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I'm just curious if this implication is correct:

$$0 \le\int_a^b \left( f(x)-S_n(x) \right)^2dx \Rightarrow 0 \le \lim_{n\rightarrow \infty}\int_a^b \left( f(x)-S_n(x) \right)^2dx.$$

What needs to be true for this to hold? Is the existence of the limit in question sufficient? This is from my attempt at proving Bessel's inequality. The sequence $$T_n = \int_a^b \left( f(x)-S_n(x) \right)^2dx =\dots= \|f\|^2-\sum_{k=0}^n \gamma_k ^2 \|\varphi_k \|^2$$

should converge (not sure to what though, not necessarily to $f$ right?) given that $f \in L^2([a,b])$ and $(\varphi_k ) \subset L^2([a,b])$ is an orthogonal sequence of functions and $\gamma_k = \dfrac{\langle f \mid \varphi_k\rangle}{\|\varphi_k\|^2}$. How to show that $T_n$ converges?

Carl Mummert
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Spine Feast
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    Note, that $$ c_n := \int_a^b(f(x) - S_n(x))^2\mathrm dx $$ is just a numerical sequence. If $c_n\geq 0$ for all $n$ and the limit $c:\lim_n c_n$ does exist, then $c\geq 0$ as well. I don't see, however, how does this help you with convergence. – SBF Jun 24 '13 at 15:08
  • Ok. As for the convergence, it's assumed that $\sum_0^n \gamma_k \varphi_k(x)$ converges pointwise to some $S(x)$. Is that enough to reason that $\sum_0^n \gamma_k ^2 ||\varphi_k||^2$ converges? – Spine Feast Jun 24 '13 at 15:24

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For Fourier series

$$f(x)\sim\dfrac{a_0}{2}+\sum_{k=1}^\infty (a_k\cos kx+b_k\sin kx)$$

for piecewise continuous functions $f(x)$ I answered a question at Mathematics: Parseval's theorem derivation using Bessel's inequality that adresses just that problem.

Community
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Reimer Brüchmann
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