A note on convolution operators on Riesz Bounded variation spaces

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions.

On Multivalent Analytic Functions Considered by a Multi-Arbitrary Differential Operator in a Complex Domain

(1) Background: There is an increasing amount of information in complex domains, which necessitates the development of various kinds of operators, such as differential, integral, and linear convolution operators. Few investigations of the fractional differential and integral operators of a complex variable have been undertaken. (2) Methods: In this effort, we aim to present a generalization of a class of analytic functions based on a complex fractional differential operator. This class is defined by utilizing the subordination and superordination theory. (3) Results: We illustrate different fractional inequalities of starlike and convex formulas. Moreover, we discuss the main conditions to obtain sandwich inequalities involving the fractional operator. (4) Conclusion: We indicate that the suggested class is a generalization of recent works and can be applied to discuss the upper and lower bounds of a special case of fractional differential equations.

Some properties of the fractal convolution of functions

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

Fast Convolution based on Winograd Minimum Filtering: Introduction and Development

Convolutional Neural Network (CNN) has been widely used in various fields and played an important role. Convolution operators are the fundamental component of convolutional neural networks, and it is also the most time-consuming part of network training and inference. In recent years, researchers have proposed several fast convolution algorithms including FFT and Winograd. Among them, Winograd convolution significantly reduces the multiplication operations in convolution, and it also takes up less memory space than FFT convolution. Therefore, Winograd convolution has quickly become the first choice for fast convolution implementation within a few years. At present, there is no systematic summary of the convolution algorithm. This article aims to fill this gap and provide detailed references for follow-up researchers. This article summarizes the development of Winograd convolution from the three aspects of algorithm expansion, algorithm optimization, implementation, and application, and finally makes a simple outlook on the possible future directions.

Wildfire Segmentation Using Deep Vision Transformers

In this paper, we address the problem of forest fires’ early detection and segmentation in order to predict their spread and help with fire fighting. Techniques based on Convolutional Networks are the most used and have proven to be efficient at solving such a problem. However, they remain limited in modeling the long-range relationship between objects in the image, due to the intrinsic locality of convolution operators. In order to overcome this drawback, Transformers, designed for sequence-to-sequence prediction, have emerged as alternative architectures. They have recently been used to determine the global dependencies between input and output sequences using the self-attention mechanism. In this context, we present in this work the very first study, which explores the potential of vision Transformers in the context of forest fire segmentation. Two vision-based Transformers are used, TransUNet and MedT. Thus, we design two frameworks based on the former image Transformers adapted to our complex, non-structured environment, which we evaluate using varying backbones and we optimize for forest fires’ segmentation. Extensive evaluations of both frameworks revealed a performance superior to current methods. The proposed approaches achieved a state-of-the-art performance with an F1-score of 97.7% for TransUNet architecture and 96.0% for MedT architecture. The analysis of the results showed that these models reduce fire pixels mis-classifications thanks to the extraction of both global and local features, which provide finer detection of the fire’s shape.

Spectra and ergodic properties of multiplication and convolution operators on the space $${\mathcal S}({\mathbb R})$$

In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ S ( R ) of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ M h : S ( R ) → S ( R ) , $$f \mapsto h f $$ f ↦ h f , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ C T : S ( R ) → S ( R ) , $$f\mapsto T\star f$$ f ↦ T ⋆ f . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

CONVOLUTION EQUATIONS ON THE ABELIAN GROUP $\cA(-1,1)$

The interval $j=[-1,1]$ turns into an Abelian group $\cA(\cJ)$ under the group operation $x+_\cJ y:=(x+y)(1+xy)^{-1},\qquad x,y\in\cJ$. This enables definition of the invariant measure $d_\cJ x=(1-x^2)^{-1}dx$ and the Fourier transform $\cF_\cJ$ on the interval $\cJ$ and, as a consequence, we can consider Fourier convolution operators $W^0_{\cJ,\cA}:=\cF_\cJ^{-1}\cA\cF_\cJ$ on $\cJ$. This class of convolutions includes celebrated Prandtl, Tricomi and Lavrentjev-Bitsadze equations and, also, differential equations of arbitrary order with the natural weighted derivative $\fD_\cJ u(x)=-(1-x^2)u’(x)$, $t\in\cJ$. Equations are solved in the scale of Bessel potential $\bH^s_p(\cJ,d_\cJ x)$, $1\leqslant p\leqslant\infty$, and H\”older-Zygmound $\bZ^\nu(\cJ,(1-x^2)^\mu)$, $0<\mu,\nu<\infty$ spaces, adapted to the group $\cA(\cJ)$. Boundedness of convolution operators (the problem of multipliers) is discussed. The symbol $\cA(\xi)$, $\xi\in\bR$, of a convolution equation $W^0_{\cJ,\cA}u=f$ defines solvability: the equation is uniquely solvable if and only if the symbol $\cA$ is elliptic. The solution is written explicitely with the help of the inverse symbol. We touch shortly the multidimensional analogue-the Abelian group $\cA(\cJ^n)$.