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6 votes

Fourier coefficients of the logarithm of a given function

What you stated is correct. The crude statement is $|d_n|\leq C(\mu)e^{-\mu n}$, for every $\mu<\mu_0$, where $$\mu_0=\min\{\lambda,\theta_0\},\; \theta_0=\min\{|\theta|: f(x+i\theta)=0\}.$$ Proof. ...
Alexandre Eremenko's user avatar
3 votes

Question regarding proof of Littlewood-Paley

Apologies if I've misunderstood the question; Grafakos comments in this paragraph that The fundamental ingredient in the proof is that $f=\sum_{\mathbf{j}\in\mathbb{Z}^n}\Delta_\mathbf{j}^\#\Delta_\...
Ben Johnsrude's user avatar
5 votes

A fractional weighted Poincaré inequality

It is not true. Start with a function $u$ which vanishes for $x<0$ and is equal to $1$ for $0 \leq x \leq \frac 12$ and then smooth from $x \geq \frac 12$. The Fourier coefficients behave like $1/n$...
Giorgio Metafune's user avatar
4 votes

Any references for generalised square functions?

For $0<p<\infty$, $0<q\le\infty$ and $s\in\mathbb R$, the Triebel-Lizorkin space $F_{pq}^s(\mathbb R^n)$ is the set of all (tempered distributions) $f$ such that $\big\||\{2^{js}\cdot P_jf\}...
Liding Yao's user avatar
0 votes

Steinhaus theorem and Hausdorff dimension

In the paper by Feng and Wu, they introduced a notion of thickness, that generalized the Newhouse thickness in the line. They showed that if a set has positive Feng-Wu thickness, then the sumset after ...
Chun-Kit Lai's user avatar

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