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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} + \dots + a_{1} x + a_{0}

(which has constants coefficients $a_{i} \in ℂ$ ) 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 $f : ℝ \to ℂ$ of class $C^{n + k}$ into the function $p (D) f$ , class at least $C^{k}$ , defined pointwise by

[p (D) f] (x) \dot{=} a_{n} f^{(n)} (x) + a_{n - 1} f^{(n - 1)} (x) + \dots a_{1} f^{'} (x) + a_{0} f (x) .

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Authors: "Mennucci , Andrea C. G." .

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