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Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

3 votes
Accepted

Reverse Markov inequality

As you noticed, for $K < 1$ (some form of) it is true. I will show that it is false for $K \ge 1$. It is enough to construct functions $f_n$ on $[0, 1]$ with $\int |f_n| = \varepsilon$ and $\int |f_n …
Aleksei Kulikov's user avatar
5 votes
Accepted

Special function in the Hardy space

First of all both of these facts are wrong. For the first one we have to assume that $f$ is not a constant (since $f(z) = i$ works). For the second one it is actually a classical and very important re …
Aleksei Kulikov's user avatar
5 votes
Accepted

Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u...

(Un?)fortunately, there are no such functions. The idea is simple, yet powerful: if the double integral $\int\int d\xi d\eta$ is finite, then for almost all $\eta$ the $d\xi$ integral is finite as wel …
Aleksei Kulikov's user avatar
2 votes
Accepted

A question on finite Fourier series

No problem, this is doable already for $N=2$. The idea is as follows: at the point of the maximum first derivative must vanish, and yet we want to separate the maximum into two by adding something sma …
Aleksei Kulikov's user avatar