Distribution (Functional)

A functional distribution, also called a Generalized Function, is a generalization of the concept of a function. Functional distributions are defined as continuous linear Functionals over a Space of infinitely differentiable functions such that all continuous functions have Schwarzian Derivatives which are themselves distributions. The most commonly encountered functional distribution is the Delta Function.

See also Delta Function, Generalized Function, Schwarzian Derivative

References

Distribution Theory

Friedlander, F. G. Introduction to the Theory of Distributions. Cambridge, England: Cambridge University Press, 1982.

Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 1: Properties and Operations. New York: Harcourt Brace, 1977.

Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 2: Spaces of Fundamental and Generalized Functions. New York: Harcourt Brace, 1977.

Gel'fand, I. M. and Shilov, G. E. Generalized Functions, Vol. 3: Theory of Differential Equations. New York: Harcourt Brace, 1977.

Gel'fand, I. M. and Vilenkin, N. Ya. Generalized Functions, Vol. 4: Applications of Harmonic Analysis. New York: Harcourt Brace, 1977.

Gel'fand, I. M.; Graev, M. I.; and Vilenkin, N. Ya. Generalized Functions, Vol. 5: Integral Geometry and Representation Theory. New York: Harcourt Brace, 1977.

Griffel, D. H. Applied Functional Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1984.

Halperin, I. and Schwartz, L. Introduction to the Theory of Distributions, Based on the Lectures Given by Laurent Schwarz. Toronto, Canada: University of Toronto Press, 1952.

Lighthill, M. J. Introduction to Fourier Analysis and Generalised Functions. Cambridge, England: Cambridge University Press, 1958.

Richards, I. and Young, H. The Theory of Distributions: A Nontechnical Introduction. New York: Cambridge University Press, 1995.

Rudin, W. Functional Analysis, 2nd ed. New York: McGraw-Hill, 1991.

Strichartz, R. Fourier Transforms and Distribution Theory. Boca Raton, FL: CRC Press, 1993.

Zemanian, A. H. Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications. New York: Dover, 1987.