By V. S. Vladimirov
This quantity offers the final concept of generalized features, together with the Fourier, Laplace, Mellin, Hilbert, Cauchy-Bochner and Poisson imperative transforms and operational calculus, with the normal fabric augmented by way of the speculation of Fourier sequence, abelian theorems, and boundary values of helomorphic services for one and several other variables. the writer addresses a number of aspects extensive, together with convolution concept, convolution algebras and convolution equations in them, homogenous generalized features, and multiplication of generalized services. This booklet will meet the desires of researchers, engineers, and scholars of utilized arithmetic, regulate thought, and the engineering sciences.
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Extra resources for Methods of the Theory of Generalized Functions (Analytical Methods and Special Functions)
Sample text
Examples. 1. Let us compute the density of charges corresponding to the dipole of the moment +1 located at the point x 0 and oriented in a given direction 1 (iI, ... ,In), III 1 (Fig. 9). Approximately corresponding to this dipole is the charge density (see Sees. 7) = = 1 1 C £ = -o(x - cl) - -o(x), Passing to the limi t here as £ 1) = -[
The antiderivative (primitive) of a generalized function. Every function f(x) continuous in an interval (a, b) has in (a, b) a unique (up to an additive constant) anti derivative j(-l)(x), f /(~) x f(-l)(X) = d€ + C, /(-1)' (x) = f(x). The last equality is what we will start with to define the antiderivative or primitive of an arbitrary generalized function / (of one variable). Suppose f E V'(a,b). pEV(a,b). 1) shows that the function j(-l) is not specified on all test functions taken from V(a J b), but only on their first derivatives.
P E V(a, b). ,p(x) 0 for x a" = min(a',xQ - c) > a ifsupp\O C [a',b'] C (a,b). Furthermore, for X> 6" max(b', Xo + c) < b, ! ,p(x) = ! 00
