Introduction to the mathematical theory of regularization

Question

I asked the question "Are there books on Regularization at an Introductory level?" at physics.SE.

I was informed that "there is (...) a mathematical theory of regularization (Cesàro, Borel, Ramanujan summations and many others) that is interesting per se".

Question: Can someone advise me on how to study one or more of the above topics and provide some reference?

Answer 1

In terms of summations of otherwise divergent series (which is what Borel summation and Cesàro summation are about), a decent reference is G.H. Hardy's Divergent Series.

In terms of divergent integrals, you may also be interested in learning about Cauchy principal values, which is related to Hadamard regularisation. (The references in those Wikipedia articles should be good enough; these two concepts are actually quite easily understood.)

Zeta function regularisation has its roots in number theory, which unfortuantely I don't know enough about to comment.

Heat kernel type regularisation techniques is closely related to the study of partial differential equations and harmonic analysis. It is related to Friedrichs mollifiers (an exposition is available in most introductory texts in generalised functions / distribution theory; and a slightly more advanced text is volume 1 of Hörmander's Analysis of Linear Partial Differential Operator). It can also be interpreted as a Fourier-space cut-off (which in physics terminology is probably called "ultraviolet cutoff" and which can be interpreted in physical space as setting a minimal length scale), so can be described in terms of, say, Littlewood-Paley Theory (another advanced text is Stein's Topics in Harmonic Analysis relating to Littlewood-Paley Theory) or the FBI transform. Unfortunately I don't know many good introduction texts in these areas. But I hope some of these keywords can aid your search.

Answer 2

I am not sure it is what you're asking but it seems that you're interested in summability methods and in regular matrix methods.

As far as I know, the pioneering work in this area was Hardy's book Divergent series. It is mentioned in the wikipedia article, too. Here is google books preview.

Classical references in this area seem to be:

• Richard G. Cooke: Infinite matrices and sequence spaces (I only have access to Russian translation Кук Р. - Бесконечные матрицы и пространства последовательностей)
• G. M. Petersen: Regular matrix transformations

Texts which tend more towards functional analysis (thus they could be called more modern) are:

• J. Boos: Classical and modern methods in summability I have written down some notes while reading a few first chapters. You can see I did not get too far. (There were some typos and minor mistakes, otherwise my impression is that it's a well-organized and interesting book.)
• A. Wilansky: Summability Through Functional Analysis