### A class of positive trigonometric sums. II.

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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 $p\>1,\phantom{\rule{0.166667em}{0ex}}$ and every set $E$ of the torus $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ 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 $\u03f5\>0$ one can find an idempotent $f$ such that the ratio ${\left({\int}_{E}{\left|f\right|}^{p}/{\int}_{\mathbb{T}}{\left|f\right|}^{p}\right)}^{1/p}$ is greater than ${C}_{p}-\u03f5$. 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]\xb2}^{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...