### A class of Fourier series

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Let $f\left(x\right)\sim \Sigma {a}_{n}{e}^{2\pi inx},f*\left(x\right)\sim {\sum}_{n=0}^{\infty}a{*}_{n}\phantom{\rule{0.166667em}{0ex}}\mathrm{cos}\phantom{\rule{0.166667em}{0ex}}2\pi nx$, where the $a{*}_{n}$ are the numbers $\left|{a}_{n}\right|$ rearranged so that ${a}_{n}^{*}\searrow 0$. Then for any convex increasing $\psi $, $\parallel \psi \left(\right|f{|}^{2}{\parallel}_{1}\le \parallel \psi \left(20\right|f*{|}^{2}{\parallel}_{1}$. The special case $\psi \left(t\right)={t}^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel}_{q}\le 5\parallel f*{\parallel}_{q}$ an equivalent of Littlewood.

We strengthen the Carleson-Hunt theorem by proving ${L}^{p}$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $r>\mathrm{\U0001d696\U0001d68a\U0001d6a1}\{{p}^{\text{'}},2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

We show that, given a set E ⊂ 𝕋 of measure zero, the set of continuous functions whose Fourier series expansion is divergent at any point t ∈ E is dense-algebrable, i.e. there exists an infinite-dimensional, infinitely generated dense subalgebra of 𝓒(𝕋) every non-zero element of which has a Fourier series expansion divergent in E.

Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in ${L}^{1}$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients....