nhaliday + fourier   48

Battle for the Planet of Low-Hanging Fruit | West Hunter
Peter Chamberlen the elder [1560-1631] was the son of a Huguenot surgeon who had left France in 1576. He invented obstetric forceps , a surgical instrument similar to a pair of tongs, useful in extracting the baby in a  difficult birth.   He, his brother, and  his brother’s descendants preserved and prospered from their private technology for 125 years. They  went to a fair amount of effort to preserve the secret: the pregnant patient was blindfolded, and all others had to leave the room.  The Chamberlens specialized in difficult births  among the rich and famous.
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may 2017 by nhaliday
pr.probability - What is convolution intuitively? - MathOverflow
I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad first approximation for this effect). I found this to be very helpful, not just for understanding convolution per se, but as a lesson that one should try to use physical intuition to model mathematical concepts whenever one can.

More generally, if one thinks of functions as fuzzy versions of points, then convolution is the fuzzy version of addition (or sometimes multiplication, depending on the context). The probabilistic interpretation is one example of this (where the fuzz is a a probability distribution), but one can also have signed, complex-valued, or vector-valued fuzz, of course.
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january 2017 by nhaliday
soft question - Why does Fourier analysis of Boolean functions "work"? - Theoretical Computer Science Stack Exchange
Here is my point of view, which I learned from Guy Kindler, though someone more experienced can probably give a better answer: Consider the linear space of functions f: {0,1}^n -> R and consider a linear operator of the form σ_w (for w in {0,1}^n), that maps a function f(x) as above to the function f(x+w). In many of the questions of TCS, there is an underlying need to analyze the effects that such operators have on certain functions.

Now, the point is that the Fourier basis is the basis that diagonalizes all those operators at the same time, which makes the analysis of those operators much simpler. More generally, the Fourier basis diagonalizes the convolution operator, which also underlies many of those questions. Thus, Fourier analysis is likely to be effective whenever one needs to analyze those operators.
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december 2016 by nhaliday

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