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Operations on generalized functions.

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posted on 19.11.2015, 08:55 by Emin. Özçag
In Chapter 1, we give some properties distributions and introduce the notions of neutrix and neutrix limit with examples, in order to study the problem of defining the convolution product and the product of distributions. The problem of defining the distribution such that the ordinary derivative formula is satisfied for all and s = 0,1,2,... is studied in Chapter 2. In Chapter 3, we define the Beta function Bp,q (,) using the neutrix limit and prove that this neutrix limit exists for all . In Chapter 4 we let f and g be distributions and let fn(x) = f(x)Tn(x), where Tn(x) is a certain function which converges to the identity function as n tends to infinity. We then define the neutrix convolution product fg as the neutrix limit of the sequence {lcub}fn * g{rcub}, provided the limit h exists in the sense that N - limn fn * g,? = h, for all in D. The neutrix convolution products In are evaluated, from which other neutrix convolution products are deduced. The neutrix convolution product of distributions in Chapter 4 is not commutative. Therefore, in Chapter 5, we consider the commutative neutrix convolution product of distributions, *, and also evaluate the neutrix convolution product. The problem of defining the product of ultradistributions is considered in Chapter 6, and the neutrix product (Ff) (Fg) in Z', where F denotes the Fourier transform, is defined as the neutrix limit of {lcub}F(fTn).F(gTn). Later, we prove that the exchange formula holds. We finally define the neutrix product F(f)0G(g) of F(f) and G(g), where F and G are distributions and f and g are locally summable functions. It is proved that if f is infinitely differentiable function with f'(x) 0 and if the neutrix product F o G exists and equals H, then the neutrix product F(f) o G(f) exists and equals H(f). We also give an alternative approach to the form F(f(x)) in D', where F and f are distributions.


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University of Leicester

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