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Another example from real analysis would be the question of the pointwise convergence of the Fourier series of a continuous function (defined on a given closed intervalinterval). Many people, including Dirichlet and even the master rigorist Weierstrass himself, believed that the Fourier series of such a function converges pointwise everywhere to the function itself. Some believed in clung on to this statement belief so strongly that they even viewed it as an infallible axiom.

Hence, one can imagine the great upset when, in 1876, Paul du Bois-Reymond proved that the existence of a continuous function exists whose Fourier series diverges at a point. The His proof is non-constructive , using and uses a method called the principle of condensation of singularities. I have absolutely no idea how the method works, but I do know of a very common proof that uses the Baire Category Theorem (using the Baire Category Theorem, one can also prove the existence of continuous functions that are not differentiable at any point).

After this upsetthe dust had settled in the wake of du Bois-Reymond's seismic discovery, people started to believe fervently believing that there should exist a continuous function whose Fourier series diverges everywhere . - an opinion that lay on the other extreme! Andrei Kolmogorov inadvertently lent support to this claim by exhibiting, in 1926, an ${L^{1}}([- \pi,\pi])$-function whose Fourier series diverges everywhere. However, there was great surprise upheaval once more in Fourier-land when the combined efforts of Lennart Carleson and Richard Hunt in the late 1960's showed that the Fourier series of any $f \in {L^{p}}([- \pi,\pi])$ converges almost everywhere to $f$, for all $p > 1$ (this result subsumes the case of continuous functions). During an interview with the AMS, Carleson revealed that he had originally been working tried to disprove his result (which proves the statement for pertaining to $p = 2$), but in the end, his failure to do so produce a counterexample convinced him that he should be working in the other direction instead.

Therefore, in the field of Fourier analysis, viewpoints have changed and cherished beliefs were have been destroyed - twice.

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Another example from real analysis would be the question of the pointwise convergence of the Fourier series of a continuous function on a given closed interval. Many people, including Dirichlet and even Weierstrass himself, believed that the Fourier series of such a function would converge converges pointwise everywhere to the function itself. Some believed in this statement so strongly that they viewed it as an infallible axiom.

Hence, one can imagine the great upset when, in 1876, Paul du Bois-Reymond proved that a continuous function exists whose Fourier series diverges at a point. The proof is non-constructive, using a method called the principle of condensation of singularities. I have no idea how the method works, but I do know of a very common proof that uses the Baire Category Theorem (using the Baire Category Theorem, one can also prove the existence of continuous functions that are not differentiable at any point).

After this upset, people started to believe that there should exist a continuous function whose Fourier series diverges everywhere. Andrei Kolmogorov inadvertently lent support to this claim by exhibiting, in 1926, an ${L^{1}}([- \pi,\pi])$-function whose Fourier series diverges everywhere. However, there was great surprise yet again once more in Fourier-land when the combined efforts of Lennart Carleson and Richard Hunt in the late 1960's showed that the Fourier series of any $f \in {L^{p}}([- \pi,\pi])$ converges almost everywhere to $f$ for all $p > 1$. During an interview, Carleson revealed that he had originally been working to disprove his result (which proves the statement for $p = 2$), but in the end, his failure to do so convinced him that he should work be working in the other direction instead.

Therefore, in the field of Fourier analysis, viewpoints changed and cherished beliefs were destroyed - twice.

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