# The composition and neutrix composition of distributions

• Brian Fisher
Conference paper

## Abstract

Let F be a distribution in $$\mathcal{D}'$$ and let f be a locally summable function. The neutrix composition F(f(x)) is said to exist and be equal to the distribution h if the neutrix limit of the sequence {F n(f(x))} is h, where F n(x) = F(x)*δ n(x) for n = 1, 2, . . . and {δ n(x)} is a certain regular sequence converging to the Dirac delta funcion. In particular, the composition F(f(x)) is said to exist and be equal to the distribution h if the sequence {F n(f(x))} converges to h in the normal sense. Some results are proved.

## References

1. [vdCo59]
van der Corput, J.G.: Introduction to the neutrix calculus. J. Analyse Math., 7, 291–398 (1959)
2. [Fis83]
Fisher, B.: On defining the distribution δ (r)(f(x)). Rostock. Math. Kolloq., 23, 73–80 (1983)
3. [Fis85]
Fisher, B.: On defining the change of variable in distributions. Rostock. Math. Kolloq., 28, 33–40 (1985)Google Scholar
4. [FJÖ02]
Fisher, B., Jolevska-Tuneska, B., Özçag, E.: Further results on the composition of distributions. Integral Transforms Spec. Funct., 13(2), 109–116 (2002)
5. [FKSN05]
Fisher, B., Kananthai, A., Sritanatana, G., Nonlaopon, K.: The composition of the distributions x ms ln x and x +rp/m. Integral Transforms Spec. Funct., 16(1), 13–20 (2005)
6. [FT05a]
Fisher, B., Taş, K.: On the composition of the distributions x +r and x +μ. Indian J. Pure Appl. Math., 36(1), 11–22 (2005)
7. [FT05b]
Fisher, B., Taş, K.: On the composition of the distributions x −1 ln |x| and x +r. Integral Transforms Spec. Funct., 16(7), 533–543 (2005)
8. [FT06]
Fisher, B., Taş, K.: On the composition of the distributions x +λ and x +μ. J. Math. Anal. Appl., 318(1), 102–111 (2006)
9. [GS64]
Gel’fand, I.M., Shilov, G.E.: Generalized Functions. Vol.I. Academic Press, New York London (1964)