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Apr
26
comment problems from the scottish book
Dan Mauldin edited a revised edition of the Scottish Book with updated commentary on many problems which was published last fall (2015). See: amazon.com/Scottish-Book-Mathematics-Selected-Problems/dp/…
Apr
22
revised Two questions on Elias Stein paper (1976)
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Apr
22
revised Two questions on Elias Stein paper (1976)
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Apr
21
answered Two questions on Elias Stein paper (1976)
Apr
10
revised Brownian motion, quadratic variation, existence of partitions?
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Apr
10
comment Wrongful conviction Bayesian argument in need of integral-solving talent
I would love to hear how estimating this integral implies that "a certain piece of forensic evidence has no inculpatory value"...
Mar
6
comment Is the number of representations as the sum of two elements of a polynomial sequence always small?
I'm not sure. I thought about that briefly but didn't get too far. This quickly leads to questions outside of my expertise.
Mar
6
comment Is the number of representations as the sum of two elements of a polynomial sequence always small?
I had also observed this after posting the question. It certainly seems that this can't be generalized in any straightforward way to sums since there are polynomials such that f(x)+f(y) does not have linear factors.
Mar
3
comment Is the number of representations as the sum of two elements of a polynomial sequence always small?
I'm curious if an improvement on the divisor bound is known for, say, $f(x)=x^5$?
Mar
3
comment Is the number of representations as the sum of two elements of a polynomial sequence always small?
@GH: if you can prove that, I'll accept it as an answer :)
Mar
3
comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
Thus, recalling the definition of convolution, $f(x)$ is upper-bounded by essentially the average of $f(x)$ over a translate of $B_{0}$ centered at $x$. In just about any harmonic analysis estimate this observation leads to the same conclusion that would be obtained if $f$ was in fact a step function constant on a latices of boxes of the shape $B_{0}$.
Mar
3
comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
Now what is formally true is that if $\hat{\eta}$ is a a smooth bump function equal to $1$ on $B$ and rapidly decaying away form B, then $f(x) = f * \eta$. In addition, it isn't hard to see that $\eta$ is now a function that is nearly constant on the $b_{0}$ and decays rapidly away from it. By nearly constant we mean bounded above and below by universal constants.
Mar
3
comment The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?
The idea in all of these "essentially constant" arguments (commonly referred to as "the uncertainty principle," as popularized by Tao) is as follows: Assume that $f$ is a function on $R^d$ such that $\hat{f}$ is supported on a box B. Then the essentially (but not formally) true claim is that $f$ is essentially constant on translations of the polar of $B$, say $B_{0}$. In the $1$-d case if $B$ is an interval of length $L$ then $B_{0}$ is an interval of length $1/L$.
Mar
2
awarded  Nice Question
Mar
2
comment Is the number of representations as the sum of two elements of a polynomial sequence always small?
@Bobby, yes the constant can depend on $f$.
Mar
2
revised Is the number of representations as the sum of two elements of a polynomial sequence always small?
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Mar
2
revised Is the number of representations as the sum of two elements of a polynomial sequence always small?
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Mar
2
revised Is the number of representations as the sum of two elements of a polynomial sequence always small?
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Mar
2
revised Is the number of representations as the sum of two elements of a polynomial sequence always small?
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Mar
2
revised Is the number of representations as the sum of two elements of a polynomial sequence always small?
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