Generalized function, derivative of a

A weak extension of the operation of ordinary differentiation. Let $ f $ be a generalized function, $ f \in \mathcal 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

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

Since the operation $ \phi \mapsto (- 1) ^ {| \alpha | } D ^ \alpha \phi $ is linear and continuous from $ \mathcal D ( O) $ into $ \mathcal D ( O) $, the functional $ D ^ \alpha f $ defined by the right-hand side of (*) is a generalized function in $ \mathcal 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 \mathcal D ^ \alpha f $ is linear and continuous from $ \mathcal D ^ \prime ( O) $ into $ \mathcal D ^ \prime ( O) $; any generalized function in $ \mathcal 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.

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 $ \mathcal 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 $ \mathcal D ^ \prime $ and it can be differentiated term-by-term in $ \mathcal 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

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5 MR0617913 Zbl 0435.46002
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001