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I'm trying to learn more about kernel machine theory and I've discovered that I need to learn a lot of background math, and so I'm looking for some good resources for this. In particular: I've got Schölkopf and Smola's Learning with Kernels book and they start discussing Fourier transformations, Green's functions, operators (e.g. I've never heard of a pseudo-differential operator before), and other such things. I have no experience working with any of this but I really want to understand it. While I can certainly google individual examples I would really prefer to have a more comprehensive treatment.

Sorry if this is too vague or specific, but I'm really struggling with finding out how to start systematically acquiring the background math so that I can comfortably work with kernels and RKHS theory. Thanks a lot.

Update: I kept my background out because I was afraid that it would make this too specific to me, but because it was asked: I've taken one course in real analysis and one course in modern algebra, as well as a standard linear algebra and multivariate calculus course. I have not studied differential equations. I've also taken a number of courses in mathematical statistics (including some measure-theoretic ones, although I've never formally studied measure theory). I'm comfortable with the narrow range of statistics that I've studied (e.g. LLN, CLT, exponential families, GLMs, mixed models, complete and sufficient statistics, ...), but I don't have much of a pure math background which I feel is starting to hurt me.

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    $\begingroup$ The terms you've mentioned all relate to differential equations. Hence, working through a relatively accessible text such as Boyce/ DiPrima's Elementary Differential Equations and Boundary Value Problems might go a long way to establishing the background math (Fourier transforms, Green's function and linear operators all receive attention in this text). $\endgroup$
    – Robert de Graaf
    Commented May 19, 2016 at 11:21
  • $\begingroup$ @RobertdeGraaf Thanks a lot for the comment. That's a very interesting point -- I've never studied differential equations, maybe that's a big missing piece. I'll definitely look into that book. $\endgroup$
    – alfalfa
    Commented May 20, 2016 at 1:52

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You haven't given us much information about your current mathematical background. Do you have the background of a typical undergraduate science or engineering student (single and multivariable calculus, ordinary differential equations and perhaps an exposure to Fourier series)? Have you taken any introductory courses in analysis?

One classic text book that introduces applied functional analysis to students who have typical engineering math backgrounds and some analysis is Optimization by Vector Space Methods by David G. Luenberger.

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    $\begingroup$ Luenberger's book is probably a good choice for someone with your background. What's really important here is not so much the concept of the RKHS as the representer theorem which gives conditions under which your machine learning problem becomes a finite dimensional optimization problem. $\endgroup$
    – Brian Borchers
    Commented May 20, 2016 at 3:17
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    $\begingroup$ I just skimmed the table of contents for that book and it seems to contain a large number of the topics that I've felt ignorant of. Thanks! $\endgroup$
    – alfalfa
    Commented May 20, 2016 at 3:23

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