Definition
27
[23Z]We formally indicate with \(D\) the operation ”computing of the derivative”. Given a polynomial \(p(x)\)
\[ p(x)=a_ n x^{n} + a_{n-1} x^{n-1} + \ldots + a_{1} x + a_ 0 \]
(which has constants coefficients \(a_ i∈ℂ\)) we formally construct the linear operator
\[ p(D)=a_ n D^{n} + a_{n-1} D^{n-1} + \dots a_{1} D + a_ 0 \]
which transforms a function \(f:ℝ→ℂ\) of class \(C^{n+k}\) into the function \(p(D) f\), class at least \(C^ k\), defined pointwise by
\[ [p(D) f] (x) {\stackrel{.}{=}}a_ n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \dots a_{1} f'(x) + a_ 0 f(x)\quad . \]