Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_n$ and the corresponding primitive $F_n$. The pointwise convergence of the integrands $f_n$ to some $f$ and the equiintegrability of the functions $f_n$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_n$ converge uniformly to $F$. In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Let T be a linear operator on a Banach space X with $supₙ\left|\right|Tⁿ/n^w\left|\right|<\infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{-1}{\sum }_{k=0}^{n-1}{T}^k$ converges uniformly; (ii) $cl(I-T)X=z\in X:li{m}_n{\sum }_{k=1}^n{T}^{k}z/kexists$.

We first give a necessary and sufficient condition for $x^{-\gamma }\varphi \left(x\right)\in L^p$, 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either ${\sum }_{k=1}^{\infty }{a}_{k}coskx$ or ${\sum }_{k=1}^{\infty }{b}_{k}sinkx$, under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x)...

We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{\epsilon }\left(x\right):={\sum }_{j\in D_{\epsilon }\left(x\right)}e{*}_j\left(x\right)e_j$, where $D_{\epsilon }\left(x\right):=j:|e{*}_j\left(x\right)|\ge \epsilon$. We study a generalized version of $T_{\epsilon }$ that we call the weak thresholding approximation. We modify the $T_{\epsilon }\left(x\right)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t,\epsilon }\left(x\right):=j:t\epsilon \le |e{*}_j\left(x\right)|<\epsilon$ and consider...

Let $\sum _{n=1}^{\infty }{a}_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $\left(a_n\right)$ is non-increasing, then $\underset{n\to \infty }{lim}n{a}_n=0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\underset{n\to \infty }{lim}n{a}_n=0$; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the $ℐ$-convergence, that is a convergence according to an ideal $ℐ$ of subsets of $\mathbb{N}$. Again, Olivier’s theorem is a consequence...

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\widehat{f}\left(n\right)$, $n\in \mathbb{N}$, of functions $f\in L^p\left(G\right)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series ${\sum }_{n=1}^{\infty }{a}_n{\left|\widehat{f}\left(n\right)\right|}^r$, where $\left\{a_n\right\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative...

We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $Gₘ\left(f\right):={\sum }_{k\in \Lambda }f̂\left(k\right)e^{i(k,x)}$, where $\Lambda \subset {\mathbb{Z}}^d$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, $C_{ℝ}\left(X\right)$) of all complex-valued (resp., real-valued) continuous functions on X, S ⊂ M a complex (resp., real) function space on X, ϕₙ a sequence of unital linear contractions from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$), and ${\varphi }_{\infty }$ a linear isometry from M into C(Y) (resp., $C_{ℝ}\left(Y\right)$). We show, under the assumption that ${\Pi }_N\subset {\Pi }_T$, where ${\Pi }_N$ is...

Given a coarse space $(X,ℰ)$ with the bornology $ℬ$ of bounded subsets, we extend the coarse structure $ℰ$ from $X\times X$ to the natural coarse structure on $(ℬ\setminus \{\varnothing \left\}\right)\times (ℬ\setminus \{\varnothing \left\}\right)$ and say that a macro-uniform mapping $f:(ℬ\setminus \{\varnothing \left\}\right)\to X$ (or $f:{\left[X\right]}^2\to X$) is a selector (or 2-selector) of $(X,ℰ)$ if $f\left(A\right)\in A$ for each $A\in ℬ\setminus \left\{\varnothing \right\}$ ($A\in {\left[X\right]}^2$, respectively). We prove that a discrete coarse space $(X,ℰ)$ admits a selector if and only if $(X,ℰ)$ admits a 2-selector if and only if there exists a linear order “$\le$" on $X$ such that the family of intervals $\left\{\right[a,b]:a,b\in X,a\le b\}$ is a base for the bornology $ℬ$.

Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$. The lamplighter chain $X^{\diamond }$ associated with $X$ is the random walk on the wreath product ${𝒢}^{\diamond }={𝐙}_2\wr 𝒢$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of ${𝐙}_2=\{0,1\}$ and $x$ is a vertex in $𝒢$. There is an edge between $(\underline{f},x)$ and $(\underline{g},y)$ in ${𝒢}^{\diamond }$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_z=g_z$ for all $z\ne x,y$. In each step, $X^{\diamond }$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$ and then...

Let $f_c(x,y)\equiv {\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }{a}_{jk}(1-cosjx)(1-cosky)$ with $a_{jk}\ge 0$ for all j,k ≥ 1. We estimate the integral ${\int }_0^{\pi }{\int }_0^{\pi }{x}^{\alpha -1}{y}^{\beta -1}\varphi \left(f_c(x,y)\right)dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].

It is well-known that a probability measure $\mu$ on the circle $𝕋$ satisfies $\parallel {\mu }^n{*f-\int f\mathrm{d}m\parallel }_p\to 0$ for every $f\in L_p$, every (some) $p\in [1,\infty )$, if and only if $|\widehat{\mu }\left(n\right)|lt;1$ for every non-zero $n\in \mathbb{Z}$ ($\mu$ is strictly aperiodic). In this paper we study the a.e. convergence of ${\mu }^n*f$ for every $f\in L_p$ whenever $pgt;1$. We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of $\mu$, for the strong sweeping out property (existence of a Borel set $B$ with $lim\; sup{\mu }^n*1_B=1$ a.e. and $lim\; inf{\mu }^n*1_B=0$ a.e.). The results are extended to general compact Abelian groups...

It is known that there is no natural Banach norm on the space $\mathrm{\mathscr{H}𝒦}$ of $n$-dimensional Henstock-Kurzweil integrable functions on $[a,b]$. We show that the $\mathrm{\mathscr{H}𝒦}$ space is the uncountable union of Fréchet spaces $\mathrm{\mathscr{H}𝒦}\left(X\right)$. On each $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space, an $F$-norm ${\parallel ·\parallel }^X$ is defined. A ${\parallel ·\parallel }^X$-convergent sequence is equivalent to a control-convergent sequence. Furthermore, an $F$-norm is also defined for a ${\parallel ·\parallel }^X$-continuous linear operator. Hence, many important results in functional analysis hold for the $\mathrm{\mathscr{H}𝒦}\left(X\right)$ space. It is well-known that every...

We extend Champernowne’s construction of normal numbers to base b to the ${\mathbb{Z}}^d$ case and obtain an explicit construction of a generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${0,1,...,b-1}^{{\mathbb{Z}}^d}$.

We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.