What is $p$-adic Fourier series?

Q1: Can we define Fourier series for a function $\mathbb{Z}_p\to \mathbb{Q}_p$?

Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion: $$B_n(\{x\})=-\frac{n!}{(2\pi i)^n}\mathop{{\sum}}_{|m|>0}\frac{e^{2\pi imx}}{m^n}\qquad(n\ge1).$$ It means that Bernoulli polynomial has fastest Fourier series among all polynomials of a given degree.

Which polynomials correspont to extremal $p$-adic series?

• See page 50 of Koblitz's intro to $p$-adic numbers, $p$-adic analytis and zeta functions. – stankewicz Dec 15 '13 at 14:44
• arxiv.org/abs/math/0102012 – Carlo Beenakker Dec 15 '13 at 14:51
• The notion of an orthogonal basis is different in $p$-adic analysis, since the sup-norm is more convenient that the $L^2$-norm. In a Hilbert space there is a close link between best approximations in a subspace and orthogonal projection. But over the $p$-adics, the concept of "best approximation" doesn't correspond to a unique vector and it's not related to an inner product. See sections 21 and 50 of Schikhof's "Ultrametric calculus". – KConrad Dec 15 '13 at 15:03

At least as I know, the first approach to $p$-adic fourier theory was done by Woodcock in the 70's. See "Fourier analysis for p-adic Lipschitz functions" (J. London Math. Soc. (2) 7 (1974), 681–693). It is interesting to note that here he defines the same "integral" that Volkenborn defined also in the mid 70's. Volkenborn was inspired by some sums that Kubota and Leopoldt used to define $p$-adic zeta functions. I don't know what was the inspiration of Woodcock, but he surely was aware of $p$-adic zeta functions.
Two more advanced references are the book of Van Rooij on non-archimedean functional analysis, and Schneider and Teitelbaum $p$-adic fourier theory. The last one has many arithmetic aplications. To mention one, thay are able to prove very nice congruences between Bernoulli-Hurwitz numbers (see Katz papers for this beatiful topic).
• The Volkenborn integral has its Wikipedia entry: en.wikipedia.org/wiki/Volkenborn_integral As a classical harmonic analyst, I view the (classical) Fourier integral as natural because it uses integration against Haar measure. But there is no Haar measure in the $p$-adic setting. Is it easy to explain to a non-expert why it is relevant to define "Fourier analysis" with respect to an integral which is NOT translation-invariant? – Alain Valette Apr 28 '17 at 19:41
• @Alain Valette: I'm not an expert, but the (number-theoretic) application I know is the following. One studies the $p$-adic dual space of a nice space of functions (continuous, locally analytic, sctrictly differentiable, etc...), and this gives you $p$-adic distributions, which are continuous linear operators but not translation-invariant (except the trivial one). For example, the Volkenborn integral is actually an element of the dual space of strctly differentiable functions (see Schikhof or Robert's books on $p$-adic analysis). – EFinat-S Apr 28 '17 at 21:45
• @Alain Valette: Now, this $p$-adic distributions let you define, for example, analogues of zeta and L-functions, attached to arithmetic objects, such as number fields or elliptic curves. Why? because they satisfy nice congruence properties that in the $p$-adic setting means continuity. A nice example is the study of the dual space of continuous functions, those linear operators being called $p$-adic measures. In the 70's Iwasawa observed that this measures are in correspondence with some $p$-adic power series, which carries arithmetic information of cyclotomic fields of prime power order. – EFinat-S Apr 28 '17 at 21:49
• @ EFinat-S: Thanks for your patient explanation! They remind me of conversations I had with Alain Robert, who was my colleague for 15 years. It seems you do harmonic analysis when you like groups, and you do $p$-adic analysis when you like number theory: equally respectable, but distinct motivations! – Alain Valette Apr 29 '17 at 6:59