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Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series).

Are there any other orthogonal and complete set of functions for square integrable functions besides e^inx?

Are there any orthogonal and complete set of functions that work for every function on a finite interval?

Is there some overview of different orthogonal complete set of functions with different conditions on the function, besides square integrable?

And the functions should be linearly independent too.

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    Orthogonality implies linear independence.2012-12-12

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The Walsh functions, square waves with periods that are dyadic fractions of the interval length, are another family. There are many more. You can even define your own-just make a new list that is a blend of two others and solve the orthogonality issue. For example, if $n$ has an odd number of factors of $2$ dividing it, we can use $e^{inx}+e^{2inx}$ and $e^{inx}-e^{2inx}$ to make another family. All the families that I know are countable, so cannot represent arbitrary functions, because we only allow countable sums.