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Suppose that $f(x)\in C[0,1]$. Can we find an infinite sequence $\{a_n\}$ such that

$$ \lim_{N\rightarrow \infty} ||f(x)-\sum_{n=0}^{N}a_n \cos3^n \pi x||_\infty=0$$?

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  • $\begingroup$ No, it is not possible: for example you cannot approximate $\cos 2\pi x$, since its $L^2$ distance to the span of $\cos 3^n\pi x$ is positive. $\endgroup$ – Alexandre Eremenko Feb 12 at 12:53
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    $\begingroup$ Wouldn't that imply $f(0)+f(1)=0$? $\endgroup$ – abx Feb 12 at 12:55
  • $\begingroup$ Even with the assumption $f(0) + f(1) = 0$ in general there cannot be absolute convergence. $\endgroup$ – Dieter Kadelka Feb 12 at 17:29