Volume 5, Issue 3

Abstract. The functions Fmup s,n(x)${\mathrm {Fmup}_{s,n}(x)}$ and their properties are considered. The asymptotic behavior of these functions as n → ∞ is studied. Generalized Fup-functions are introduced and their asymptotic behavior is investigated.

Abstract. In this paper error estimates of approximations in the complex domain for the Mellin transform are given for functions f which vanish beyond a finite domain [a,b]⊂[0,∞〉${[ a,b] \subset [ 0,\infty \rangle }$ and whose derivatives belong to L p [ a , b ]. These estimates enable us to obtain two associated numerical quadrature rules and error bounds of their remainders.

Abstract. The reconstruction property for Banach spaces was introduced by Casazza and Christensen. In this paper we give a type of the reconstruction property in Banach spaces which is generated by the Toeplitz matrices and we call it the Toeplitz reconstruction property. It is proved that the standard reconstruction property in a Banach space can generate the Toeplitz reconstruction property from a given Toeplitz matrix but not conversely. Sufficient conditions on infinite matrices to have the reconstruction property for a discrete signal space are given.

Abstract. Given a digraph G=(V(G),A(G))${G=(V(G),A(G))}$, or more simply ( V , A ). Two digraphs G and G ' are hemimorphic if G ' is isomorphic to G or to its dual G*=(V,{(x,y):(y,x)∈A})${G^*=(V,\lbrace (x,y):(y,x)\in A\rbrace )}$. For x∈V${x\in V}$, denote dG+(x)=|{y∈V:(x,y)∈Aand(y,x)∉A}|$d_G^{+}(x)=\vert \lbrace y\in V: (x, y)\in A \text{and } {(y, x)\notin A\rbrace }\vert $ and dG-(x)=|{y∈V:(x,y)∉Aand(y,x)∈A}|${d_G^{-}(x)=\vert \lbrace y\in V: (x, y)\notin A \text{ and } (y, x)\in A\rbrace \vert }$. Given two digraphs G and H with the property that V(G)∩V(H)=∅${V(G)\cap V(H)=\emptyset }$, let v0∈V(G)${v_0\in V(G)}$. We say that we dilate v 0 by H if we replace v 0 by H , obtaining then a new digraph R satisfying: for all z∈V(G)∖{v0}${z\in V(G)\setminus \lbrace v_0\rbrace }$ and for all h∈V(H)${h\in V(H)}$ is (z,h)∈A(R)⇔(z,v0)∈A(G)${(z,h)\in A(R)\iff (z,v_0)\in A(G)}$ and (h,z)∈A(R)⇔(v0,z)∈A(G)${(h,z)\in A(R)\iff (v_0,z)\in A(G)}$. We say that the digraph R is obtained from G by dilating the vertex v 0 by the digraph H . Our main result is: let σ be an isomorphism from a digraph M onto a digraph M ' and v0∈V(M)${v_0\in V(M)}$ such that dM+(v0)≠dM-(v0)${d_M^{+}(v_0)\ne d_{M}^{-}(v_0)}$ and H , H ' be two digraphs. Consider the digraph R (resp. R ') obtained by dilating v 0 (resp. σ(v0)${\sigma (v_0)}$) in M (resp. in M ') by H (resp. H '). If R and R ' are hemimorphic and H and H ' are hemimorphic, then R and R ' are isomorphic and H and H ' are isomorphic. The main result generalizes those of Y. Boudabbous and J. Dammak and of M. Bouaziz and Y. Boudabbous.

Abstract. We continue the description of the C * -algebras of low-dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra. We show here that the C * -algebras of the three 6-dimensional nilpotent Lie groups having the group G 5,2 as quotient group modulo the centre have norm controlled dual limits.