Wavelet and Linear Algebra

Abstract In this paper, we study the excess of continuous $K$-$g$-frames and give some results about this notion.
Also, we extend the concept of atomic system to continuous version and study its relations by continuous $K$-$g$-frames. Indeed, we give some equivalent  characterizations for continuous $K$-$g$-frames.  As well as, the relationship of a continuous $K$-$g$-frame and the range of operator $K$ will be verified. Finally, we study the induced $cK$-frames by continuous $K$-$g$-frames.

Abstract     Trace and some other interesting properties of Haar wavelets matrix of size $2^K$ are studied by Shiralashetti and Kumbinarasaiah \cite{ref6}, results related to the trace of Haar wavelets  matrix derived based on $K$ is even or odd and the same is concluded in Theorem $3.2$. This article deals with the trace of Haar wavelets matrix in depth by identifying and overcoming the pitfalls occur in the proof of Theorem 3.2.

Abstract     In this note, we intend to introduce the concept of weaving continuous g-frames in Hilbert spaces. In addition, we present some new result for these frames and also we show that it is enough to check that on smaller measurable space than the given measurable space. We investigate the relationship between these frames and c-woven also, the sufficient and condition will be given. Finally, we verify the perturbation of weaving c-g-frames.

Abstract
Consider a locally compact group $G$ with two compact subgroups $H$ and $K$. Equip the double coset space $K\setminus G/H$ with the quotient topology. Suppose that $\mu$ is an $N$-relatively invariant measure, on $K\setminus G/H$.
We define a multiplication on $L^1(K\setminus G/H,\mu)$ such that  this space becomes a Banach algebra  that possesses a left (right) approximate identity.

Abstract     In this paper, we reintroduce the concept of controlled K-frames and then, we show that this definition is equivalent
with the concept that has been recently introduced in \cite{Nouri}. Meanwhile, we correct
one of the results which was obtained in the mentioned paper. In the sequel, we construct some new controlled K-frames
by some operator theory tools. Finally, we provide some conditions under which the sum of two controlled K-frames remains
a controlled K-frame.

Abstract It is well known that if a real valued convex function on a compact convex domain
contained in the real numbers attains its maximum,
then it does so at least at one extreme point of its domain.
In this paper,
we consider a matrix convex function on a compact and $C^*$-convex set generated by self--adjoint matrices.
An important issue is so that this function on a compact and $C^*$-convex domain attains its maximum at a $C^*$-extreme point.