bio | website | cornellmath.wordpress.com |
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location | Ithaca, NY | |
age | 30 | |
visits | member for | 3 years, 11 months |
seen | yesterday | |
stats | profile views | 859 |
I was a graduate student at Cornell University studying harmonic analysis with Camil Muscalu. Now I'm a postdoc at Washington University in St. Louis.
Sep 22 |
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What is the naming reason of poles in complex analysis?
@Martin: According to the online etymological dictionary, the English word pole derives from the Latin word palus meaning stake. So the term could be handed down to us from when mathematicians still wrote in Latin. The term could also derive from Greek, polos (axis of a sphere), or it could be a coincidence. Simply because the direct cognate of a word in German doesn't correspond to the current meaning of an English word doesn't mean the words didn't have similar meanings long ago. The English word table is related to the German word Tafel even though Tafel only infrequently means table. |
Jul 22 |
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Spherical Harmonics - a bunch of questions about them
There are nice sections about spherical harmonics in two of Stein's books Singular Integrals and Differentiability Properties of Functions and Introduction to Fourier Analysis on Euclidean Spaces, but if you don't know anything about $L^2$, these treatments could very well be too difficult. |
Jul 18 |
answered | Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory. |
Jul 13 |
awarded | Enthusiast |
Jul 10 |
answered | Splines, harmonic analysis, singular integrals. |
Jul 10 |
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Splines, harmonic analysis, singular integrals.
Your statement about functions of compact support is the correct problem with defining the Fourier transform of distributions with smooth compactly supported functions as the test functions. |
Jul 10 |
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Splines, harmonic analysis, singular integrals.
These are good ideas, but I think your concluding sentence is a bit too strong. One could easily define the space of test functions to be the space of compactly supported smooth functions. The Fourier transform of such a function, being a very special kind of Schwarz function, is also smooth with extreme decay at infinity. The problem is that while any locally integrable function defines a linear functional over $C_c^{\infty}$, the formula $\hat{T}(\phi)=T(\hat{\phi})$ does not make sense since $\hat{\phi}$ will never be compactly supported when $\phi$ is compactly supported. |
Jul 2 |
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Demystifying complex numbers
I think your example with the Fourier expansion somewhat misses the fact that the Fourier transform is one of the most important mathematical tools in analysis and involves complex numbers. |
Jun 15 |
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Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?
I think it's not that these spaces are the ones that matter but rather these spaces are the only ones where it is convenient to perform computations. Like I said in a comment to another answer, $L^2$ would be just as exotic if not for access to great Hilbert space tools. |
Jun 14 |
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Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity?
I tend to agree with you to some extent. With $L^\infty$ and $L^1$, things are much simpler computationally, and with $L^2$ you have the immense advantage of Hilbert space tools like Plancherel's identity. If not for Hilbert space techniques, one would likely find $L^2$ just as exotic as the others. The interpolation theorems basically say that understanding two of these spaces is enough to understand everything in between. So long as the adjoint of the operator is similarly behaved, just two is enough to understand all $p\ge 1$. This is the case with the Hilbert transform, for instance. |
Jun 14 |
answered | Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity? |
Jun 12 |
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Weighted Hardy Inequality for bounded domains
At least in the case where the function is zero at L, I suppose you've tried the change of variable x to v/(v+1), so that one can attempt to use the inequality in your link (after some algebra) and then do some more algebra to simplify things a bit? It seems like that would work to at least give you some control over the function u(x)-u(L). Would this be at all helpful for your purposes? Why do you need such an inequality? |
Jun 7 |
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What are your experiences of handouts in mathematics lectures?
Are you not free to disallow the use of laptops in class? |
Jun 7 |
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What are your experiences of handouts in mathematics lectures?
I completely agree with these sentiments, Terry. Such a system is extremely rewarding to students who work hard. Students all have varying levels of raw ability; the only thing you can do as an instructor is make hard work as valuable to them as possible. |
Jun 7 |
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Real analysis has no applications?
Greg, I completely agree with you regarding BC calculus. I took BC calculus in high school, and it was a wonderful experience. We did avoid some of the more rigorous details --- I think delta-epsilon proofs of continuity were completely absent, for instance --- but I think I still came out of that course with a great understanding of analysis. I clearly understood the delicate interplay between approximation and error terms. Once deltas and epsilons came, I needed only understand the framework; the tools were already there. |
May 19 |
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Why is Lebesgue integration taught using positive and negative parts of functions?
Fatou's lemma is pretty useful. |
May 16 |
awarded | Commentator |
May 16 |
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Differentiable structures on R^3
The inverse of $x^3$ is not smooth, but the term diffeomorphism as it pertains to differentiable structures does not require this: a bijective map $f:\mathbb{R}\rightarrow\mathbb{R}$ is a diffeomorphism of $(\mathbb{R},A)$ and $(\mathbbf{R},B)$ if $x\in A$ iff $x(f)\in B$. Here A and B are maximal atlases. |
May 15 |
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Beginning a sentence with a mathematical symbol
Greg, do you have any particular examples worth sharing, or are you simply finding your prose has become a monotonous glob of ideas punctuated by thus and therefore? |
May 15 |
answered | Beginning a sentence with a mathematical symbol |