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Properties of the Radon Transform and Inversion Formulas
Published in A.G. Ramm, A.I. Katsevich, The RADON TRANSFORM and LOCAL TOMOGRAPHY, 2020
Throughout this section (unless specified otherwise) we assume that f(x) is from the Schwartz space S consisting of infinitely differentiable functions which rapidly decay with all their derivatives. More precisely, this space consists of all f ∈ C∞ (ℝn) such that supx∈ℝn|x|β|∂xγf(x)|<∞,
Generalizations of Shannon Sampling Theorem
Published in Ahmed I. Zayed, Advances in Shannon’s Sampling Theory, 2018
Provided with appropriate topologies, these vector spaces become testing-function spaces in the sense of Zemanian [123]. Let E*, D*, K*(σ), S*, Z*(σ) denote their dual spaces respectively. The space S is known as the Schwartz space of C∞ rapidly decreasing functions and its dual S* is the space of tempered distributions. D* is the space of Schwartz distributions.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
We note some basic properties of the Fourier transform in Theorem 4.2. There we write C(ℝ) for the continuous functions from ℝ to ℂ. Many of the more difficult results in the theory are proved by first working in a space, called the Schwartz space, of smooth functions with rapid decay at infinity.
On the approximate inverse method in SPECT image reconstruction
Published in Applicable Analysis, 2020
Next, we give some definitions of spaces and norms needed in the subsequent sections. Namely, denote Ω as an open subset of , the space of square integrable functions on a set with the -norm and the scalar product , the space of tempered distributions and the space the Schwartz space.
On a class of fractional Schrödinger equations in
Published in Applicable Analysis, 2018
Manassés de Souza, Yane Lísley Araújo
where , , denotes the fractional Laplacian, defined for all function belongs to the Schwartz space, by
Normalized solutions to the fractional Kirchhoff equations with a perturbation
Published in Applicable Analysis, 2023
Lintao Liu, Haibo Chen, Jie Yang
In this paper, we are concerned with the existence of solutions to the following fractional Kirchhoff equation: under the normalized constraint where , a, b, c>0, and is a parameter. The fractional Laplaction can be defined by for , where is the Schwartz space of rapidly decaying function, denote an open ball of radius ε centered at x and the normalization constant . For , the fractional Laplaction can be defined by the Fourier transform , being the usual Fourier transform. The application background of operator can be founded in several areas such as fractional quantum mechanics [1, 2], physics and chemistry [3], obstacle problems [4], optimization and finance [5], conformal geometry and minimal surfaces [6] and so on. Note that on is a nonlocal operator. In the remarkable work of Caffarelli and Silvestre [7], they introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions. More precisely, for a , the extension be defined by Then, it follows from [7] that where is a constant depending on N and s. This extension method has been successfully applied to nonlinear equations involving a fractional Laplacian, and a series of significant results have been obtained, we refer the readers to see [8–13] and the references therein.