On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

On a Generalized Convolution Operator

Recently in the paper [Mediterr. J. Math. 2016, 13, 1535–1553], the authors introduced and studied a new operator which was defined as a convolution of the three popular linear operators, namely the Sǎlǎgean operator, the Ruscheweyh operator and a fractional derivative operator. In the present paper, we consider an operator which is a convolution operator of only two linear operators (with lesser restricted parameters) that yield various well-known operators, defined by a symmetric way, including the one studied in the above-mentioned paper. Several results on the subordination of analytic functions to this operator (defined below) are investigated. Some of the results presented are shown to involve the familiar Appell function and Hurwitz–Lerch Zeta function. Special cases and interesting consequences being in symmetry of our main results are also mentioned.

On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

Classification of Airborne Laser Scanning Point Cloud Using Point-Based Convolutional Neural Network

In various applications of airborne laser scanning (ALS), the classification of the point cloud is a basic and key step. It requires assigning category labels to each point, such as ground, building or vegetation. Convolutional neural networks have achieved great success in image classification and semantic segmentation, but they cannot be directly applied to point cloud classification because of the disordered and unstructured characteristics of point clouds. In this paper, we design a novel convolution operator to extract local features directly from unstructured points. Based on this convolution operator, we define the convolution layer, construct a convolution neural network to learn multi-level features from the point cloud, and obtain the category label of each point in an end-to-end manner. The proposed method is evaluated on two ALS datasets: the International Society for Photogrammetry and Remote Sensing (ISPRS) Vaihingen 3D Labeling benchmark and the 2019 IEEE Geoscience and Remote Sensing Society (GRSS) Data Fusion Contest (DFC) 3D dataset. The results show that our method achieves state-of-the-art performance for ALS point cloud classification, especially for the larger dataset DFC: we get an overall accuracy of 97.74% and a mean intersection over union (mIoU) of 0.9202, ranking in first place on the contest website.

Certain Properties of a Generalized Class of Analytic Functions Involving Some Convolution Operator

We use the concept of convolution to introduce and study the properties of a unified family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$, $(0\leq\gamma\leq1,\,k\geq0)$, consisting of uniformly $k$-starlike and $k$-convex functions of complex order $b\in\mathbb{C}\setminus\{0\}$ and type $\alpha\in[0,1)$. The family $\mathcal{TUM}_\gamma(g,b,k,\alpha)$ is a generalization of several other families of analytic functions available in literature. Apart from discussing the coefficient bounds, sharp radii estimates, extreme points and the subordination theorem for this family, we settle down the Silverman's conjecture for integral means inequality. Moreover, invariance of this family under certain well-known integral operators is also established in this paper. Some previously known results are obtained as special cases.