Guide New Mathematical Forms For Generalized Functions

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Fourier Transforms of Test Functions 1. The Case of Several Variables 1.

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Analytic Functionals 1. Fourier Transforms of Functions in S 2.

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Fourier Transforms of Generalized Functions. A Single Variable 2. Definition 2. Examples 2. Fourier Transforms of Analytic Functionals 3. Several Variables 3. Definitions 3. Fourier Transform of the Direct Product 3. Fourier Transforms and Differential Equations 4.


Introductory Remarks 4. Introductory Remarks on Differential Forms 1. Example: Derivation of Green's Theorem 1. Multiplet Layers 1. Generalized Functions Associated with Quadratic Forms 2. Elementary Solutions of Linear Differential Equations 2. Generalized Functions Associated with Bessel Functions 2.

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Homogeneous Functions 3. Introduction 3. Generalized Homogeneous Functions of Degree —n 3. Generalized Homogeneous Functions of Degree —n — m 3. Reducible Singular Points 4. Generalized Functions of Complex Variables Bl. Homogeneous Functions of a Complex Variable B1. Generalized Functions of m Complex Variables B2.

Homogeneous Generalized Functions B2.

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Associated Homogeneous Functions B2. The Residue of a Homogeneous Function B2. Homogeneous Generalized Functions of Degree —m,—m B2. Powered by. You are connected as.

Algebra of generalized functions (Shirokov) - Wikisource, the free online library

The product of a generalized function in and a function is defined by the equation. Here , and for ordinary functions in , the product coincides with the ordinary product of the functions and. However, this product operation cannot be extended to arbitrary generalized functions in such a way that it is associative and commutative, otherwise there would be the contradiction:.

In order to define the product of two generalized functions and , it is sufficient for them to possess, roughly speaking, the following properties: "non-regularity" of in a neighbourhood of any point must be compensated by corresponding "regularity" of , and conversely; for example, if see Support of a generalized function. A product can be defined in certain classes of generalized functions, but it may turn out not to be uniquely determined. In fact,. But on test functions for which ,.

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Hence it is natural to put if. Extending this functional to all test functions in , one obtains 4. The function does not belong to , but it defines regular generalized functions: in , , and in ,. They can be consistently extended to generalized functions in , for example, by taking the finite Hadamard part of the divergent integral renormalizing it.

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