oscillating and non-oscillating functions

[Tania]

Can someone direct me to a rigorous proof that an oscillating function cannot be represented using a FINITE number of non-oscillating functions.

Example;

cos(x) cannot be represented using a FINITE number of non-oscillating functions [excluding non-real functions like exp(ix) ]

I have had some thoughts on increasing/decreasing functions and that combining a FINITE number of them will not produce a function which oscillates for all x. A full proof (or where to find it) would be most helpful

Thanks...Tania.

 

[Opalg]

Can someone direct me to a rigorous proof that an oscillating function cannot be represented using a FINITE number of non-oscillating functions.

What do you mean by a "non-oscillating function"? And what do you mean by "represented"?

I would say that the function f(x) = cos(x) - x is non-oscillating. So is the function g(x) = x. Yet their sum is cos(x). Does this count as an example of "representing" an oscillating function using "non-oscillating functions"?

 

[Tania]

oscillating function

Thanks for the reply:

cos(x)-x does not oscillate but it is made up of an oscillating part. By represent I mean 'equals'

I guess I really want to know if functions like cos(x) can be equal a combination of a finite number of rational functions, polynomials, exp(x), ln(x) functions but not complex exponentials.

Thanks to all for reading and thinking about this.