Wavelet and Linear Algebra

Abstract In this paper, we prove some new results about cK-frames. Also, we introduce the concept of cK-Riesz basis and we provide a necessary and sufficient condition under which $F$ is a cK-Riesz basis. Finally, for the closed range operator $K \in B(\mathcal{H})$, we prove that under some conditions, $\pi_{R(K)}F$ is a cK-Riesz basis if and only if it has only one dual, where $\pi_{R(K)}$ is the orthogonal projection from $\mathcal{H}$ onto $R(K)$, i.e., the range of $K$.

Abstract In this paper,  we discuss some generalizations coming from wavelet transform on Sobolev spaces. In particular, we introduce the bounded localization operators on Sobolev spaces which are related to multi-dimensional wavelet transform on Sobolev spaces. Moreover, we propose the localization operators on Sobolev spaces are in $p$-Schatten class and they are compact. Finally, we give the boundedness and compactness of localization operators on Sobolev spaces with two admissible wavelets.

Abstract  One of the popular methods of data clustering is spectral clustering. The main step of this method is constructing a graph representation of the data set and its similarity matrix. The similarity matrices which are constructed based on some important points not all data points, are among the main approaches. In this paper, the stationary distribution for a random walk on a weighted graph $G$ is considered to find anchor points of the data set. Then we build the similarity matrix based on the anchor nodes and the weighted random walk transition matrix. After that, spectral clustering is applied on the gained similarity matrix. We propose the theoretical discussions and then we evaluate our method on benchmarks.        

Abstract In this paper, the notion of $C-$spectral norm is introduced for operators; it was defined and studied for matrices before. Here, some $C-$spectral norm inequalities between operator matrices and their operator entries, for $2\times 2$ and $n \times n$ operator matrices, are studied. Also, some $C-$spectral norm equalities between operator matrices are brought.

Abstract In this paper, we study Hermite-Hadamard type inequalities for sub-topical (increasing and plus sub-homogeneous) functions in the framework of abstract convexity. Some examples of such inequalities for functions defined on special domains are given.

Abstract As a generalization of convexity,  $ E $-convexity has been defined and studied in many publications. In this study, we recall the class of $ E $-$\mathbb {Q} $-convex sets, $ E $-$ \mathbb {Q}  $-convex and $ E $-additive functions and proved some properties of $ E $-$ \mathbb {Q}  $-convex functions.  Also, we develop the classical theorems of Jensen and Bernstein-Doetsch on $ E $-$ \mathbb {Q}  $-convex functions when vector spaces are over the rational numbers $ \mathbb {Q} $.

Abstract T. Ando characterized linear preservers of majorization in  [Linear Algebra Appl. 118 (1989) 163-248]. In this note, we present a method to state a simple proof of Ando's theorem. By using this method, we state an equivalent condition for matrix representations of linear preservers of $G$-majorization on matrices, where  $G$ is a finite subgroup of orthogonal  group $O(\mathbb{R}^n)$.
Moreover, we introduce reflective majorization and characterize all its linear preservers.