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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2412.17466 |
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| _version_ | 1866910759928201216 |
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| author | Clément, François Steinerberger, Stefan |
| author_facet | Clément, François Steinerberger, Stefan |
| contents | The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2πa}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2πb}{n} k \right) \right)\right|,$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Failure of Orthogonality of Rounded Fourier Bases Clément, François Steinerberger, Stefan Classical Analysis and ODEs The purpose of this note is to prove estimates for $$ \left| \sum_{k=1}^{n} \mbox{sign} \left( \cos \left( \frac{2πa}{n} k \right) \right) \mbox{sign} \left( \cos \left( \frac{2πb}{n} k \right) \right)\right|,$$ 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. |
| title | Failure of Orthogonality of Rounded Fourier Bases |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2412.17466 |