Nice applications for Schwartz distributions I am to teach a second year grad course in analysis with focus on Schwartz distributions. Among the core topics I intend to cover are:


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*Some multilinear algebra including the Kernel Theorem and Volterra composition,

*Some Fourier analysis including the Bochner-Schwartz Theorem,

*An introduction to wavelets with a view to structure theorems for spaces of distributions or function spaces,

*Probability theory on spaces of distributions including the Lévy Continuity Theorem,

*A study of homogeneous distributions and elementary solutions to linear PDEs.
My question is: What "cool topics/applications" would it be nice to include in such a course? I am particularly interested in examples with a high return on investment, i.e., that would not take too long to cover yet would provide the students with valuable tools for eventually a future research career in analysis. Please provide references where I can learn more about your suggestions. I would like some variety if possible. I got suggestions pertaining to probability, PDEs and mathematical physics, but it would be nice to get apps related to other areas of math.
 A: One of my favourite applications of 'basic' distribution theory, which actually requires most of the tools in your list to be fully apreciated, is the Malgrange--Ehrenpreis Theorem on the local solvability of arbitrary constant coefficients PDE. There are a few proofs, but the proof in M.Taylor's book on Pseudodifferential Operators (last Section of first Chapter) is especially suited for your course I think. It is a substantial theorem though, so it might or might be not a good idea to include it, depending on the length and size of the course.
A: Unsurprisingly, the topics that occur to me have various connections to number theory (and related harmonic analysis) (and unclear to me what might have already been done in your course...):
EDIT: inserted some links... EDIT-EDIT: one more...


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*Genuinely distributional proof of Poisson summation: http://www.math.umn.edu/~garrett/m/fun/poisson.pdf

*Meromorphic continuation of distributions $|x|^s$ and ${\mathrm sgn}(x)\cdot |x|^s$ http://www.math.umn.edu/~garrett/m/fun/notes_2013-14/mero_contn.pdf

*(Fancier version of the previous: mero cont'n of $|\det x|^s$ on $n\times n$ matrices (if this is interesting, I have some notes, or maybe it's a fun exercise). In particular, stimulated by a math-overflow question of A. Braverman some time ago, there are equivariant distributions (e.g., on two-by-two matrices) such that both they and their Fourier transform are supported on singular matrices... Wacky! http://www.math.umn.edu/~garrett/m/v/det_power_distn.pdf

*Decomposition of $L^2(A)$ for compact abelian topological groups $A$ (by Hilbert-Schmidt, hence compact, operators). http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06c_cpt_ab_gps.pdf

*Reconsideration of Sturm-Liouville problems (with reasonable hypotheses), to really prove things that are ... ahem... "suggested" in usual more-naive discussions.

*Quadratic reciprocity over number fields (and function fields not of char=2) ... cf. http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf
(This presumes Poisson summation for $\mathbb A/k$...)

*Explanation that Schwartz' kernel theorem is a corollary of a Cartan-Eilenberg adjunction (between $\otimes$ and $\mathrm{Hom(,-)}$), when we know that there are genuine (i.e., categorically correct) tensor products for "nuclear Frechet spaces", ... which leads to the issue of suitable notions of the latter. http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/06d_nuclear_spaces_I.pdf

*The idea that termwise differentiation of Fourier series is "always ok" (with coefs that grow at most polynomially), if/when the outcome is interpreted as lying in a suitable Sobolev space on the circle. And that polynomial-growth-coefficiented Fourier series _always_converge_... if only in a suitable Sobolev space.
http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/03b_intro_blevi.pdf
** Possibility of ranting about the limitations of pointwise convergence... especially when placed in contrast to convergence in Sobolev spaces...


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*Use of Snake Lemma to talk about mero cont'n of the Gamma function, via mero cont'n of $|x|^s$. :) http://www.math.umn.edu/~garrett/m/v/snake_lemma_gamma.pdf

*Peetre's theorem that any (not necessarily continuous!) linear functional on test functions that does not increase support is a differential operator. (I have a note on this, which may be more palatable to beginners than Peetre's paper.) http://www.math.umn.edu/~garrett/m/v/peetre.pdf

*Uniqueness of invariant functionals... As the easiest case (which is easy, but cognitive-dissonance-provoking, in my experience), proving that $u'=0$ for a distribution $u$ implies that $u$ is (integration-against) a constant. (Maybe you'd do this along the way...) http://www.math.umn.edu/~garrett/m/v/uniq_of_distns.pdf

*... this is not a stand-alone topic, but: the usual discussions of pseudo-differential operators (e.g., "symbols") somehow shrink from talking about quotients of TVS's in a grown-up way... If that hadn't been done earlier, and/or people had a (reasonable!) feeling of discomfort about the usual style of chatter in the psi-DO world, perhaps this could be happy-making.

