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Yes, but only on a finite open interval, that is (a, b) with a, b real numbers. In the post the sum of periodic functions is equal to x^2 on the whole real lin
by ot 13y ago
Yes, but only on a finite open interval, that is (a, b) with a, b real numbers.
In the post the sum of periodic functions is equal to x^2 on the whole real line. This is impossible with sums of continuous periodic functions, such as Fourier series, because continuous periodic functions are bounded, while x^2 is unbounded.
- vbuterin 13y ago> This is impossible with sums of continuous periodic functions, such as Fourier series, because continuous periodic functions are bounded, while x^2 is unbounded. But infinite sums of continuous periodic functions are not necessarily bounded: sin(x) + sin(x/2) + sin(x/4) + ... is an example. So there has to be something else to the proof.
- ot 13y agoYou are right, but I was referring to either finite sums or Fourier series, your example is not one because sum of the squares of the coefficients is infinite. I don't know what can be said about arbitrary countable/uncountable sums of continuous periodic functions.