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The Fourier series is a decomposition of an arbitrary function into a superposition of sinusoidal functions. Some time back, I asked a question about whether it is possible to decompose functions using other families of functions and indeed, it is.

Today's question is much simpler: What do you call such an expansion and the associated transform? Is there a standard name for "a transform whose counterpart for finite domains decomposes a function into the sum of several other functions"?

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I don't think Fourier transform is what you mean, you probably mean Fourier series decomposition. – Ben Millwood Jul 2 '12 at 12:43
How abou "transform"? – akkkk Jul 2 '12 at 12:46
@Auke If that's the correct term, then so be it. I was just wondering whether there's some more specific term for this sort of thing. – MathematicalOrchid Jul 2 '12 at 12:54
@MathematicalOrchid: well, "domain transform" (as in, moving from time domain to frequency domain) would be more specific, but I'm not sure people would understand what you mean by that. – akkkk Jul 2 '12 at 12:55
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There's a concept called an integral transform that generalizes the Fourier transform -- but I don't think that's exactly what you're looking for.

Alternatively, you could be looking at a Schauder basis for the function space in question -- but that name just corresponds to an assertion that a series form for every function exists and doesn't say anything about how to find it.

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I had assumed that "integral transform" simply means any operation that involves an integral somehow. But it appears that it means something rather more specific. Indeed, I think this is probably about the right answer for my question. – MathematicalOrchid Jul 2 '12 at 13:55

I guess Integral Transforms would be the right word.

These are generally used to transform PDEs into separable PDEs, an example of which is the Fourier Transform.

If the domain is finite or periodic, an infinite sum of solutions (eg:superposition of sinusoidal functions) is okay, but an integral of solutions is generally required for infinite domains.

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