# The Schwartz-Sobolev Theory of Distributions

• Ram P. Kanwal

## 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 = (x 1, x 2,…, 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 = (k 1, k 2,…, k n ), the so-called multiindex of order n; then we define
$$\begin{gathered} \left| k \right| = k_1 + k_2 + \cdot \cdot \cdot k_n , x^k = x_1 ^{k_1 } x_2^{k_2 } \cdot \cdot \cdot x_n^{k_n } , \hfill \\ k! = k_1 !k_2 \cdot \cdot \cdot k_n !, \left( {\begin{array}{*{20}c} k \\ p \\ \end{array} } \right) = \frac{{k!}} {{k!\left( {k - p} \right)}} \hfill \\ and \hfill \\ D_k = \frac{{\partial ^{\left| k \right|} }} {{\partial x_1^{k_1 } \partial x_2^{k_2 } \cdot \cdot \cdot \partial x_n^{k_n } }} = \frac{{\partial ^{k_1 + k_2 + \cdot \cdot \cdot + k_n } }} {{\partial x_1^{k_1 } \partial x_2^{k_2 } \cdot \cdot \cdot \partial x_n^{k_n } }} = D_2^{k_2 } \cdot \cdot \cdot D_n^{k_n } , \hfill \\ \end{gathered}$$
(1)
where D j = ∂/∂x j , j = 1,2,…, n. For the one-dimensional case D k reduces to d/DK. Furthermore, if any component of k is zero, the differentiation with respect to the corresponding variable is omitted. For instance, in R 3, with k = (3, 0, 4), we have
$$D^k = \partial ^7 /\partial x_1^3 \partial x_3^4 = D_1^3 D_3^4 .$$
(2)

## Keywords

Differential Operator Integrable Function Delta Function Regular Distribution Null Sequence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.