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It's tough. I used to work in an area of applied probability where some statistical-mechanics principles were applicable. I'd read papers where authors were ma

It's tough. I used to work in an area of applied probability where some statistical-mechanics principles were applicable. I'd read papers where authors were making analogies of a large neural network to a stat-mech system, using an applicable stat-mech approximation, and then differentiating that approximation to get a probability bound. It gave interesting results, and did show you something about the original problem that was hard to get by sticking to the original formalism. But at the end of the day, you really would not bet the farm on the truth of those approximations... On the other hand, Fourier analysis was originally doubted and scorned by mathematics, but (if I'm remembering the story correctly) ended up being used so much that theory was developed to explain in what sense the Fourier transform approximates the original function. Another example of the interplay between physics and mathematics is the percolation problem, where there was a kind of archipelago of physics-motivated results that probabilists have been trying to tidy up for decades now. E.g., sec. 1.2 of: https://www.unige.ch/~duminil/publi/2018ICM.pdf https://www.unige.ch/~duminil/publi/2018ICM.pdf

graycat 4y ago
There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more. Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions. I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions. I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey. There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics. But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup). I'm no teacher. I do not now nor have I ever had any desire to be a prof. For more, get copies of Rudin's books and dig in. Wonder of wonders, not all physics profs have done that.
graycat 4y ago
There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more. Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions. I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions. I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey. There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics. But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup). I'm no teacher. I do not now nor have I ever had any desire to be a prof. For more, get copies of Rudin's books and dig in. Wonder of wonders, not all physics profs have done that.

graycat 4y ago

There is both Fourier series and the Fourier transform. For each, need to establish that they exist. Then will usually want to know that the inverse transform DOES approximate the original function. Details are in 4 books by W. Rudin. One of these, on groups, I ignored. I did too much group theory, e.g., wrote my honors paper on it, and didn't want to do more. Principles does Fourier series and does not use measure theory. Real and Complex Analysis does the Fourier transform and uses the Lebesgue integral, that is, measure theory. Functional Analysis does Fourier theory with distributions. I had serious courses from Principles and R & CA. Also Royden and Neveu. All from a star student of Cinlar, long at Princeton. I read FA, quickly. I don't much care to bother with distributions. I got into applications via the fast Fourier transform and power spectral estimation as in Blackman and Tukey. There is an intuitive view that sort of works: The given function is a point in a vector space, with an inner product. The sine waves are coordinate axes. They are orthogonal. The Fourier things are projections of the point onto the coordinate axes. The inverse Fourier thing reconstructs the original function. If use only some of the coordinate axes, then get a least squares approximation of the original function. This all works exactly in ordinary linear algebra, e.g., as in Halmos, Finite Dimensional Vector Spaces, sometimes given to physics students studying quantum mechanics. But these Fourier things have infinitely many coordinate axes, countably infinite for Fourier series and uncountably infinite for the transform, and there the finite dimensional things don't always work. So, Rudin has to be very careful in presenting what does work and proving it -- so with the details it's not easy reading. What fails is a fairly general situation in a fully general Hilbert space. E.g., are not locally compact, but can get some help from a clever use of the parallolgram inequality (somewhat relevant in my startup). I'm no teacher. I do not now nor have I ever had any desire to be a prof. For more, get copies of Rudin's books and dig in. Wonder of wonders, not all physics profs have done that.

graycat 4y ago