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It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq 0$) or $f(t) = \sin (at+b)$ (or any finite superposition of such functions, even with incommensurable periods) is $0$; for instance, introduce a suitable kernel $g(t)$ such as $\exp(-t^2)$ and define $\int_{\bf R} f$ to be $\lim_{s \rightarrow \infty} \int_{-\infty}^{\infty} f(t) \: g(t/s) \: dt$. But I have trouble when I try to do this for integrals over ${\bf R}^2$, and things go seriously amiss over ${\bf R}^3$. It seems that all mollifiers fail, basically because the surface measure of bodies in ${\bf R}^d$ is "too big" compared to the interior measure when $d > 1$ (a problem that gets worse and worse as $d$ increases).

What I want is a theory of generalized integration over ${\bf R}^d$ such that compactly supported functions integrate to their usual value, while periodic functions with mean value $0$ integrate to $0$; and I want this theory to apply to a vector space of functions (that is, I want arbitrary linear combinations of finitely many generalized-integrable functions to be generalized-integrable).

Is there (or could there be) such a theory?

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    $\begingroup$ Can you give a specific example of an integral that you're having trouble with? ${\bf R}^d$ is still of polynomial growth, and generally one can handle oscillation via mollifiers, using repeated integration by parts to control error terms. Alternatively one can take the distributional Fourier transform of $f$ and hope that it is equal to a continuous function near the origin. Zeta regularisation type methods based on analytic continuation should also work. $\endgroup$ – Terry Tao Apr 4 '16 at 17:29
  • $\begingroup$ Here's an example: Let $f_1$ and $f_2$ be the indicator functions of two (unrelated) Leech-lattice packings of balls in ${\bf R}^{24}$. (By "unrelated" I mean that they differ by some generic isometry of ${\bf R}^{24}$.) The theory should have the property that $f_1 - f_2$ integrates to 0. Actually, it'd be nice if the theory could handle distributions, since for my purposes it might be nicer to work with the difference between the Dirac combs associated with the centers of the balls in the packing. $\endgroup$ – James Propp Apr 4 '16 at 18:17
  • $\begingroup$ In the theory of Sebastião e Silva mentioned in my answer below, the presence of combinations of Dirac functions in the integrand poses no problems. $\endgroup$ – oeiras Apr 4 '16 at 19:03
  • $\begingroup$ In your example it appears from the Poisson summation formula that the distributional Fourier transform of $f_1-f_2$ vanishes near zero, so it seems one could just take the second option I mentioned in my earlier comment. (Poisson summation should also make the mollifier method work, I think.) $\endgroup$ – Terry Tao Apr 4 '16 at 19:21
  • $\begingroup$ The question as asked admits the following trivial answer: if $f=g+p$ is a sum of a compactly supported integrable function and a periodic function, define $I(f)=\int g+\overline{p}$. $\endgroup$ – Christian Remling Apr 4 '16 at 23:28
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There is a concept of definite integrals for distributions which has three of the features you request:

continuous functions of compact support are integrable---even functions which are integrable in the sense of Lebesgue;

the trigonometric functions you describe integrate over the real line to zero;

the space of integrable distributions forms a vector space and the integral is a linear function.

The theory was developed about 50 years ago by the portuguese mathematician J. Sebastião e Silva and is completely elementary, being based on his axiomatic approach. All of his writings are available online and can easily be found using a search machine.

The definition is based on two simple facts:

every distribution on the line has a primitive;

the existence of an elementary definition of the concept of the limit of a distribution at a point (which can be $\pm \infty$).

Sebastião e Silva also considered the case of multivariate distributions.

Many of the articles mentioned are in portuguese but there is an english version "Integrals and orders of growth of distributions" in the proceedings of a conference on the theory of distributions (Lisbon, 1964) in which he discusses applications to the Fourier transform of distributions. This and his other writings are available at the site: www.sebastiaoesilva100anos.org.

Information added as an edit since the system rejected a comment. Sebastião was working on an encyclopaedic monograph which would have given a complete presentation of his theory with applications to mathematical physics but his early passing prevented its completion.

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  • $\begingroup$ Thanks, oeiras! While I wait for my library to get me a copy of the English version, can you tell me whether e Silva's definition is isometry-invariant ($\int f = \int (f \circ T)$ for all isometries $T$ of ${\bf R}^n$)? If (as I suspect from what I've seen) e Silva's definition for multivariate distributions uses induction on dimension, proving isometry-invariance could be nontrivial. Also, I'm hoping his definition satisfies $\int (f - f \circ T) = 0$ for a broad class of functions $f$ with $\int f = \infty$; without that, I doubt my Leech-lattice-packing integral would be tractable. $\endgroup$ – James Propp Apr 5 '16 at 14:52
  • $\begingroup$ Also, I'd be very interested to know of follow-up literature that builds on e Silva's theory. The book "Introduction to the Theory of Distributions" (by J. Campos Ferreira, R. F. Hoskins, and Jose Sousa-Pinto) looks like it has a lot of what I want, but it defers the treatment of the multdimensional theory to "a later volume" that as far as I can tell was never published (though maybe I'm just googling ineptly). $\endgroup$ – James Propp Apr 5 '16 at 14:59
  • $\begingroup$ Firstly, the english version is also available online as are the whole Proceedings (published by the Gulbenkian Foundation) which are well worth a look, also for articles by Schwartz, Lions, Martineau ... on the site I quoted. There is alo a more detailed account of his integration theory under "Textos Didacticos", but in portuguese. I will have a look there to see if there is anything on the isometry you mention. The book by Campos Ferreira is based on a course given in Lisbon which I also had the privilege to attend. I stll have my notes but only the univariate case was covered. $\endgroup$ – oeiras Apr 6 '16 at 5:56
  • $\begingroup$ I found the online link oeiras mentioned; it is on page 4 of the pdf file sebastiaoesilva100anos.org/Publicacoes/Lista/Publicacoes.pdf, or one can go to sebastiaoesilva100anos.org/Publicacoes/publicacoes.html and click on the leftmost book on the lowest shelf. The former links gives just the article; the latter gives the whole book. $\endgroup$ – James Propp Apr 6 '16 at 15:03
  • $\begingroup$ It's worth mentioning that for the Heaviside function on ${\bf R}$, $\int (f - f \circ T)$ is NOT equal to 0 when $T$ is a nontrivial translation. So some restrictions apply to the sort of functions $f$ that satisfy this equality. Come to think of it, isometry-invariance is not what I need for my intended application. $\endgroup$ – James Propp Apr 6 '16 at 15:33

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