Rather unexpected I found in the problem corner on page 42 of the Mathematical Intelligencer vol. 13.1 (1991) the following statement, which attracts me in a way to finding a proof for.
Theorem: Let f be an arbitrarily often differentiable function defined on an open real interval X such that for any x in X there exists a nonnegative integer n such that Dnf ( x ) = 0 . Then the function f is already a polynomial.
Because I didn't see an immediate approach for a proof, I tried to construct a counterexample. This trial was only almost successful. It led me to an idea for proving this theorem. The proof only needs standard tools of analysis.
It should be mentioned that the theorem stated above remains true, if X is an arbitrary interval containing more than one single point and the condition holds for all points except of the endpoints.
The construction of the function, which has been expected to serve as
a counterexample will be given in the following:
Let X be the whole real line and
f ( x ) = 0 for
all x < 0 and
x > 1.
Moreover let
f ( x ) = 1
for any 1/3 < x < 2/3 .
In all of these inequalities the operator
"less than" can be replaced by "less than or equal to"
and "greater than" by "greater than or equal to"
due to the continuity conditions of
f 2).
From the restrictions
f ( x ) = f ( 1-x ) ,
for all x and
Df ( x ) = 3/2 f ( 3x ) ,
for x < 1/3 it follows - provided the function f exists - that f is linear within the intervals [1/9, 2/9] and [7/9, 8/9]. Consequently ist coincides with a quadratic polynomial int he intervals [1/27, 2/27], [7/27, 8/27], [19/27, 20/27] und [25/27, 26/27] etc.
From this construction it follows that for any point of the real line - except of a subset of Cantor's set - this function possesses a vanishing derivative of certain order. (This set of point, where the requirements of the theorem stated above are not satisfied is exactly Cantor's set [Alexandroff, P. S.: Einführung in die Mengenlehre und die Theorie der reellen Funktionen. 5. Aufl., VEB Deutscher Verlag der Wissenschaften, Berlin 1971] except of denumerably many point which are exactly the endpoints of the intervals where the function coincides with a polynomial (see above).
The proof about the existence of this function I've never done. But I looked immediately for another function which is closely related to that just constructed. One gets the new one if the length of the mean interval ([1/3, 2/3] in the construction presented above) tends to zero. (I named this function by the greek letter sigma.) Clearly this function does not longer possess the desired properties to behave polynomially3) almost everywhere. It shows this behavior only on the dense but countable set of dyadic points4). The graph of the function sigma gives an impression of its beauty.
In the first article, which is concerned with this function sigma, the existence is proved [Volk, W.: Properties of subspaces generated by an infinitely often differentiable function and its translates, ZAMM . Z. angew. Math. Mech. 76 (1996) Suppl. 1, S. 575 - 576] (cf. also Publikationen). Moreover it is proven that all polynomials up to a certain degree are contained in function spaces, which are spanned by sigma and its translates.
The function sigma looks very suitable for some applications in applied mathematics. Hence I plan to publish papers on the following topics:
1) The preparation of the HTML-documents have been restricted to simple features. Especially they have been designed without using symbol fonts because those are not available for all platforms. Hence greek letters are represented by they names emphasized in italics. The disadvantage in readability looks for me to be the best alternative. (back to the reference)
2) Theses symbols are not contained in the standard ASCII-font. Consequently they aren't used in this document (cf. also footnote 1) ). (back to the reference)
3) Definition: A real-valued arbitrarily often differentiable function f is called polynomially at a point x within its domain, if there exists a positive integer n0 such that for any integer n > n0 the identity Dnf ( x ) = 0 holds. (back to the reference)
4) Definition The set
{j/2k | j, k in
Z}
is called the set of dyadic points. Here
Z denotes the set of all integers.
(cf. Pollen, D.: Daubechies' scaling function on [0,3],
in: A Tutorial in Theory and Applications,
edited by: C. K. Chui, Academic Press, San Diego 1992,
3 - 13) (back to the reference)
| Back to the root | Created by W. Volk in December 1998 Last correction on January 14th, 2003 |