What conditions must a sequence of functions$ \{f_n\}_{n=1}^{\infty} $ must have in order to generate any "nice" function $F(x)$ as $F(x)=\sum\limits_{-\infty}^\infty a_n f_n(x)$ .
For example:
- The monomials $ x^n $ can represent many infinitely differentiable functions via Taylor series.
- The exponential functions $ e^{inx} $ form the basis for Fourier series and allow us to represent periodic functions (under suitable integrability or smoothness conditions).
My question is:
What kind of general conditions must a sequence $ \{f_n\}$ satisfy so that any “nice” function can be represented as a (possibly infinite) linear combination of the $ f_n $?
A “nice function” here I mean the one that is continuous or even infinitely differentiable, depending on the context. One might also impose additional conditions, such as restricting the domain or requiring the function to be periodic or satisfy other properties.
Another related question is:
Are there any other types of series expansions—besides Fourier or Taylor—that use other kinds of functions (not just $x^n $ or $ e^{inx}$) to represent functions in a similar way?
