Generalized function, derivative of a
A weak extension of the operation of ordinary differentiation. Let
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 |
Comments
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 |
Generalized function, derivative of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_function,_derivative_of_a&oldid=52283