semigroup Forum

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Section: Notation and Preliminary Resultsmentioning

confidence: 99%

On the relative solvability of certain inverse monoids

Cordeiro

Delgado

2010

Semigroup Forum

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“…For more of the standard background on Q p , see [11,16,23]; see [24,25] for more advanced expositions.…”

Section: Proposition 23 (From [2]) Let G Be a Locally Compact Abelimentioning

confidence: 99%

“…The study of Fourier analysis on local fields and on adèle groups arises in class field theory and in the study of certain zeta functions; see [5,6,23,27], for example. However, no prior knowledge of such topics are needed to read this paper.…”

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confidence: 99%

Examples of wavelets for local fields

Benedetto¹

2004

Contemporary Mathematics

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Abstract. It is well known that the Haar and Shannon wavelets in L 2 (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 on G, 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 F 2 ((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 2 (R d ). The group R d has been an appropriate setting both because of its use in applications and because of its special property of containing lattices such as Z d which induce discrete groups of translation operators on L 2 (R d ). Of such wavelets, the easiest to describe are the Haar wavelet and the Shannon wavelet in L 2 (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 d . J. Benedetto and the author [2] have presented a theory of wavelets on L 2 (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 2 (G), but the two are the same. Moreover,2000 Mathematics Subject Classification. 11S85, 42C40.

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“…This will necessitate some algebraic background; accordingly this section requires some familiarity with the concepts involved. However, it is not essential to the rest of the paper, and readers may wish to skip directly to Section 2.3, or instead to consult [93], [142] for definitions and examples.…”

Section: Hecke's Treatment Of Number Fields (1916)mentioning

confidence: 99%

“…The above construction can be generalized to an arbitrary number field -or even "global field" -F to obtain its adele ring A F (see [93], [142]). Most constructions involving A Q generalize to A F , though we will mainly focus on F = Q for expositional ease.…”

Section: Generalizations To Adele Groupsmentioning

confidence: 99%

Riemann’s zeta function and beyond

Gelbart

Miller

2003

Bull. Amer. Math. Soc.

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Abstract. In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of L-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

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Fourier Analysis on Number Fields

Cited by 157 publications

References 0 publications

On the relative solvability of certain inverse monoids

On the relative solvability of certain inverse monoids

Examples of wavelets for local fields

Riemann’s zeta function and beyond

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