Ingham [6] ha migliorato un risultato precedente di Wiener [23] sulle serie di Fourier non armoniche. Modificando la sua funzione di peso noi otteniamo risultati ottimali, migliorando precedenti teoremi di Kahane [9], Castro e Zuazua [3], Jaffard, Tucsnak e Zuazua [7] e di Ullrich [21]. Applichiamo poi questi risultati a problemi di osservabilità simultanea.

We study weighted $L_p$-integrability (1 ≤ p < ∞) of trigonometric series. It is shown how the integrability of a function with weight $x^{-\alpha }$ depends on some regularity conditions on Fourier coefficients. Criteria for the uniform convergence of trigonometric series in terms of their coefficients are also studied.

Integrability and $L^1-$convergence of modified cosine sums introduced by Rees and Stanojević under a class of generalized semi-convex null coefficients are studied by using Cesàro means of non-integral orders.

We show that, if the coefficients (an) in a series $a_0/2+{\sum }_{n=1}^{\infty }{a}_{n}cos\left(nt\right)$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\sum }_{m=0}^{\infty }{{\sum }_{j=1}^{\infty }[{\sum }_{n=j{2}^m}^{(j+1)2^m-1}|a_n-a_{n+1}\left|\right]²}^{1/2}<\infty$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ${\sum }_{n=1}^{\infty }{b}_{n}sin\left(nt\right)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if ${\sum }_{n=1}^{\infty }\left|b_n\right|/n<\infty$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences...

In the setting of spaces of homogeneous-type, we define the Integral, $I_{\phi }$, and Derivative, $D_{\phi }$, operators of order $\phi$, where $\phi$ is a function of positive lower type and upper type less than $1$, and show that $I_{\phi }$ and $D_{\phi }$ are bounded from Lipschitz spaces ${\Lambda }^{\xi }$ to ${\Lambda }^{\xi \phi }$ and ${\Lambda }^{\xi /\phi }$ respectively, with suitable restrictions on the quasi-increasing function $\xi$ in each case. We also prove that $I_{\phi }$ and $D_{\phi }$ are bounded from the generalized Besov ${\dot{B}}_p^{\psi ,q}$, with $1\le p,q<\infty$, and Triebel-Lizorkin spaces ${\dot{F}}_p^{\psi ,q}$, with $1<p,q<\infty$, of order $\psi$ to those of order $\phi \psi$ and $\psi /\phi$ respectively,...