Numbers Matter

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A Few Infinite Series Questions

A few ideas I’m exploring.

Orthogonality. Fourier series can be viewed in a more vector space context: Let $V$ be the inner product space (though we want a Hilbert space) of real-valued continuous functions defined on an interval, say $[-\pi, \pi]$ . The inner product is given by the integral over the interval $[-\pi, \pi]$ of the product of the two functions. Then, the trigonometric polynomials $\sin(nx)$ and $\cos(nx)$ for $n\geq 1$ form an orthogonal (Hilbert) basis for $V$ , so we can express any function as an infinite series of sines and cosines. Now, what if we do the same thing with other bases apart from the trigonometric polynomials? Is it possible to give a characterization for all such orthogonal bases?
The zeta function. Further, we could tweak the inner product in a standard way by multiplying the product of the two functions in the integral by a weighting function $w(x)$ . What if we select the $w(x)$ to be the Riemann-zeta function? One could try to compute an orthogonal basis in this inner product space by applying Gram-Schmidt to $\{1, x, x^2, \cdots\}$ and then equate the Fourier expansion of a function, say $e^x$ , and its expansion in this new basis. Would equating the two give identities involving the Riemann-zeta function?
Mirrors. Let’s define a mirror to be a function $f$ defined on an interval of $\mathbb{R}$ that is symmetric about the $y$ -axis. Further, it must satisfy a decay condition (if the interval is the whole of $\mathbb{R}$ ) that I have to make precise. Now, let’s place a laser at the point $(b,0)$ that sends rays parallel to the $y$ -axis, so that it reflects at $f$ . Now, the ‘focus’ of $f$ would be the set of possible $y$ -intercepts of the reflected rays. One can show, by forming an elementary differential equation, that the only time when the ‘focus’ is truly a focus is when the mirror is a parabola.

To make other mirrors $f$ (such as circular mirrors) have a single focus, let’s consider another mirror, $f'$ , placed some distance beneath $f$ , facing towards it. We then want to solve for $f'$ such that after the ray fired from $(b,0)$ reflects from both $f$ and $f'$ , it has a $y$ -intercept independent of $b$ . That is, the combined system $(f,f')$ has a single focus. Given a $f$ , solving for $f'$ involves solving a differential equation that doesn’t have a nice elementary solution, at least according to Wolfram Alpha (I’ll check this). How do you show that this has no solution? I’m exploring the potential use of Differential Galois Theory here.

To bring in infinite series, let’s say that we don’t want the system $(f,f')$ to have a single focus, but rather a norm-bounded condition on the focus function of the system $(f,f')$ . (The focus function takes in $b$ , and outputs the coordinates of the doubly reflected ray.) Basically, we want to create a sequence of mirrors: $f, (f,f'), (f,f',f''), \cdots$ such that the corresponding focus functions converge (in what sense?) Note that $f^{n \text{ primes}}$ and $f^{(n+1) \text{ primes}}$ lie on opposite sides of the $x$ -axis, and each $f^{n \text{ primes}}$ is smaller in absolute value is smaller than that of $f^{n-2 \text{ primes}}$ . One could also ask a similar question in higher dimensions.