In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_d\ast g^{\ast d}+a_{d-1}\ast g^{\ast (d-1)}+\cdots +a₁\ast g+a₀=0$, where $a₀,...,a_d:\mathbb{N}\to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form ${\sum }_{x\in X}f\left(x\right)e^{-sx}$ ($s\in {ℂ}^k$), where $X\subseteq {[0,\infty )}^k$ is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied,...

By a general Franklin system corresponding to a dense sequence of knots 𝓣 = (tₙ, n ≥ 0) in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots 𝓣, that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is a characterization of sequences 𝓣 for which the corresponding general Franklin system is a basis or an unconditional basis in H¹[0,1].

We show that each general Haar system is permutatively equivalent in $L^p\left([0,1]\right)$, 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^p\left([0,1]\right)$, 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^p\left({[0,1]}^d\right)$, 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases in $L^p\left({[0,1]}^d\right)$ for 1...

In this paper we give general methods of construction of various equivalent minimal pairs of compact convex sets that are not translates of one another.

By means of two general operations + and x, called pan-operations'', we build a new kind of integral. This formulation contains, as particular cases, both Choquet's and Sugeno's integrals.

Cet article est consacré à l’étude des espaces $A_{\omega }=ℱL^2({\mathbf{R}}^n;\omega \left(x\right)dx)$ qui sont des algèbres de Banach. On démontre que les multiplicateurs ponctuels de $A_{\omega }$ sont les fonctions qui appartiennent localement et uniformément à $A_{\omega }$ si et seulement si $A_{\omega }$ contient des fonctions à support compact.

We introduce the space $GBD$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $GBD$, which leads to a compactness result for the space $GSBD$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...

The Newman-Shapiro Isometry Theorem is proved in the case of Segal-Bargmann spaces of entire vector-valued functions (i.e. summable with respect to the Gaussian measure on ℂⁿ). The theorem is applied to find the adjoint of an unbounded Toeplitz operator $T_{\phi }$ with φ being an operator-valued exponential polynomial.

We study subalgebras of $C_b\left(X\right)$ equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.

The generalized notion of weak amenability, namely $(\varphi ,\psi )$-weak amenability, where $\varphi ,\psi$ are continuous homomorphisms on a Banach algebra $𝒜$, was introduced by Bodaghi, Eshaghi Gordji and Medghalchi (2009). In this paper, the $(\varphi ,\psi )$-weak amenability on the measure algebra $M\left(G\right)$, the group algebra $L^1\left(G\right)$ and the Segal algebra $S^1\left(G\right)$, where $G$ is a locally compact group, are studied. As a typical example, the $(\varphi ,\psi )$-weak amenability of a special semigroup algebra is shown as well.