Generalized function, derivative of a

A weak extension of the operation of ordinary differentiation. Let $f$ be a generalized function, $f \in D ^ \prime ( O)$. The generalized (weak) derivative

$$D ^ \alpha f = \ \frac{\partial ^ {| \alpha | } f }{\partial x _ {1} ^ {\alpha _ {1} } \dots \partial x _ {n} ^ {\alpha _ {n} } } ,\ \ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} ,$$

of order $\alpha = ( \alpha _ {1} \dots \alpha _ {n} )$ is defined by the equation

$$\tag{* } ( D ^ \alpha f , \phi ) = \ ( - 1 ) ^ {| \alpha | } ( f , D ^ \alpha \phi ) ,\ \ \phi \in D ( O) .$$

Since the operation $\phi \mapsto (- 1) ^ {| \alpha | } D ^ \alpha \phi$ is linear and continuous from $D ( O)$ into $D ( O)$, the functional $D ^ \alpha f$ defined by the right-hand side of (*) is a generalized function in $D ^ \prime ( O)$. If $f \in C ^ {p} ( O)$, then $D ^ \alpha f \in C ^ {p - | \alpha | } ( O)$ for all $\alpha$ with $| \alpha | \leq p$.

The following properties hold for the derivatives of a generalized function: the operation $f \mapsto D ^ \alpha f$ is linear and continuous from $D ^ \prime ( O)$ into $D ^ \prime ( O)$; any generalized function in $D ^ \prime ( O)$ is infinitely differentiable (in the generalized sense); the result of differentiation does not depend on the order; the Leibniz formula is valid for the differentiation of a product $af$, when $a \in C ^ \infty ( O)$; and $\supp D ^ \alpha f \subset \supp f$.

Let $f \in L _ { \mathop{\rm loc} } ^ {1} ( O)$. It may happen that a certain generalized derivative can be identified with some $L _ { \mathop{\rm loc} } ^ {1} ( O)$- function. In this case $D ^ \alpha f ( x)$ is a generalized derivative of function type.

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Examples.

1) $\theta ^ \prime = \delta$, where $\theta$ is the Heaviside function and $\delta$ is the Dirac function (cf. Delta-function for both).

2) The general solution of the equation $u ^ \prime = 0$ in the class $D ^ \prime$ is an arbitrary constant.

3) The trigonometric series

$$\sum _ {k = - \infty } ^ \infty a _ {k} e ^ {ikx} ,\ \ | a _ {k} | \leq A ( 1 + | k | ) ^ {m} ,$$

converges in $D ^ \prime$ and it can be differentiated term-by-term in $D ^ \prime$ infinitely many times.

References

 [1] L. Schwartz, "Théorie des distributions" , 1 , Hermann (1950) MR0035918 Zbl 0037.07301 [2] S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian) MR0165337 Zbl 0123.09003