Generalized function, derivative of a
of order is defined by the equation
Since the operation is linear and continuous from into , the functional defined by the right-hand side of (*) is a generalized function in . If , then for all with .
The following properties hold for the derivatives of a generalized function: the operation is linear and continuous from into ; any generalized function in 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 , when ; and .
Let . It may happen that a certain generalized derivative can be identified with some -function. In this case is a generalized derivative of function type.
1) , where is the Heaviside function and is the Dirac function (cf. Delta-function for both).
2) The general solution of the equation in the class is an arbitrary constant.
3) The trigonometric series
converges in and it can be differentiated term-by-term in infinitely many times.
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Generalized function, derivative of a. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Generalized_function,_derivative_of_a&oldid=28201