Category Archives: Fourier analysis

Matrix multiplication as a convolution

1. The product of two power series/polynomials is The coefficients given by is sometimes called the Cauchy product. This is a convolution. 2. Let be a finite group and be the group algebra with complex coefficients. Let be two elements … Continue reading

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Fourier coefficients as eigenvalues/spectrum

In this post I want to make a connection between Fourier coefficients and eigenvalues/spectrum. Let me put the claim up front: If , then is not invertible with respect to the convolution product. Please feel free to jump directly to … Continue reading

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Arithmetic progression in some subsets of $\mathbb{Z}_N$ (Part 2)

Last time I have mentioned the idea of the proof (of theorem 2). Now I continue and give a full detail here.

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Arithmetic progression in some subsets of $\mathbb{Z}_N$ (Part 1)

This post is based mainly on the paper “Arithmetic progression in sumsets” by Ben Green, which talks about the length of arithmetic progressions of some special kind of subsets of , where is a sufficiently big odd prime. I will … Continue reading

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Fourier Analysis and Number Theory III: The Poisson summation formula as a trace formula

Yin and Yang are important concept in the Chinese philosophy. They usually describe two opposite things which are not only complement of each other but also have deep interrelationship. In mathematics, we called them duality. For example (1) primes vs … Continue reading

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Fourier Analysis and Number Theory II: The Poisson Summation Formula and The Functional Equation of The Riemann-Zeta Function

Riemann’s ten-page-long paper “Über die Anzahl der Primzahen unter einer gegebener Gröβe” has great influence on modern number theory. In the paper, he established two important properties of the Riemann-zeta function (the summation is absolutely convergent for ).

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Fourier Analysis and Number theory I: The self-dual function e^{-\pi x^2}

During a lecture at UCLA, Serge Lang asked what is the most important function in mathematics. This question is quite personal and every person certainly has his own opinion. In fact, a professor spoke out loud that the constant function … Continue reading

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