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The Carleson-Hunter theorem states that the Fourier series of a function $f\in L_p$ converges almost everywhere to $f$ if $p>1$. Suppose we know that a trigonometric series converges almost everywhere to $f\in L_p$ for some $p\ge 1$. Does it follow that this trigonometric series is the Fourier series of $f$? This of course is true if the partial sums of the trigonometric series are dominated by a function in $L_1$. But is it true without any conditions on the nature of the convergence of the trigonometric series?

Let me provide some background for this question. Again it is something which arose in an undergraduate analysis course I was teaching. A student posed the following question: suppose by manipulating infinite series and trigonometric identities one obtains some trigonometric series which converges to some function except possibly at a discrete set of exceptional points. Is it automatically true that this gives the Fourier series expansion of that function? Here is an illustrative example. By substituting $z=e^{i\theta}$ into the power series for $\log(1+z)$ and comparing the real and imaginary parts, one obtains $$\log\left(2\cos\left(\frac{\theta}{2}\right)\right)=-\sum_{n=1}^\infty\frac{(-1)^n\cos(n\theta)}{n}$$ and $$\frac{\theta}{2}=-\sum_{n=1}^\infty\frac{(-1)^n\sin(n\theta)}{n},$$ valid for $\theta\in(-\pi,\pi)$. Can one IMMEDIATELY conclude that these are Fourier series expansions, WITHOUT performing any further analysis?

If there was some general result, along the lines of a converse to the Carleson-Hunter theorem, which precludes the existence of exotic trigonometric series expansions for reasonably regular functions, then one would be safe in drawing such a conclusion. However the example in Zygmund that David Mitra cites in his second comment below shows that this is not the case if one allows the set of exceptional points to be an arbitrary set of measure 0.

So let me rephrase the question: is it true if the set of exceptional points is required to be discrete (or equivalently finite within a period interval)? (David's first comment shows that this is true if the exceptional set is empty.) Or is there some other way of finessing this issue (eg. checking if the coefficients are square summable)?

Note that the answer provided in fourier-series-of-log-sine-and-log-cos is another example illustrating this issue. Contrast this with the answer provided in compute-the-fourier-coefficients-and-series-for-log-sinx which avoids this issue by direct computation of the Fourier coefficients.

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  • $\begingroup$ In Zygmund's Trigonometric Series, vol. 1, Theorem IX (3.1), it's proven that the answer to your question is "yes", if the series converges pointwise (to a finite sum) everywhere to an integrable $f$. $\endgroup$ – David Mitra May 12 at 17:52
  • $\begingroup$ Also in Zygmund (IX (6.14)): there exists a perfect set $M$ of measure zero and a trigonometric series that vanishes off $M$ but does not vanish identically. $\endgroup$ – David Mitra May 12 at 18:22
  • $\begingroup$ The whole topic is called the theory of trigonometric sets of uniqueness and it's been treated (but not fully solved as it seems to have a deep arithmetic connection) by the Russian school of the 1920-30's led by N Bary - books by Zygmund (above) and Bary $\endgroup$ – Conrad May 13 at 14:28

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