Is the Cesàro sum of the Fourier series of $f$ a best approximation of $f$ in any sense?

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 $$1, with the surprising fact that the result is false in general if $$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...

• 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 $m\to \infty.$ – DanielWainfleet Jun 3 '20 at 10:31