Functional Analysis – Convergence of Fourier Series

fa.functional-analysisfourier analysislimits-and-convergence

Say $f \in L^p[a,b]$, with $p \in \mathbb{N}, p > 1 $. Does its Fourier Series converge in the metric space $L^p[a,b]$? Does the series converge pointwise? And at which conditions?
Say now $p = 1$, Does its Fourier Series converge in the metric space $L^1[a,b]$? Does the series converge pointwise? And under which conditions?

Best Answer

  1. Convergence in $L^p$, $p>1$.

True, by M. Riesz's Theorem. This is a standard topic in every harmonic analysis course, with several readable proofs.

  1. Convergence pointwise almost everywhere, $p>1$.

True, by the Carleson-Hunt Theorem. This result is over 50 years old, but famously difficult. Though the techniques used are now standard, there aren't really any easy proofs, even for continuous functions.

  1. Convergence in $L^1$.

False, and not very difficult to prove. Suppose that Fourier series converged in $L^1$. Let $S_N$ be the operator that takes a (Schwartz, say) function and returns the $N$th partial Fourier sum. I claim that these operators are not bounded uniformly in the $L^1$ norm, which is enough to give a contradiction by the uniform boundedness principle. For indeed applying it to the Fejer kernel $K_M$ gives $$\|S_NK_M\|_{L^1}\to\|D_N\|_{L^1}$$ as $M\to\infty$. Where $D_N$ is the Dirichlet kernel. But $\|D_N\|_{L^1}$ is unbounded, so $S_N$ cannot be uniformly bounded.

  1. Convergence pointwise almost everywhere, $p=1$.

False, due to Kolmogorov almost 100 years ago. In fact, it is possible to find an $L^1$ function whose Fourier series diverges everywhere. This is not an easy counterexample, but it is presented in some courses in harmonic analysis.

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