Meaning of differential operator | Babel Free

Definitions

An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).

Equivalents

Examples

“Define P(D) to be the differential operator given by P(D)#61;D²#43;5D#43;6, where D is the differentiation operator. Then P(D)(#92;sin(x))#61;-#92;sin(x)#43;5#92;cos(x)#43;6.”

“[…]these conditions appeared in the very first papers on ordinary differential operators.”

“A differential operator P#92;left(D#92;right) is hypoelliptic if for any domain ⊂ #92;mathbb#123;R#125;ⁿ any solution u of the equation P#92;left(D#92;right)u#61;0 from the class #92;mathfrak#123;D#125;'#92;left(#92;Omega#92;right) is a function from C#92;infty(ω) for any open set ω ⊂⊂ #92;Omega. A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].”

“Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.”

CEFR level