# Questions tagged [fourier-analysis]

The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.

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### Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
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### Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem: Recall that ...
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### What are the necessary/sufficient conditions for a Fourier transform to have at least $k$ roots?

Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform. What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ ...
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### Vanishing of the product of a function and its own Fourier transform

I have found the following question to be surprisingly hard: Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that $$f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},$$ ...
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### When can one bound the Hilbert transform on the torus in $L^1$?

If $f$ is a function on $\mathbb T$ then the conjugate $\tilde f$ of $f$ satisfies Zygmund's bound $$\int |\tilde f|\leq A \int |f|\log(e+|f|)+B.$$ I am curious if there are any conditions where one ...
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### Smallest regular $m$-gon covering a regular $n$-gon

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question. Let us fix a regular $n$-gon with area $1$. What is the smallest ...
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### Functions whose Fourier coefficients satisfy $\sum_{k=1}^\infty |c_k| < 1$?
Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
### Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers
When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used: n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...