We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belong to one of the generalized Lipschitz classes Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if $|f(x+h)-f\left(x\right)|\le C{h}^{\alpha }L(1/h)$ for all x ∈ , h >...

We show the results corresponding to some theorems of S. Lal and H. K. Nigam [Int. J. Math. Math. Sci. 27 (2001), 555-563] on the norm and pointwise approximation of conjugate functions and to the results of the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] also on such approximations.

For a locally compact Hausdorff group $G$ we investigate what functions in $L_{\infty }\left(G\right)$ give rise to completely continuous multipliers $T_g$ from $L_1\left(G\right)$ into $L_{\infty }\left(G\right)$. In the case of a metrizable group we obtain a complete description of such functions. In particular, for $G$ compact all $g$ in $L_{\infty }\left(G\right)$ induce completely continuous $T_g$.

A sum of exponentials of the form $f\left(x\right)=exp\left(2\pi i{N}_{1}x\right)+exp\left(2\pi i{N}_{2}x\right)+\cdots +exp\left(2\pi i{N}_{m}x\right)$, where the $N_k$ are distinct integers is called an idempotent trigonometric polynomial (because the convolution of $f$ with itself is $f$) or, simply, an idempotent. We show that for every $p\>1,$ and every set $E$ of the torus $𝕋=ℝ/\mathbb{Z}$ with $\left|E\right|\>0,$ there are idempotents concentrated on $E$ in the $L^p$ sense. More precisely, for each $p\>1,$ there is an explicitly calculated constant $C_p\>0$ so that for each $E$ with $\left|E\right|\>0$ and $ϵ\>0$ one can find an idempotent $f$ such that the ratio ${\left({\int }_E{\left|f\right|}^p/{\int }_{𝕋}{\left|f\right|}^p\right)}^{1/p}$ is greater than $C_p-ϵ$. This is in fact...

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...