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# Fourier Analysis on Local Fields. (MN-15):

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## Overview

This book presents a development of the basic facts about harmonic analysis on local fields and the *n*-dimensional vector spaces over these fields. It focuses almost exclusively on the analogy between the local field and Euclidean cases, with respect to the form of statements, the manner of proof, and the variety of applications.

The force of the analogy between the local field and Euclidean cases rests in the relationship of the field structures that underlie the respective cases. A complete classification of locally compact, non-discrete fields gives us two examples of connected fields (real and complex numbers); the rest are local fields (p-adic numbers, p-series fields, and their algebraic extensions). The local fields are studied in an effort to extend knowledge of the reals and complexes as locally compact fields.

The author's central aim has been to present the basic facts of Fourier analysis on local fields in an accessible form and in the same spirit as in Zygmund's *Trigonometric Series* (Cambridge, 1968) and in *Introduction to Fourier Analysis on Euclidean Spaces* by Stein and Weiss (1971).

Originally published in 1975.

The **Princeton Legacy Library** uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

## Product Details

ISBN-13: | 9780691618128 |
---|---|

Publisher: | Princeton University Press |

Publication date: | 03/08/2015 |

Series: | Princeton Legacy Library Series |

Pages: | 308 |

Product dimensions: | 6.00(w) x 9.10(h) x 0.70(d) |

## Read an Excerpt

#### Fourier Analysis on Local Fields

**By M. H. Taibleson**

**Princeton University Press and University of Tokyo Press**

**Copyright © 1975 Princeton University Press**

All rights reserved.

ISBN: 978-0-691-08165-6

All rights reserved.

ISBN: 978-0-691-08165-6

#### Contents

Preface,Introduction,

Chapter: I. Introduction to local fields, 1,

Chapter II. Fourier analysis on K, the one-dimensional case, 20,

Chapter III. Fourier analysis on Kn, 115,

Chapter IV. Regularization and the theory of regular and sub-regular functions, 168,

Chapter V. The Littlewood-Pale) function and some applications, 195,

Chapter VI. Multipliers and singular integral operators, 217,

Chapter VII. Conjugate systems of regular functions and an Fo and M. Riesz theorem, 241,

Chapter VIII. Almost everywhere convergence of Fourier series, 262,

Bibliography, 286,

CHAPTER 1

Introduction to local fields

This chapter contains a description of several examples of local fields and a review of basic facts about the classification and structure of local fields.

1. Rademacher functions and the Walsh-Paley group

The Rademacher functions are defined on [0,1] by the rule, φk(x) = sgn(sin 2k+1 πx x), k = 1, 2, 3,..... It is easy to see that the sequence {φk}∞k=1 is orthonormal on [0,1] with respect to ordinary Borel-Lebesgue measure. In the language of probability theory we would say that the sequence of Rademacher functions is a sequence of uncorrelated random variables, each with mean zero and variance 1. While it is not crucial for the development that follows, it is of interest to note that these functions are also equi-distributed and independent.

{φk}∞k=1 is not complete. It is extended to a complete system as follows: Let ψ0(x) [equivalent to] 1, and if n is a positive integer we write n uniquely as, n = a0 + a1 2 + ... + al2l, where al = 1 and ak = 0 or 1, k = 0, 1, ..., l-1. Then we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The system {ψn}∞n=0 is a complete orthonormal sequence on [0,1]. It is known as the Walsh-Paley system.

For f [member of] L1[0, 1] we define the Fourier coefficients and partial sums of the Fourier series of f with respect to the Walsh-Paley system in the usual way. That is, we let ck = ∫10 f(x)ψk(x)dx, k = 0, 1, ..., and Sn(x;f) = [summation]n-1k=0 ckψk(x). These partial sums have a most curious property. Let x [member of] [0,1], Ik, x be the dyadic interval of length 2-k that contains x and assume for the sake of simplicity that x is not a dyadic rational. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This phenomenon is explained by the following: If x [member of] [0,1] we may write x = [summation]∞k=0 ak2-k, ak = 0 or 1. There Is also a space of sequences {a0, a1, ...} of zeros and ones. We identify x with the sequence [a0, a1, ...]. If x is a dyadic rational, 0 < x < 1, then this representation is not unique, but it only fails to be 1:1 on this countable set. In this sequence space an addition, usually denoted x + y J is defined as addition of sequences coordinatewise, mod 2. When this collection of sequences is given the obvious metric topology, it is easily seen that it is a compact abelian group. This group is called the dyadic group, 2ω, or the Walsh-Paley group. A remarkable fact is that the Borel ring and Haar measure on 2ω agree with the usual Borel-Lebesgue structure of [0,1].

The sequence of Walsh-Paley functions is a complete set of characters on 2ω.

2. The 2-adic numbers. and 2-series numbers

The 2-adic norm, |·|2, is defined on the (rational) integers as follows: |0|2 = 0. If n ≠ 0 we write n = 2ka where (a, 2) = 1 and let |n|2 = 2-k. It is easy to see that x -> |x|2 is a norm on the integers. Moreover, |n+m|2 ≤ max[|n|2, |m|2] for n and m integers. If we keep the usual arithmetic for the integers and define a metric by d(n,m) = |n-m|2 we see that, with this metric, the integers form a metric space. Its completion is called the ring of 2-adic integers. It is easily identified with the collection of formal power series {x = [summation]∞k=0 ak2k} where each ak = 0 or 1, and where add it ion and multiplication is done formally, carrying from left to right. In this collection we see that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We also see that the non-negative (rational) integers are exactly that collection of 2-adic integers with finite expansions. It is a useful exercise to verify that -1 = 1 + 2 + 22 + ... 1/(1-2). (Note that |2|2 =1/2.)

