Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 3 down vote accepted

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.