this paper is an investigation of $l$-dual frames with respect to a function-valued inner product, the so called $l$-bracket product on $l^{2}(g)$, where g is a locally compact abelian group with a uniform lattice $l$. we show that several well known theorems for dual frames and dual riesz bases in a hilbert space remain valid for $l$-dual frames and $l$-dual riesz bases in $l^{2}(g)$.

In this paper, we introduce the notion of a frame in a 2- inner product space and give some characterizations. These frames can be considered as a usual frame in a Hilbert space, so they share many useful properties with frames.

In quantum mechanics, physical states are represented by rays in Hilbert space $\mathcal{H}$, which is a vector imbued an inner product $\ensuremath{\langle}\phantom{\rule{0.16em}{0ex}}|\phantom{\rule{0.16em}{0ex}}\ensuremath{\rangle}$, whose meaning arises as the overlap $\ensuremath{\langle}\ensuremath{\phi}|\ensuremath{\psi}\ensuremath{\rangle}$ for $|\ensuremath{\psi}\ensuremath{\rangle}$ p...

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

This paper is an investigation of $L$-dual frames with respect to a function-valued inner product, the so called $L$-bracket product on $L^{2}(G)$, where G is a locally compact abelian group with a uniform lattice $L$. We show that several well known theorems for dual frames and dual Riesz bases in a Hilbert space remain valid for $L$-dual frames and $L$-dual Riesz bases in $L^{2}(G)$.

This article addresses the problem of cardinality estimation in inner product spaces. Given a set high-dimensional vectors, query, and threshold, this estimates number vectors such that their products with query are not less than threshold. is an important for recent machine-learning applications maintain objects, as users items, by using matrices. The requirements solutions high efficiency acc...

In this paper, we introduce the concept of family of local atoms in a 2-inner product space and then this concept is generalized to an atomic system. Besides, a characterization of an atomic system lead to obtain a new frame. Actually this frame is a generalization of previous works.

The notion of the sub-exact sequence is generalization exact in algebra especially on a module. A module over ring R vector space field F. Refers to special F when we have complete inner product space, it called Hilbert space. if every Cauchy converges. Now, introduce which can later be useful statistics. This paper aims investigate properties and their relation direct summand As result, get tw...