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Large Sieve Inequalities for Algebraic Trace Functions

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Philippe Michel, Ping Xi
2017

Abstract

The large sieve inequalities for algebraic trace functions are considered in this article. A fundamental iterative relation is established by classical Fourier analysis, and l-adic Fourier analysis and multiplicative convolutions of sheaves are also required to guarantee the iterations, from which some large sieve inequalities are established for general (non-Kummer type) trace functions.

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https://infoscience.epfl.ch/entities/publication/edf1eefd-cbff-44dd-9b25-2e7b4cae945c

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Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.

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