# Fejer's theorem and convergence of Fourier series in measure

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 $$d(f,g) := \inf\lbrace\varepsilon \mid \mu(\lbrace x \mid |f(x)-g(x)|\geq \varepsilon \rbrace \leq \varepsilon \rbrace.$$

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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.

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Can you give more details on the argument? I do not understand how you apply the Maurey-Nikishin factorization theorem. –  Andreas Thom Oct 15 '11 at 7:09
You might be more familiar with the application that says that a Banach subspace of $M$ embeds into $L_p$ for all $p<1$, but actually it is a factorization result for operators into $M$. I don't have sources here to give you a precise reference. I think the place that it was first spelled out was Maurey's thesis, but something close enough is probably in Diestel-Jarchow-Tonge or Wojtasczyzk. –  Bill Johnson Oct 15 '11 at 16:25
Do you have a good reference for the subspace version? I could not locate it in the literature. What I found is: Let $u$ be a continuous linear operator from a quasi-normed linear space $E$ into $L^0(\Omega,\mu)$. The following statements are equivalent: (a) For each $\alpha\in(0,1)$ there exists a measurable subset $\Omega_\alpha\subset\Omega$ with $\mu(\Omega-\Omega_\alpha)\leq\alpha$ and a constant $K_\alpha$ such that $(\int_{\Omega_\alpha}|u(x)|^p\,d\mu)^{1/p}\leq K_\alpha\|x\|$ for $x\in E$; –  Andreas Thom Oct 17 '11 at 6:47
(b) $u$ admits a factorization $u=T_g\circ v$, where $v$ is a continuous linear map from $E$ into $L^p(\Omega,\mu)$, and $T_g\colon L^p(\Omega,\mu)\rightarrow L^0(\Omega,\mu)$ is the operator of multiplication by a measurable function $g$. –  Andreas Thom Oct 17 '11 at 6:47
Did you look in Maurey's thesis? It is in Asterisque. Also, check out references in Kalton's paper Kalton, N. J.(1-MO) Banach spaces embedding into L0. Israel J. Math. 52 (1985), no. 4, 305–319. 46B25 (46E30). Also Kalton, N. J.; Koldobsky, A. Banach spaces embedding isometrically into Lp when 0<p<1. Proc. Amer. Math. Soc. 132 (2004), no. 1, 67–76. This one I can view. They say that Nikisin-Maurey is in Wojtasczyzk's book, p. 257ff. –  Bill Johnson Oct 17 '11 at 20:46