In this article we prove for $1<p<\infty$ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega$. More precisely, $\Omega$ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega$ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection...

Let $a₁,...,a_k$ be arbitrary nonzero real numbers. An $(a₁,...,a_k)$-decomposition of a function f:ℝ → ℝ is a sum $f₁+\cdots +f_k=f$ where $f_i:ℝ\to ℝ$ is an $a_i$-periodic function. Such a decomposition is not unique because there are several solutions of the equation $h₁+\cdots +h_k=0$ with $h_i:ℝ\to ℝa_i$-periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a₁,...,a_k)$-decomposition is essentially unique. We characterize those periods for which essential...

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

For 1 ≤ q < ∞, let ${}_q\left(\right)$ denote the Banach algebra consisting of the bounded complex-valued functions on the unit circle having uniformly bounded q-variation on the dyadic arcs. We describe a broad class ℐ of UMD spaces such that whenever X ∈ ℐ, the sequence space ℓ²(ℤ,X) admits the classes ${}_q\left(\right)$ as Fourier multipliers, for an appropriate range of values of q > 1 (the range of q depending on X). This multiplier result expands the vector-valued Marcinkiewicz Multiplier Theorem in the direction...

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

We identify the torus with the unit interval [0,1) and let n,ν ∈ ℕ with 0 ≤ ν ≤ n-1 and N:= n+ν. Then we define the (partially equally spaced) knots $t_j$ = ⎧ j/(2n) for j = 0,…,2ν, ⎨ ⎩ (j-ν)/n for for j = 2ν+1,…,N-1. Furthermore, given n,ν we let $V_{n,\nu }$ be the space of piecewise linear continuous functions on the torus with knots $t_j:0\le j\le N-1$. Finally, let $P_{n,\nu }$ be the orthogonal projection operator from L²([0,1)) onto $V_{n,\nu }$. The main result is $li{m}_{n\to \infty ,\nu =1}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=su{p}_{n\in \mathbb{N},0\le \nu \le n}\left|\right|P_{n,\nu }:L^{\infty }\to L^{\infty }\left|\right|=2+(33-18\surd 3)/13$. This shows in particular that the Lebesgue constant of the classical...

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

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...

Consider the following higher order difference equation $$x(n+1)=f(n,x(n\left)\right)+g(n,x(n-k\left)\right),n=0,1,\cdots$$ where $f(n,x)$ and $g(n,x):\{0,1,\cdots \}\times [0,\infty )\to [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\left\{\tilde{x}\left(n\right)\right\}$ under certain conditions, and then establish a sufficient condition for $\left\{\tilde{x}\left(n\right)\right\}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also...

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately...

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

In this paper, we propose an extension of a periodic $GARCH$ ($PGARCH$) model to a Markov-switching periodic $GARCH$ ($MS$-$PGA$ $RCH$), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well...

In this paper, we introduce and study the notion of completely bounded ${\Lambda }_p$ sets (${\Lambda }_p^{cb}$ for short) for compact, non-abelian groups G. We characterize ${\Lambda }_p^{cb}$ sets in terms of completely bounded $L^p\left(G\right)$ multipliers. We prove that when G is an infinite product of special unitary groups of arbitrarily large dimension, there are sets consisting of representations of unbounded degree that are ${\Lambda }_p$ sets for all p < ∞, but are not ${\Lambda }_p^{cb}$ for any p ≥ 4. This is done by showing that the space of completely bounded $L^p\left(G\right)$...

Let $j\ge 2$ be a given integer. Let $H_k^{*}$ be the set of all normalized primitive holomorphic cusp forms of even integral weight $k\ge 2$ for the full modulo group $\mathrm{SL}(2,\mathbb{Z})$. For $f\in H_k^{*}$, denote by ${\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)$ the $n$th normalized Fourier coefficient of $j$th symmetric power $L$-function ($L(s,{\mathrm{sym}}^{j}f)$) attached to $f$. We are interested in the average behaviour of the sum $$\sum _{\genfrac{}{}{0pt}{}{n=a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x}{(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}}{\lambda }_{{\mathrm{sym}}^{j}f}^2\left(n\right),$$ where $x$ is sufficiently large, which improves the recent work of A. Sharma and A. Sankaranarayanan (2023).

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent...

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...

Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪\left(E_0\right)$ and suppose that $ℱ=𝒪\left(E_0\right)/ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $\mu$ with support on the variety associated to $ℱ$ such that $\phi$ is in $ℐ$ if and only if $\mu \phi =0$. Such a current $\mu$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....