# Is it possible to use $\sum_{n=0}^{\infty}a_n \cos3^n \pi x$ to approximate any $f(x)\in C[0,1]$? [closed]

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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• 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. – Alexandre Eremenko Feb 12 at 12:53
• Wouldn't that imply $f(0)+f(1)=0$? – abx Feb 12 at 12:55
• Even with the assumption $f(0) + f(1) = 0$ in general there cannot be absolute convergence. – Dieter Kadelka Feb 12 at 17:29