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arXiv · 2412.17466

Failure of Orthogonality of Rounded Fourier Bases

François Clément·Stefan Steinerberger
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Abstract

The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2\pi a}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2\pi b}{n} k \right) \right)\right|,$$ when $n$ is prime and $a,b \in \mathbb{N}$. We show that the expression can only be large if $a^{-1}b \in \mathbb{F}_n$ (or a small multiple thereof) is close to $1$. This explains some of the surprising line patterns in $A^T A$ when $A \in \mathbb{R}^{n \times n}$ is the signed discrete cosine transform. Similar results seem to exist at a great level of generality.

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François Clément, Stefan Steinerberger. 2024-12-23. Failure of Orthogonality of Rounded Fourier Bases. https://arxiv.org/abs/2412.17466

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