The Schwartz-Sobolev Theory of Distributions

Ram P. Kanwal
Additional contact information
Ram P. Kanwal: The Pennsylvania State University, Department of Mathematics

Chapter Chapter 2 in Generalized Functions Theory and Technique, 1998, pp 18-48 from Springer

Abstract: Abstract Let R n be a real n-dimensional space in which we have a Cartesian system of coordinates such that a point P is denoted by x = (x1, x2,..., x n ) and the distance r, of P from the origin, is r = |x| = (x 1 2 + x 2 2 + ... + x n 2 )1/2. Let k be an n-tuple of nonnegative integers, k = (k1, k2,..., k n ), the so-called multiindex of order n; then we define $$\begin{matrix}|k|={{k}_{1}}+{{k}_{2}}+\cdots +{{k}_{n}}, {{x}^{k}}=x_{1}^{{{k}_{1}}}x_{2}^{{{k}_{2}}}\cdots x_{n}^{{{k}_{n}}}, \\ k!={{k}_{1}}!{{k}_{2}}!\cdots {{k}_{n}}!, \left( \begin{matrix}k \\ p \\ \end{matrix} \right)\frac{k!}{k!\left( k-p \right)!} \\ \end{matrix}$$ and 1 $${{D}^{k}}=\frac{{{\partial }^{|k|}}}{\partial x_{1}^{{{k}_{1}}}\partial x_{2}^{{{k}_{2}}}\cdots \partial x_{n}^{{{k}_{n}}}}=\frac{{{\partial }^{{{k}_{1}}+{{k}_{2}}+\cdots +{{k}_{n}}}}}{\partial x_{1}^{{{k}_{1}}}\partial x_{2}^{{{k}_{2}}}\cdots \partial x_{n}^{{{k}_{n}}}}=D_{1}^{{{k}_{1}}}D_{2}^{{{k}_{2}}}\cdots D_{n}^{{{k}_{n}}}$$ where D j = ∂/∂x j , j = 1, 2,..., n. For the one-dimensional case D k reduces to d/dx. Furthermore, if any component of k is zero, the differentiation with respect to the corresponding variable is omitted. For instance, in R3, with k = (3, 0, 4), we have $${{D}^{K}}={{\partial }^{7}}/\partial x_{1}^{3}\partial x_{3}^{4}=D_{1}^{3}D_{3}^{4}$$

Keywords: Differential Operator; Integrable Function; Regular Distribution; Null Sequence; Analytic Operation (search for similar items in EconPapers)
Date: 1998
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4684-0035-9_2

Ordering information: This item can be ordered from
http://www.springer.com/9781468400359

DOI: 10.1007/978-1-4684-0035-9_2

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().