Hi Matt! Are you placing any kind of restriction on $\mu$? Are you allowing any measure space at all? It's rather a long shot, but you know the Nagy-Foias functional model for contraction operators between Hilbert spaces? That might change your question into an equivalent question about Hankel and Toeplitz operators - but it probably will be equally difficult.

I imagine this question will forever remain impossibly difficult for finite mathematicians; and even if it is someday solved, I personally am sure I will never understand the proof [or the criteria on $E$]. Having said that, it would be nice to be proved wrong! I mean, I don't understand it, but I thought that the Carleson-Hunt theorem tells us nothing whatsoever about the detailed structure of $E$, apart from having zero measure. If even this horribly difficult theorem gives no information, I am not optimistic. The $L^p$ spaces cannot distinguish between different $E$ with $m(E) = 0$.

Use Fubini's Theorem to reduce to a single integral, i.e. interchange the order of integration, so you are integrating a function of $\theta$ from $0$ to $\pi/2$. This requires the indefinite integral of $1/ \sin x$. Now you are in the standard realm of the Calculus of Variations; I'll leave you to fill in the integration details...! There is no guarantee that the Euler-Lagrange equations will actually have a closed form solution. But, you could at least solve them numerically, as you asked.

OK, so is $a_n(\lambda) = f(\lambda)b_n$ allowed, or not? If not, why not? Put more conditions on this $\lambda$ dependence! To be honest, this question looks like you're just making up random equations for its own sake; this can be great fun, but I don't see its relevance to mathematical research (unless you have a specific problem which needs this kind of construction, and even then it seems a bit too localised to me).

...of course those integrals don't have any particular significance, it's just for fun. But, e.g. why not take $a_n(\lambda) = f(\lambda)b_n$ for any $f$ you like, and any $b_n$ for which $\sum b_n$, $\sum b_n^2$ both converge; and you can change $b_n$ in various ways (just changing the first term is probably enough) to get various $\rho$. If this is not what you want, why not?

From the original question: "non-trivially dependent" - sorry, this is just too vague and meaningless. Why don't you give some "trivially dependent" examples first and say why you don't accept them. Better yet, say why you want such a series; what will you do with it? By the way, just for fun, $\int_0^\infty \frac{\sin x}{x} dx = \int_0^\infty \left( \frac{\sin x}{x} \right)^2 dx$.

You could go through your favourite proof that $e^{2 \pi i} = 1$ and substitute everything back into the integral directly and see if it works, but obviously that would be truly horrible! If there is a good answer NOT involving exp, this would be interesting (and would, presumably, provide a new proof for exp). Maybe you could choose some properties of exp, the most obvious being the addition formula, and see if you can prove them directly from the integral formula; that might give some idea.

I added the Functional Analysis tag; apologies if you think it's not appropriate.

Some (probably stupid) comments: maybe it would help if you explicitly gave the formula for $\varphi_n$, since there seem to be several different versions in use. Anyway, if you only need an upper bound, you could find an interval $[-R_n, R_n]$ where $\varphi_n$ is concentrated, and use Cauchy-Schwarz to bound the $L^1$ norm on this interval using the $L^2$ norm, since the $L^2$ norm is much easier to use. The only trick you'd need to make this work is an inequality telling you how rapidly the functions decay away from zero. Hopefully, $R_n$ would not grow too fast with $n$ for this to work.

Is it the case that all the "truly deep and interesting aspects" of numerical analysis are too complicated, or long, to explain in an ordinary course? e.g. how do meteorologists/computational physicists/etc. solve huge systems of equations? Is it really just using the same algorithms that we see in the books, but with expensive supercomputers, or are there fundamentally better techniques which are too difficult to cover?

...Maybe I'm getting confused by the term "categorical"; I would say it's unreasonable to ask for arbitrary infinite products without some kind of extra condition. We don't ask for infinite products and sums of numbers without an extra convergence property, so why should we expect one with spaces? After all, we can't do sums $\oplus X_\alpha$ unless we impose extra conditions (e.g. $l^p$ sums).

...So, you're saying that $P$ (with the finite supremum condition) is a product, but that it's not "categorical", correct? [Just to check I'm not even more stupid than I think I am!]

...Sorry, I am being stupid, I misread the question! $\lambda$ being bounded is equivalent to the condition $\sup_\alpha \| f_\alpha \| < \infty$. It's pretty clear, I think, that $P$ will define a bounded operator if and only if the supremum is finite; then the norm equals the supremum exactly.

But, Yemon, $\| f_\alpha \| = | \lambda(\alpha)|$ as an operator from $Y$ to $X_\alpha$, so the condition $\sup_\alpha \| f_\alpha \| < \infty$ means that $\lambda$ can't be unbounded. Or am I being stupid (very possible)?

confluential: you say: "The solution to this difference differential equation is 0-th row of matrix exponential..." Therefore, why are you asking this question on MO? Why don't you directly ask "How do we compute exp(M) for a tri-banded matrix M" or similar (after first trying to look it up yourself, of course; I'm sure almost every good numerical analysis book will discuss this at length).

If memory serves correctly, a classic old book: Bromwich, Infinite Series, has a lot about products as well as sums [but disclaimer: I haven't even looked at this book for at least 15 years!] You might have some luck with various English mathematics books written before 1950, since this kind of thing was a lot more in fashion back then, at least in England [presumably due to G.H.Hardy's influence, although I am no historian].

@Qiaochu Yuan, @Gerhard Paseman Thanks for your comments. Maybe I've completely missed the point and don't understand what's going on here, so should shut up! I seem to be getting a bit confused between "real numbers" and "real closed fields"!

I'm an analyst, not a logician, so maybe I will write nonsense; but your question seems very strange to me. For example, the numbers 0,1 are always present in any field, not just the real numbers; and from these, a copy of the integers (maybe modulo N) can be constructed. So, what do you mean by "we'll need axioms that explain what an integer is"? You also say "the usual Archimedean property is not the kind of axiom we can include..." But the real numbers definitely DO have the Archimedean property, whether you take it an an axiom, or deduce it from other axioms; so what do you mean by this?