Rearrangements of functions on a locally compact abelian. An application of fourier transforms on finite abelian. We will come to this generalisation later in this set of. There is a vast literature on the theory of fourier transform on groups, including fourier transform on locally compact abelian groups 27, on compact groups 34, and nite groups 73. As observed elsewhere 2, 24, fourier analysis on other. Fourier analysis on nite abelian groups april 1, 2012 1. For the case of general n, t, and k, we saw how the enumeration can be cast as a discrete fourier transform on a finite abelian group g n,t. C by writing them as linear combinations of multiplicative charactersgroup homomorphisms g. On a nite abelian group, the fourier expansion of a complexvalued function frepresents f, in the sense that, for every g2g, fg 1 jgj x. Fourier transform on grouplike structures and applications. Representation theory for finite nonabeliangroups in last weeks notes we obtained a satisfactory theory of the fourier transform on. In the nonabelian setting we recapture this property if we define the right inne. Fourier transform and fourier series are two manifestations of a similar idea, namely, to write general functions as superpositions whether integrals or sums of some special class of functions. On a finite abelian group, the fourier expansion of a.
Fourier analysis on finite abelian groups with an emphasis on uncertainty principles cameron lavigne december 18, 20. Studying the fourier transform of lca groups allows us to explain many of the properties we take for granted about the everyday fourier transform of. The fourier transform on compact groups is a major tool in representation theory and noncommutative harmonic analysis. Standard abelian fourier analysis aims to study functions f. To reduce a general problem to a numerical computation, sampling and periodization is necessary. Fourier series and finite abelian groups william c. What is the difference between fourier series and fourier. C such that convolutions are transformed into pointwise products. Fourier transform turns convolution into matrix multiplication 1. Notes from a course given by victor ostrik in luminy, 2010. Fourier series on finite abelian groups have applications in engineering. Readings in fourier analysis on finite nonabelian groups radomir s.
Fourier analysis on finite abelian groups bao luong. The quantum fourier transform and extensions of the. The goal of the work to be described in these lectures is. The fourier transform and equations over finite abelian groups an introduction to the method of trigonometric sums lecture notes by. The fourier transform considered as an operator from l. Usually, we are interested in the following two cases.
The level is intended for physics undergraduates in their 2nd or 3rd year of studies. Introduction the purpose of this paper is to illustrate how fourier series and the fourier transform appear as generalizations of natural activities related to the group. Some remarks on the nonabelian fourier transform in. The fourier transform and equations over finite abelian groups. Fourier analysisonfinite abeliangroups we have been using fourier analysis on the euclidean group rd and to a lesser extent, on the toral group td for some time now. In the context of the nonabelian fourier transform, the natural extension of the amplitude and phase response to a convolution by a given filter mask are shown to be the polar decompositions of the fourier transform matrices.
Studying the fourier transform of lca groups allows us to explain many of the properties we take for granted about the everyday fourier transform of real numbers. Given a finite abelian group g with n elements, we want to study the fourier transform over it. Forthecaseofn 2200 and t 17, we characterized g n,t in an explicit fashion, and described how to generate its elements in an e. The notion of fourier transform is among the more important tools in analysis, which has been generalized in abstract harmonic analysis to the level of abelian locally compact groups. In 8, 9 biggs uses the fourier transform on an abelian group qof order qto exhibit. The twomodular fourier transform of binary functions. September 26, 2015 fourier analysis on nite abelian groups. Abelian dual group and finite fourier transform let gbe a nite abelian group, and gb homg. Readings in fourier analysis on finite nonabelian groups. Fourier analysis on nite abelian groups september 26, 2015 1. The norm of the fourier transform on compact or discrete abelian groups article pdf available november 2016 with 45 reads how we measure reads. Define fourier transform pair or define fourier transform and its inverse transform. Abelian varieties, theta functions and the fourier transform.
The complex or infinite fourier transform of fx is given by. We find in this paper the equimeasurable hulls and kernels of some function classes on a locally compact abelian group. Vazirani, chair the quantum fourier transform qft has emerged as the primary tool in quantum al. C, assigning to fthe element p g2g fgg so g corresponds to the \point mass at g. As an example, fast quantum fourier transforms for all 4 classes of nonabelian 2groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. Finite or discrete fourier transforms fft are essential tools in engineering disciplines based on signal transmission, which is the case in most of them. Fourier transform fft is essential for highspeed computing 8. Let x be a compact abelian group with dual group y, and let s be a subset of y. Fourier transform of invariant differential operators on a locallycompact abelian group s. The quantum complexity of the fourier transform for these groups of size 2n is on2 in all cases.
Fast fourier transforms ffts are fast algorithms, i. Fourier transform on locally compact quantum groups byungjay kahng communicated by editor abstract. Fourier transform of invariant differential operators on a. Akbarov 1 mathematical notes volume 56, pages 852 855 1994 cite this article. An excellent discussion of a modern application of. Fast fourier analysis for abelian group extensions core. A brief introduction to the fourier transform this document is an introduction to the fourier transform.
There are also generalizations in noncommutative geometry, see quantum group fourier transform. This means the fourier transform on a nonabelian group takes values as hilbert space operators. Pdf the norm of the fourier transform on compact or. Character sheaves character sheaves were invented by lusztig. Biggs uses the fourier transform on an abelian group q of order q to exhibit the duality between interaction models defined in terms of vertex q colourings. Fourier analysis on finite abelian groups april 1, 2012. It turns out that fourier analysis can in fact be formalised on any locally compact hausdorff abelian group. This selfcontained book provides a thorough look at the fourier transform, one of the most useful tools in applied mathematics.
Schulz department of mathematics and statistics, northern arizona university, flagsta. April 1, 2012 fourier analysis on nite abelian groups. Let g be a finite abelian group of order n, written additively. Abelian dual group and finite fourier transform let g be. We identify the group algebra cg with the set of functions f. Then the function fx is the inverse fourier transform of fs and is given by. The fourier transform can thus be generalized to sets other than the real line, such as the circle, the integers, and in fact any locally compact abelian group. Akavia, goldwasser, safra adi akavia, massachusetts institute of technology, theory. G c, where g can be an arbitrary group not only abelian. With countless examples and unique exercise sets at the end of most sections, fourier analysis on finite abelian groups is a perfect companion for a first course in fourier analysis.
In the abelian setting the fourier transform is a unitary linear transformation proper scaling required. It turns out that fourier analysis can in fact be formalised on any locally compact hausdor. Character sheaves, tensor categories and nonabelian fourier transform abstract. Readings in fourier analysis on finite non abelian groups radomir s. C that records the coe cients of each character in the decomposition of f. Fft are related with circulant matrices, which can be viewed as group matrices of cyclic groups. The groups r, t rz, z and zn are all lcas of great interest, but numerical computations are almost always done on the nite group znusing the fast fourier transform. In this regard, we introduce a generalization of the previous investigations to the case of finite groups, abelian or not. Lecture notes for thefourier transform and applications. These classes consist of all functions for which the fourier transform belongs to a given lorentz space on the dual group. The quantum fourier transform and extensions of the abelian hidden subgroup problem by lisa ruth hales doctor of philosophy in logic and the methodology of science university of california at berkeley professor umesh v. Fast quantum fourier transforms for a class of nonabelian. For example, consider g z n, the group of integers under addition modulo n. Fourier transform, fourier analysis, signal processing, approximation the.
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