Let $(e_n)_{n\in\mathbb{\mathbb{Z}}} = (t\mapsto e^{int})_{n \in \mathbb{Z}}$ the Fourier orthonormal base of $L^2(\mathbb{T})$ with the scalar product $\langle f,g\rangle = \int_{-\pi}^\pi f(t)\bar{g}(t)\frac{\operatorname{d}t}{2\pi}$. Define for each $N\in\mathbb{N}$ the space $V_N = \operatorname{span}(e_{-N},...,e_N)$. It is immediate to see that $\sum_{n=-N}^N \hat{f}(n)e_n$ is the unique best approximation in $V_N$ of $f$ in $L^2(\mathbb{T})$-norm and also that $\|\sum_{n=-N}^N \hat{f}(n)e_n -f\|_2 \to 0, N\to \infty$. The same holds true with respect the Sobolev norm $H^k(\mathbb{T})$, for each $k\in\mathbb{N}$. In particular, we have that the Cesàro means $\frac{1}{N+1} \sum_{n=0}^N \sum_{k=-n}^n \hat{f}(k)e_k$ are (besides trivial cases) a strictly worse approximation in $V_N$ of $f$ in $L^2(\mathbb{T})$-norm (or in $H^k(\mathbb{T})$-norm) w.r.t. $\sum_{n=-N}^N \hat{f}(n)e_n$. Despite of this fact, due to summability kernel theory, it is easy to see that if $f\in L^p(\mathbb{T})$ for $1\le p < \infty$ then $\|\frac{1}{N+1} \sum_{n=0}^N \sum_{k=-n}^n \hat{f}(k)e_k - f\|_p \to 0, N\to\infty$, while it is much more difficult (Hilbert transform theory if $p \neq 2$) to prove the same thing for $\| \sum_{n=-N}^N \hat{f}(n)e_n - f\|_p \to 0, N\to\infty$ for $1<p<\infty$, with the surprising fact that the result is false in general if $p = 1$ (Banach-Steinhaus + unboundedness). Analogous results and problems also hold and arise for Sobolev spaces $W^{k,p}(\mathbb{T})$. Not to mention that it's not so difficult to prove pointwise a.e.-convergence of $\frac{1}{N+1}\sum_{n=0}^N \sum_{k=-n}^n \hat{f}(k)e_k$ to $f$ if $f \in L^1(\mathbb{T})$ (Lebesgue theorem), while the corresponding result for $\sum_{n=-N}^N \hat{f}(n)e_n$ is false in general (Kolmogorov counterexample) and it is an heroic quest to prove the pointwise a.e. convergence if $f \in L^p(\mathbb{T})$ for some $p>1$ (Carleson-Hunt theorem). I found these facts very puzzling: what seems the "natural" best approximation of $f$ seems to have a worse behaviour as we moving away from $p=2$ w.r.t. Cesàro-means of the same quantity. This seems also more wierd noticing that $p \mapsto \|f\|_p$ is continuous and $\|\sum_{n=-N}^N \hat{f}(n)e_n -f\|_2 < \|\frac{1}{N+1} \sum_{n=0}^N \sum_{k=-n}^n \hat{f}(k)e_k - f\|_2$ if $f$ is non-constant.

With this in mind, I started wondering if maybe there's some reason to intuitively justify these discrepancies: are also Cesàro means $\frac{1}{N+1} \sum_{n=0}^N \sum_{k=-n}^n \hat{f}(k)e_k$ a best approximation of $f$ in some sense? (maybe in $V_N$ w.r.t. some norm? or maybe elsewhere?) Does anyone know something in this direction?

Also any suggestion in explaining (what to me seem) the oddities above will be very appreciated...

  • $\begingroup$ Theorem (Fejer). Let $f:[-\pi,\pi]\to \Bbb R$ be continuous with $f(-\pi)=f(\pi).$ Let $F_n(x)=\sum_{j=0}^n (A_j\cos jx+B_j\sin jx)$ where the $A_j,B_j $ are the Fourier co-efficients of $f.$ Then $(1/m)\sum_{n=0}^m F_n$ converges uniformly to $f$ as $m\to \infty.$ $\endgroup$ – DanielWainfleet Jun 3 '20 at 10:31

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