Wavelet and Linear Algebra

Abstract In this paper, we employ the generalized Guignard's constraint qualification to present the dual cone characterizations of the constraint set $S$ with set valued constraints in $\R^n.$ The obtained results provide sufficient conditions for which the ``strong conical hull intersection property`` (strong CHIP, in short) holds. Moreover, we establish necessary and sufficient conditions for characterizing ``perturbation property`` of the constrained best approximation to any point $x \in \R^{n}$ from a convex set $\tS:=K \cap S$ by the strong CHIP of $K$ and $S$ at a reference point, where $K$ is a non-empty closed convex set in $ \R^{n}.$ Finally, under the generalized Guignard's constraint qualification we derive the Lagrange multipliers characterizations of the constrained best approximation with set valued constraints. The clarification of our results is illustrated by the numerical experiments.

Abstract In this paper a class of Gabor frames with time shift parameter $a>0$, frequency shift parameter $b>0$ and bounded compactly supported generator function $g$ such that $supp\ g\subseteq\left[\left(k+2\right)a-\frac{2}{b},ka+\frac{1}{b}\right]$ or $supp\ g\subseteq\left[\left(k+1\right)a-\frac{1}{b},ka+\frac{1}{b}\right]$, where $k$ is an integer number is introduced. In particular, a sufficient condition on a function $g\in C_c^+\left( \mathbb{R}\right) $ with $supp\ g\subseteq\left[\left(k+2\right)a-\frac{2}{b},ka+\frac{1}{b}\right]$ and positive decreasing derivative $g^\prime$ on $\left(ka-\frac{1}{b},\left( k+2\right)a \right)$, that make $\left\{E_{mb}T_{na}g\right\}_{m,n\in\mathbb{Z}}$ into a Gabor frame, is given.

Abstract Tensor completion is one of the ecient methods for restoring data
such that minimizing the rank of the tensor leads to an appropriate solution.
However, it gives a non-convex objective function, which generates an NPhard
problem. To overcome this problem, instead of using the rank function,
the trace norm is applied. To solve this problem, Simple Low Rank Tensor
Completion (SiLRTC) can be used. In the methods based on trace norm, the
Singular Value Decomposition (SVD) is used, which increases computational
complexity of these methods with increasing dimensions. In order to reduce
the computational complexity of SVD, the approximate SVD can be utilized.
In this paper, to accelerate the convergence speed of SiLRTC Algorithm, the
new combined method FTF-SiLRTC is presented. On the other hand, the
images recovered using the mentioned algorithms are generally accompanied
by horizontal and vertical noise lines and have low accuracy. To solve this
diculty, the total variation (TV) regularization is added to the problem and
the FTF-SiLRTC-TV Algorithm is introduced to solve it with higher accuracy.

Abstract In this paper‎, ‎we explore the properties of spectral functions from the perspective of convex analysis and monotone operator theory‎. ‎Specifically‎, ‎we examine the $\e$-subdifferential and $\e$-enlargement of a spectral function‎. ‎Also‎, ‎we study representative functions associated with the subdifferential of the spectral function‎. ‎In addition‎, ‎Fitzpatrick function has been studied due to its significance as one of the most important representative functions.

Abstract This article uses the rational Haar wavelet and the successive method
to solve the nonlinear Fredholm integral differential equation. Additionally, we
have proved the convergence and order of convergence in this method by using
the fixed point Banach theorem. In this way, numerical integration is not used.
We also talk about two examples. We solved, drawing the absolute error, and
plot of the exact and numerical solution. Finally, the results show that the
proposed method is powerful for solving this equation.

Abstract Abstract: In this paper, we establish extensions of Jensen’s discrete inequality for the class of p-convex functions. Also, we give lower and upper bounds for this inequality. We apply these results in information theory and obtain new and strong bounds for Shannon’s entropy of a probability distribution. Also, We give some applications.