*Meromorphic/holomorphic vector-valued functions (cf. Grothendieck c. 1953-4, and also various of my notes) e.g., meromorphic families of distributions... E.g., the $|x|^s$ family on $\mathbb R$ has residues which are the even-order derivatives of $\delta$, and ${\mathrm {sgn}}(x)\cdot |x|^s$ has as residues the odd-order derivatives of $\delta$. http://www.math.umn.edu/~garrett/m/fun/Notes/09_vv_holo.pdf
Depending on context, there are somewhat-fancier things that I do find entertaining and also useful. Comments/correspondence are welcome.
A: No so much as applications, but I would love to learn about the basics of hyperfunctions (of the Japanese school) in such a course, if only in one dimension. Specifically I think it gives an easier way to define the wavefront set, which is what's required to explain why we can't square the delta function (referring to a comment above). See also the question Is square of Delta function defined somewhere?
For what it's worth, I found the book


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*Kato & Struppa, Fundamentals of Algebraic Microlocal Analysis
particularly well written. The article in EoM is also great
https://www.encyclopediaofmath.org/index.php/Hyperfunction
A: Two (edit: now four) not-so-usual examples come to my mind:


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*There is the proof of the central limit theorem using Fourier analysis, as done in Chapter 7 of Hörmander's book. It's a cornerstone of probability theory and the Fourier-analytic proof presented by Hörmander is a nice showcase of the power of distribution theory: one gets a conceptually deep and useful theorem in full generality with relatively little effort;

*Something I have been meaning to try is to work out some "mathematical toy examples" of continuum limits of distributions supported on a lattice, as done e.g. in Chapter 15 of the book by R. Fernández, J. Fröhlich and A.D. Sokal, Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory (Springer, 1992). I do not know if it is a feasible task, but since you are a specialist is constructive QFT, you should be able to do it... Edit 3: a classical and rather charming example of this has just came to my attention: the so-called Bernstein polynomials, used by their namesake Sergei N. Bernstein (the same mathematician who found the family of inequalities - also named after him - for smooth functions with compactly supported Fourier transform which underlie the Littlewood-Paley theory discussed in my Edit 1 below) to provide a probabilistic proof of the Weierstrass approximation theorem through the weak law of large numbers. Taking formal adjoints of the sequence of Bernstein polynomial operators provides a general lattice approximation for distributions on compact rectangular domains. A rather extensive discussion can be found in the short and beautiful book of the same title by G. G. Lorentz (the mathematician who invented the so-called Lorentz interpolation spaces, not the Dutch physicist).

*(Edit 1) A third one, which I am particularly fond of, is the dyadic Fourier analysis of Littlewood and Paley, which is strongly connected to point (3). The connection to renormalization group ideas is quite obvious, and it has become a standard tool for obtaining important inequalities in nonlinear analysis (Gagliardo-Nirenberg, Moser, Schauder), all the way to the monumental Nash-Moser implicit function theorem (as shown by Hörmander - him again - in the 70's). A nice pedagogical exposition of these ideas can be found the book by S. Alinhac and P. Gérard, Pseudo-Differential Operators and the Nash-Moser Theorem (AMS, 2007) - just watch out for the numerous typographic errors!

*(Edit 2) A fourth one, which is not actually unusual in itself but rather in its standpoint, is that many interesting examples of distributions (principal values, partes finies, extensions of homogeneous distributions to the origin, etc.) are obtained as extensions of distributions initially defined in (often finite-codimensional) subspaces of test functions, collectively called renormalizations. What would be interesting, also regarding applications to physics, is to treat such examples from this unified viewpoint. Indeed, as it can be seen e.g. by (Bogoliubov-Parasiuk-)Hepp's and Epstein-Glaser's treatments of perturbative renormalization in QFT, the latter really amounts to renormalizing distributions in the above sense. As a famous dictum of K. Hepp goes (Proof of the Bogoliubov-Parasiuk Theorem on Renormalization, Commun. Math. Phys. 2 (1966) 301-326):


"The renormalization theory (of Dyson) is in this framework a constructive 
  form of the Hahn-Banach theorem." (pp. 302 - parentheses mine)

A: Let me mention two nice applications to Sobolev spaces:
The first result shows that if $f\in \mathcal{D}'$ and $\nabla f\in L^p_{\rm loc}$, then $f\in L^p_{\rm loc}$. For a short proof of this classical result see
https://mathoverflow.net/a/296464/121665
For applications of this result to higher order Sobolev spaces see:https://mathoverflow.net/a/297392/121665
