Fejer's theorem says that for any continuous function $f \colon S^1 \to \mathbb C$ with Fourier coefficients $a=(a_n)_{n \in \mathbb Z}$ the sequence
$$\sigma_n(a) := \frac1n \sum_{k=1}^n \sum_{l=-k}^k a_l \exp(2\pi i \cdot l\phi)$$
convergences uniformly to $f$. Moreover, by the Riemann-Lebesgue Lemma, the sequence $a=(a_n)_{n \in \mathbb Z}$ is in $c_0(\mathbb Z)$.
Question: Let $a=(a_n)_{n \in \mathbb Z}$ be in $c_0(\mathbb Z)$.
Does $\sigma_n(a)$ converge in measure to some measurable function on $S^1$?
More generally, is there any summing procedure (or in fact any assignment whatsoever), which leads to a linear map $\Phi \colon c_0(\mathbb Z) \to M(S^1)$, which extends Fourier summation on finitely supported functions (and preferably also Fejer summation for Fourier series of continuous functions). Here, $M(S^1)$ denotes the space of measurable functions on $S^1$ (up to measure zero) with the usual measure topology given by the metric
$$d(f,g) := \inf\lbrace\varepsilon \mid \mu(\lbrace x \mid |f(x)-g(x)|\geq \varepsilon \rbrace \leq \varepsilon \rbrace.$$
Best Answer
There is no continuous linear operator from $c_0$ to $M(S^1)$ that maps the unit vector basis to the characters. In fact, any continuous linear operator from $c_0$ to $M(S^1)$ maps the unit vector basis to a sequence which converges to zero at a good rate. To see this, note that by Maurey-Nikishin, the operator factors through $L_p$ for all $0<p<1$, and these spaces have cotype 2.
Does this answer your question? If not, I don't see a good formulation for your question. If there is anything like a summation method, the Banach-Steinhaus theorem would imply that the operator is continuous.