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 ψ_n at the origin satisfies

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

involving the supremum norm of ψ_n 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 ψ_n 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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