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There is what I call infinite polynomials, which, when done right, can create any real to real function. These polynomials are made up of various things times 1, x, x squared, etc. Similarly sine and cosine functions can be added together in a Fourier series, and can also approximate any function. Are there other, distinct kinds of functions that can do this, like perhaps exponential functions?

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    $\begingroup$ Your "infinite polynomials" sound like analytic functions. $\endgroup$
    – user403337
    Jun 20, 2019 at 4:01
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    $\begingroup$ Fourier series cannot be added (even countably) to create any real-to-real function. However, given a square-integrable function, Fourier series can have a sum that agrees with the given almost everywhere (or everywhere, when the function is continuous). Unfortunately, there are just too many functions from $\Bbb{R}$ to $\Bbb{R}$ to be expressed as countable linear combinations of a countable set of functions. $\endgroup$ Jun 20, 2019 at 4:16

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Yes! See https://en.wikipedia.org/wiki/Orthogonal_functions.

What you're looking for is a complete set of orthogonal functions; in other words, a collection of independent functions that, when put together, can approximate any continuous function as accurately as you like.

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    $\begingroup$ Being about to approximate any continuouus function doesn't have anything to do with orthogonality. $\endgroup$ Jun 20, 2019 at 4:04
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    $\begingroup$ Yes but if a complete set of orthogonal functions spans a Hilbert space that is dense in $C[a,b]$ and they can be decomposed into simpler functions like OP is mentioning, then this certainly applies $\endgroup$
    – whpowell96
    Jun 20, 2019 at 4:09
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Maybe you are looking for this:

https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem

In its original form, it states that polynomials can approximate continuous functions (as you might already know). This has been generalized to many families of functions. Basically, you only need your family to be a C*-algebra that separates points (i.e, there are functions that take different values on different points). This not only works on real functions, it also works on a much broader kind of space (compact Haussdorf spaces)

Note: This works on real closed intervals as big as you want, but not on all $\mathbb{R}$

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  • $\begingroup$ I think this is a better answer than the accepted one. It's the conditions of the Stone-Weierstrass theorem that are the key, not orthogonality. $\endgroup$
    – bubba
    Jun 24, 2019 at 9:02
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A general context for your question is that of a Schauder basis for a Banach space. The famous Haar sequence is an example of a type of function you are describing, and is the foundation for the theory of wavelets.

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