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.

Read more about this topic:  Non-analytic Smooth Function

Famous quotes containing the words application to, application, taylor and/or series:

    ““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)

    “I think that a young state, like a young virgin, should modestly stay at home, and wait the application of suitors for an alliance with her; and not run about offering her amity to all the world; and hazarding their refusal.... Our virgin is a jolly one; and tho at present not very rich, will in time be a great fortune, and where she has a favorable predisposition, it seems to me well worth cultivating.”
    —Benjamin Franklin (1706–1790)

    “Make me thy Loome: thy Grace the warfe therein,
    My duties Woofe, and let thy word winde Quills.
    The shuttle shoot. Cut off the ends my sins.
    Thy Ordinances make my fulling mills,
    My Life thy Web: and cloath me all my dayes
    With this Gold-web of Glory to thy praise.”
    —Edward Taylor (1645–1729)

    “Personality is an unbroken series of successful gestures.”
    —F. Scott Fitzgerald (1896–1940)