I'm reading Stein & Shakarchi's Fourier Analysis, but had a question about a small part of the proof of the mean-square convergence of fourier series of a function $f$.
To show the convergence, they first proved the Best Approximation Lemma, saying that fourier series is the best trigonometric polynomial estimating f for degree at most N. Then on page 79, they used a corollary 5.4 in Chapter 2 saying that there exists a polynomial $P$ of degree $M$ s.t. $|f(\theta)-P(\theta)|<\epsilon$ for all $\theta$. This is basically the contents of that corollary, and I'm totally okay with that. After some algebra, and using best approximation theorem, we can certainly get $$||f(\theta)-S_M(f)||<\epsilon$$ Note that the best approximation theorem has both sides to have the same degree. How can we conclude that $$||f(\theta)-S_N(f)||<\epsilon,\ \forall N \geq M\ ?$$
Why is it true for $N > M$ as well?
Definition:
1. $||f(\theta)-S_M(f)||^2 = \frac{1}{2\pi}\int_{0}^{2\pi}|f(\theta)-S_M(f)|^2 d\theta$.
2. $S_N(f) = \sum_{-N}^{+N} a_ne^{in\theta}$ is the $Nth$ partial sum of $f$, where $a_n$ is the $Nth$ fourier coefficient of $f$.
Thanks in advance for any insights.
Best Answer
The lemma doesn't require equal degrees on both sides - it says that the Fourier partial sum is at least as good as any approximation of equal or lesser degree.
So, then, we have by other means a trig polynomial of degree $M$ that approximates to within $\epsilon$. By the lemma, for all $N\ge M$, the $N$th Fourier partial sum is at least as good as everything else with degree $\le N$ - including that degree-$M$ approximation we found.