User:Maproom/like Fourier

I understand that if a function satisfies certain conditions, specifically it is cyclic, and maybe some others, then it can be expressed as the sum of a series of sine functions. I can learn much more about this by reading Fourier analysis.

I vaguely recall a similar concept. If a function satisfies some other set of conditions (essentially, it's zero-valued or otherwise boring except in one neighbourhood), then it can be expressed as the sum of a series of exponentials. Did I just dream this? Or if it's so, where can I read about it? Maproom (talk) 19:05, 10 June 2016 (UTC)

Well technically any function that can be expressed as the sum of a series of sine functions can be expressed as a series of exponentials using the identity ${\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}$ 2001:630:12:2428:A016:2B2E:370F:6610 (talk) 19:23, 10 June 2016 (UTC)

If you look at the picture sideways you could say that if a function is periodic (and with other nice properties like continuity) with period 2πi then it's the sum of a series of functions of the form e^nx, where n ranges over the integers. There's also the Fourier transform which uses an integral instead of a sum and may be closer to the idea in the original question. --RDBury (talk) 19:46, 10 June 2016 (UTC)

The "otherwise boring except in one neighborhood" condition may be that it is a "Schwarz function" in the Schwartz space. The Fourier transform is an automorphism of this space, which is used to define Tempered distributions & define the Fourier transform for them.John Z (talk) 23:44, 10 June 2016 (UTC)

I think you want Laplace transform which, rather loosely speaking, is to the exponential function what the Fourier transform is to the sine wave. — Preceding unsigned comment added by 82.46.116.9 (talk) 19:03, 12 June 2016 (UTC)

My thanks to all of you. I now have enough leads to find out all I want to know. Maproom (talk) 08:05, 13 June 2016 (UTC)

compact support is the term for a function that is only non-zero in a sort of finite neighborhood. SemanticMantis (talk) 15:53, 13 June 2016 (UTC)

Strictly speaking, compactness is stronger than boundedness.--Jasper Deng (talk) 20:45, 13 June 2016 (UTC)