Volume 5, Issue 1

Abstract. In this paper, a general theorem concerning the ϕ-|C,1;δ|k${\varphi -{|{C,1;\delta }|}_k}$-summability factors of infinite series has been proved.

Abstract. Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra 𝔫${\mathfrak {n}}$ is an n -dimensional vector space over the reals. Moreover, 𝔫=𝔷⊕𝔟⊕𝔞${\mathfrak {n=z}\oplus \mathfrak {b}\oplus \mathfrak {a}}$, 𝔷${\mathfrak {z}}$ is the center of 𝔫${\mathfrak {n}}$, 𝔷=ℝZn-2d⊕ℝZn-2d-1⊕⋯⊕ℝZ1${\mathfrak {z} =\mathbb {R}Z_{n-2d}\oplus \mathbb {R}Z_{n-2d-1}\oplus \cdots \oplus \mathbb {R}Z_{1}}$, 𝔟=ℝYd⊕ℝYd-1⊕⋯⊕ℝY1${\mathfrak {b} =\mathbb {R}Y_{d}\oplus \mathbb {R} Y_{d-1}\oplus \cdots \oplus \mathbb {R}Y_{1}}$, 𝔞=ℝXd⊕ℝXd-1⊕⋯⊕ℝX1${\mathfrak {a} =\mathbb {R}X_{d}\oplus \mathbb {R}X_{d-1}\oplus \cdots \oplus \mathbb {R}X_{1}}$. Next, assume 𝔷⊕𝔟${\mathfrak {z}\oplus \mathfrak {b}}$ is a maximal commutative ideal of 𝔫${\mathfrak {n}}$, [𝔞,𝔟]⊆𝔷${[ \mathfrak {a,b}] \subseteq \mathfrak {z}}$, and det ([Xi,Yj])1≤i,j≤d${\mathrm {det}([X_i,Y_j])_{1\le i,j\le d}}$ is a non-trivial homogeneous polynomial defined over the ideal [𝔫,𝔫]⊆𝔷${[ \mathfrak {n,n}] \subseteq \mathfrak {z}}$. We do not assume that [𝔞,𝔞]${[\mathfrak {a},\mathfrak {a}]}$ is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of L2(N)${L^2(N)}$ with respect to some discrete set Γ${\Gamma }$. The set Γ${\Gamma }$ is explicitly constructed by fixing a strong Malcev basis for 𝔫${\mathfrak {n}}$. We provide sufficient conditions for which a function f is determined from its sampled values on (f(γ))γ∈Γ${(f(\gamma ))_{\gamma \in \Gamma }}$. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

Abstract. In this paper, we derive new integral inequalities for functions with h -convex and h -concave first derivatives. As a consequence, we give new estimates for the remainder term of the midpoint, trapezoid, and Simpson formulae for functions whose derivatives in absolute value at certain power are h -convex and h -concave and we point out the results for some special classes of functions.

Abstract. The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the “flat” Euclidean space ℝn${\mathbb {R}^n}$. In this paper, we present a generalization of the method of periodic unfolding applicable to structures defined on certain compact Riemannian manifolds. While many results known from unfolding in domains of ℝn${\mathbb {R}^n}$ can be recovered, for the unfolding of gradients a transport operator has to be defined. This operator connects vector fields on the manifold and in the reference cell, which allows for the formulation of general two-scale problems. We illustrate the use of the new unfolding technique with a simple elliptic model-problem.

Abstract. We consider the integration operator Ig,ϕ(n)f(z)=∫0zf(n)(ϕ(ζ))g(ζ)dζ,$ I^{(n)}_{g,\hspace*{0.56905pt} \varphi }f(z)=\int _0^z f^{(n)}(\varphi (\zeta ))g(\zeta )d\zeta , $ where g is a holomorphic function on the unit disk 𝔻${\mathbb {D}}$, ϕ${\varphi }$ is a holomorphic self-map of 𝔻${\mathbb {D}}$ and n∈ℕ∪{0}${n\in \mathbb {N} \cup \lbrace 0\rbrace }$. The boundedness and compactness of Ig,ϕ(n)${I^{(n)}_{g,\hspace*{0.56905pt} \varphi }}$ between the space of Cauchy integral transforms and the Dirichlet space are characterized.