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Banach Spaces
Published in Hugo D. Junghenn, Principles of Analysis, 2018
An application of the CBS inequality shows that Kf2≤k2f2 $ \left\Vert Kf\right\Vert_2 \le \left\Vert k\right\Vert_2\left\Vert f\right\Vert_2 $ , hence K is a bounded linear operator on L2(μ) $ L^2(\mu ) $ with K≤k2 $ \left\Vert K\right\Vert \le \left\Vert k\right\Vert_2 $ . The operator K is called an integral operator with kernel k . We show that K is compact.
Examples of linear inverse problems
Published in Mario Bertero, Patrizia Boccacci, Christine De Mol, Introduction to Inverse Problems in Imaging, 2021
Mario Bertero, Patrizia Boccacci, Christine De Mol
The function K(x,x′) is the space-variant PSF of the system, already defined in Section 2.1. In the theory of integral equations this function is also called the kernel (or integral kernel) of the integral operator. However, we must point out that in other examples considered in this chapter, such as computed tomography, the kernel is a distribution and not a function. In this section we assume that it is a function.
Integral Equations
Published in Ronald B. Guenther, John W. Lee, Sturm-Liouville Problems, 2018
Ronald B. Guenther, John W. Lee
Integral equationThe theory of integral equations was developed in part as a powerful tool for studying problems originally formulated in terms of ordinary or partial differential equations. It is natural that a problem formulated in terms of differential equations can be converted into an integral equation because differentiation and integration are inverse processes. One advantage of converting to an integral equation is that the integral operator that arises is better behaved than the differential operator in the original problem. Another advantage is that boundary conditions are incorporated directly into the integral equation and do not have to be treated separately.
Real-time simulation and performance of DSTATCOM using an improved load current detection-based control technique for compensation of current harmonics and load transients
Published in EPE Journal, 2021
Trilochan Penthia, Anup Kumar Panda, Mrutyunjaya Mangaraj
Kernel learning method: The Kernel trick or method is very popular in the field of machine learning, signal processing and data mining [21,22]. The unique features and advantages of the Kernel method have been well explained in the literatures [23,24]. Moreover, the learning method uses a function known as Kernel function, which operates in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space. The Kernel function was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations. Considering the advantages of Kernel learning concept (i.e. based on the Reproducing Kernel Hilbert Space (RKHS) [24,25]), the weights of the proposed Icosθ control algorithm are updated in this paper, as explained in Subsection-3.3.