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Suppose we have a finite-norm vector $X$ in a Hilbert space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum_k a_k V_k$. If the basis is orthonormal, then we know that $a_k=<X,V_k>$ and also $||X||^2=\sum_k a_k^2$. In particular, we can say from that that $\sum_k a_k V_k$ converges at least as fast as $a_k^2$ drops for large $k$'s.

Now the question is this, if the expansion is in a basis $V_k$ that is not orthogonal, are there known results about the speed of convergence of such a series, $X=\sum_k a_k V_k$?

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    $\begingroup$ You may want to use LaTeX formatting (just use the usual $symbols) for a better diplay next time... $\endgroup$
    – leo monsaingeon
    Commented Jun 18, 2014 at 8:07
  • $\begingroup$ well ok, this assumes we know $a_k$, but the situation we have is that we have $X$ and $\{V_k\}$. Since in nonorthogonal case finding $a_k$'s requires solving a system of general linear equation, we can't get much information about the behavior of $a_k$. Perhaps I should have said, are there any known results about convergence that rely on the properties of $X$ and $\{V_k\}$. $\endgroup$
    – Yuriy M
    Commented Jun 18, 2014 at 8:41
  • $\begingroup$ If the dimension of the Hilbert space is infinite, you can find a vector whose expansion converges with the error going to $0$ slower than any prescribed function that goes to $0$, even in the orthogonal basis setting. However, useful things can be said depending on the particular situation. For example, if your space $X$ is $L^2$, and the basis elements $V_k$ are some kind of piecewise polynomials, then the convergence is faster for smoother functions. $\endgroup$
    – timur
    Commented Jun 18, 2014 at 12:43
  • $\begingroup$ @Kevin Ventullo I guess I know how to derive a few things like Riemann Hypothesis (or its negation, if you prefer) from the inequality you wrote (the most likely reason is, of course, that you meant something completely different and your fingers just slipped on the keypad when typing :-)) $\endgroup$
    – fedja
    Commented Jun 18, 2014 at 22:18
  • $\begingroup$ Hmm, yeah that was nonsense. Not sure what I was thinking $\endgroup$
    – Kevin Ventullo
    Commented Jun 19, 2014 at 5:04

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