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Suppose $f \in L^2[-\pi,\pi]$. I want to show that $\sum_{-\infty}^{\infty} a_n^2 < \infty$ where $a_n$ is the $n$-th Fourier coefficient of $f$. I saw Rudin's real and compleax book. I was unable to make out the proof. Any concrete solution will be appreciated.

TRUSKI
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This is Bessel's inequality. For an integer $N$ let $$g_N(x)=\sum_{n=-N}^N a_n e^{in x}$$ and $$h_N(x)=f(x)-g_N(x).$$ Prove that $g_N$ and $h_N$ are orthogonal: $\int_{-\pi}^\pi\overline{g_N(x)}h_N(x)\,dx=0$. Then prove that $$\int_{-\pi}^\pi|f(x)|^2\,dx =\int_{-\pi}^\pi|g_N(x)|^2\,dx+\int_{-\pi}^\pi|h_N(x)|^2\,dx \ge \int_{-\pi}^\pi|g_N(x)|^2\,dx$$ and this is $2\pi\sum_{n=-N}^N|a_n|^2$.

Angina Seng
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  • Thanks @ Shark. – TRUSKI Jul 28 '17 at 06:44
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    It might be worthwhile to mention that in the OP's context there is actually equality in Bessel's inequality, that it is called Parseval's identity, and that the sum of the squares of the coefficients is precisely the square of the norm of the element $f$. – uniquesolution Jul 28 '17 at 07:02