Non-analytic Smooth Function - Application To Taylor Series

Application To Taylor Series

For every sequence α0, α1, α2, . . . of real or complex numbers, the following construction shows the existence of a smooth function F on the real line which has these numbers as derivatives at the origin. In particular, every sequence of numbers can appear as the coefficients of the Taylor series of a smooth function. This result is known as Borel's lemma, after Émile Borel.

With the smooth transition function g as above, define

This function h is also smooth; it equals 1 on the closed interval and vanishes outside the open interval (−2,2). Using h, define for every natural number n (including zero) the smooth function

which agrees with the monomial xn on and vanishes outside the interval (−2,2). Hence, the k-th derivative of ψn at the origin satisfies

and the boundedness theorem implies that ψn and every derivative of ψn is bounded. Therefore, the constants

involving the supremum norm of ψn and its first n derivatives, are well-defined real numbers. Define the scaled functions

By repeated application of the chain rule,

and, using the previous result for the k-th derivative of ψn at zero,

It remains to show that the function

is well defined and can be differentiated term-by-term infinitely often. To this end, observe that for every k

\sum_{n=0}^\infty\|f_n^{(k)}\|_\infty \le \sum_{n=0}^{k+1}\frac{|\alpha_n|}{n!\,\lambda_n^{n-k}}\|\psi_n^{(k)}\|_\infty +\sum_{n=k+2}^\infty\frac1{n!} \underbrace{\frac1{\lambda_n^{n-k-2}}}_{\le\,1} \underbrace{\frac{|\alpha_n|}{\lambda_n}}_{\le\,1} \underbrace{\frac{\|\psi_n^{(k)}\|_\infty}{\lambda_n}}_{\le\,1} <\infty,

where the remaining infinite series converges by the ratio test.

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