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....

Let 0 < p ≤ 1, let ω: ℤ → [1,∞) be a weight on ℤ and let f be a nowhere vanishing continuous function on the unit circle Γ whose Fourier series satisfies ${\sum }_{n\in \mathbb{Z}}{\left|f̂\left(n\right)\right|}^p\omega \left(n\right)<\infty$. Then there exists a weight ν on ℤ such that ${\sum }_{n\in \mathbb{Z}}{\left|\widehat{(1/f)}\left(n\right)\right|}^p\nu \left(n\right)<\infty$. Further, ν is non-constant if and only if ω is non-constant; and ν = ω if ω is non-quasianalytic. This includes the classical Wiener theorem (p = 1, ω = 1), Domar theorem (p = 1, ω is non-quasianalytic), Żelazko theorem (ω = 1) and a recent result of Bhatt and Dedania (p = 1). An analogue of...

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left(j^{-\alpha }\right)$ for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

Denote by $f_{ss}(x,y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that $f_{ss}$ belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

If $x$ is a bounded function on $\mathbf{Z}$, the multiplier with symbol $x$ (denoted by $M_x)$ is defined by $(M_{x}f)\widehat{}=x\widehat{f}$, $f\in L^2\left(\mathbf{T}\right)$. We give some conditions on $x$ ensuring the “interpolation inequality” $\parallel M_{x}f{\parallel }_{L^p}\le C\parallel f{\parallel }_{L^1}^{\alpha }\parallel M_{x}f{\parallel }_{L^q}^{1-\alpha }$ (here $1\&lt;p\&lt;q$ and $\alpha =\alpha (p,q,x)$ is between 0 and 1). In most cases considered $M_x$ fails to have stronger $L^1$-regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E\subset \mathbf{Z}$ every positive sequence in ${\ell }^2\left(E\right)$ can be majorized by the sequence $\{$$...$