A note on Roe’s characterization of the sine function

Abstract:

Let ${f^{(n)}}n = 0, \pm 1, \pm 2, \ldots$ be a sequence of complex valued functions on the real line with $(d/dx){f^{(n)}} = {f^{(n + 1)}}$ and satisfying inequalities $|{f^{(n)}}(x)| \leq {M_n}{(1 + |x|)^k}$ where as $n \to \infty$ the growth conditions $\underline {\lim } {M_n}{(1 + \varepsilon )^{ - n}} = 0$ and $\underline \lim {M_{ - n}}{(1 + \varepsilon )^{ - n}} = 0$ hold for all $\varepsilon > 0$. Then ${f^{(0)}}(x) = p(x){e^{ix}} + q(x){e^{ - ix}}$ where $p$ and $q$ are polynomials of degree at most $k$.