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Trial And Error.

A portrait of a famous Mathematician who was often criticized for not being rigorous enough in his new inventions.

"Anyone who has never made a mistake has never tried anything new." - Albert Einstein


1d
comment Shift operators and c0 semigroups
If it were $C0$, then, for every $f$ and $\epsilon > 0$, there would exist $\delta > 0$ such that $|f(y)-f(x)| < \epsilon$ whenever $0 \le y -x < \delta$. Can you think of a function that might not satisfy such a condition?
1d
answered Redundancy in the Laplace transform and Mellin's inverse formula
1d
comment Properties of Orthonormal Systems and Projections
@Jamil_V : Because $f \perp F^{\perp}$ and $u \perp F^{\perp}$, then $f-u \perp F^{\perp}$. However $v=f-u$ and $v \in F^{\perp}$. Therefore $v \perp v$ which gives $v=0$ and $f=u \in F$.
2d
comment Finding rotation axis and angle to align two 3D vector bases
If you want a linear transformation that maps the first three vectors to the second one, then you don't need to find an axis of rotation and a rotation. Finding an axis of rotation is not generally stable as the transformation approaches the identity matrix where there are 3 eigenvectors with eigenvalue 1. You may want to rethink your approach.
2d
revised Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?
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2d
revised Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?
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2d
comment Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?
@mickep : J. Dieudonne "A History of Functional Analysis" credits Fourier with origin of Spectral Theory. E. C. Titchmarsh focuses on the complex analysis in "Eigenfunction Expansions ...". Titchmarsh gave the first general approach for proving pointwise convergence of general Fourier expansions as I described. General Titchmarsh-Weyl theory focuses on Complex Analysis aspects. Teschl in "Mathematical Methods in Quantum Mechanics" uses a circa 1905 representation theorem for holomorphic functions to prove/study the Spectral Theorem for sefaldjoint operators. See T. Kato "Perturbation Theory".
2d
comment Approximation Property: Hilbert Spaces
@Freeze_S : This reminds of the tower of Babel where man is trying to build a tower to God. Eventually their language splits. A modern twist is that the time that it takes to move another stone to the top can consume a lifetime. I heard a String Theorist talk about coming to end of knowledge because, by the time someone was ready for research in the field, they were also ready for retirement. :) Barry Simon quotes Ecclesiates in one of his books: "Of making many books there is no end, and much study wearies the body." It's an important goal to try to organize and condense. Towers fall.
2d
answered Why are functional analysts interested in not only the point spectrum of $f$, but also, its spectrum?
Jan
28
revised Understanding a proof from Conway: showing existence of idempotents using functional calculus
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Jan
28
comment Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?
@Winther : That's a great discussion. I upped your answer and took note of the full book. Interesting. Based on that, I'm betting there is no simpler proof known at this point than Karamata's. Some things just can't be simple.
Jan
28
answered Understanding a proof from Conway: showing existence of idempotents using functional calculus
Jan
28
comment Approximation Property: Hilbert Spaces
@WillJagy : I like many of the MathJax extensions, and that's why I was asking. Thanks for the info. Anyway, it's a good thing to save in a Latex file; saving things here is probably not going to work in the long run.
Jan
28
comment Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?
@Winther : Thank you for that PDF. The proof there is 6+ pages long and uses Weierstrass approximation. That's not so simple, but the fact that your PDF is not ancient, and is full of cleverness makes me think that may be the state-of-art proof.
Jan
28
comment Approximation Property: Hilbert Spaces
@WillJagy : Does your version of Latex understand the MathJax formatting, or you do make a few mods when saving?
Jan
28
asked Proof that $\sum_{n} a_{n}$ converges if $a_{n}=O(1/n)$, $\lim_{x\uparrow 1}\sum_{n}a_{n}x^{n}$ exists?
Jan
28
comment Approximation Property: Hilbert Spaces
@WillJagy : Your Latex document is probably a better idea in general. There's no guarantee that the information placed here will be around later. Even if it is around, there's no guarantee that you'll be easily able to locate it later. And, it's a little easier to search single document than to search the archives here. It's hard to find good systems for keeping and locating information, especially if you want to make the information available to others in some systematic way. But I don't think Freeze_S can be accused of cluttering the 300,000+ questions on this site. :)
Jan
27
comment Why heat equation is not time-reversible? (Time arrow in mathematics)
@NickThompson : Averaging an initial distribution with a Gaussian isn't something you can undo in general.
Jan
27
comment Positivity of the Fourier transform of a certain function
Have you tried to analyze $\int_{0}^{\infty}\cosh(x)^{-\nu}\cos(sx)dx$ as an alternating series? You end up only have to consider the first few 1/2 cycles of $x\mapsto \cos(sx)$ for $s > 0$ to know the ultimate sign of the alternatively series.
Jan
27
answered L1 norm less than BV norm