It is easily verified that the ring of 2-adic integers, under addition, is a compact abelian group. Its field of quotients is called the 2-adic numbers. The 2-adic numbers can also be obtained by extending the 2-adic norm to the rational numbers (write n/m = 2k x a/b with a and b relatively prime to 2 and set |n/m|2 = 2-k) and then completing the rationals as a metric space with the induced metric. In either case we obtain a locally compact topological field of characteristic zero, that is totally disconnected. Its elements are identified as formal Laurent series, {x = [summation]∞k=l ak2k}, where each ak = 0 or 1 and we "carry" in the arithmetic. Note also that if al = 1 in the representation above, then |x| = 2-l.

If we consider the same set of formal symbols, but now do the addition and multiplication with the coefficients {ak} viewed as elements of GF(2), and use the same topology and norm as we used for the 2-adic numbers, we also obtain a locally compact topological field that is totally disconnected. However, this field is of characteristic 2. It is called the Z-series field. The set of power series elements in the 2-series field (namely the collection of elements x = [summation]∞k=0 akpk) is seen to be a compact ring. It is not difficult to verify that the power series ring of the 2-series field coincides, under addition, with the Walsh-Paley group, 2ω. The group 2ω is thus endowed, in a natural way, with a multiplication.

3. p-adic and p-series numbers

In a manner that is directly analogous to that for 2-adic and 2-series numbers we may construct p-adic and p-series fields where p is any rational prime. These fields will be locally compact fields that are totally disconnected and are either of characteristic zero (p-adic) or of characteristic p (p-series).

4. Classification of local fields

Let K be a field and a topological space. Then K is a locally compact field if K+ and K* are locally compact abelian groups (where K+ and K* are the additive and multiplicative groups of K).

If K is any field and is endowed with the discrete topology then K is a locally compact field. Therefore we exclude the discrete topological fields from consideration. (The only "naturally" discrete fields are the finite fields.)

We consider fields K that are locally compact, non-discrete and (redundantly but emphatically) topologically complete.

If K is connected then K is either R or C.

If K is not connected then K is totally disconnected.

If K is of finite characteristic, then K is a field of formal power series over a finite field GF(pc). If c = 1 it is a p-series field. If c ≠ 1 then K is an algebraic extension of degree c of a p-series field.

If K is of characteristic zero then K is either a p-adic number field or a finite algebraic extension of such a field.

In the sequel, a local field K is always a locally compact, non-discrete. totally disconnected field.

5. Properties of local fields

Let K be a fixed local field. Since K+ is a locally compact abelian group we may choose a Haar measure dx for K+. If α ≠ 0 (α [member of] K), then d(αx) is also a Haar measure. Let d(αx) = |α|dx and call |α| the absolute value or valuation of α. Let |0| = 0.

Comment. The mapping α -> |α| is simply the modular function of the endomorphism x -> αx.

The mapping x -> |x| has the following properties: |x| = 0 iff x = 0; |xy| = |x y|; and |x+y| ≤ max[|x|, |y|]. The last of these properties is called the ultrametric inequality and a norm or valuation which satisfies it is said to be ultrametric or a nonarchimedian norm.

Note. If |x| ≠ |y| then |x+y| = max[|x|, |y|].

Proof. Suppose |x| ≠ |y| and |x + y| < |x|. Then |x| = |(x+y)-y| ≤ max[|x+y|, |y|] < |x|, a contradiction.

Fact. A Haar measure on K* is dx/|x|.

Notation. D = (x [member of] K: |x| ≤ 1}. D is called the ring of integers in K.

The ring of integers 1n a local field K is the unique maximal compact subring of K.

Example. If K is the 2-adic number field then D is the ring of 2-adic integers. If K is the 2-series field then the additive group of D is the Walsh-Paley group.

Notation. D = {x [member of] K: |x| < 1}. D is called the prime ideal in K.

The prime ideal in K is the unique maximal ideal in D. It is principal and prime. The fact that K is totally disconnected implies that the valuation is discretely valued. That is, the set of absolute values is of the form {sk}+∞k= - ∞ [union] {0} for some s > 0. Thus, there is an element of D of maximum absolute value.

Notation. Let p be a fixed element of maximum absolute value. p is called a prUne element of K. As an ideal in D, B = (p) = pD.

Example. If K is the 2-adic numbers as defined in §2 then x [member of] D iff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and p = 2 (p0 = 1).

Fact. D is compact and open. Hence B is compact and open.

It follows from these facts that D/B is isomorphic to a finite field GF(q), where q = pc for some prime p and positive integer c. To see this note that B compact implies that D/B is compact, that B open implies that D/B is discrete and since B is maximal D/B is a field and hence a finite field. In the sequel we assume that the prime power, q = pc is fixed.

Notation. For E a measurable subset of K let |E| = ∫K [xi]E(x)dx where [xi]E is the characteristic function of E and dx is a Haar measure normalized so |D| = 1.

Facts. |B| = q-1. |p| = q-1.

Proof. Decompose D into q costs of T. Thus 1 = |D| = q|T|, and so |B| = q-1. But B = p D. Thus q-1 = |B| = |p D| = |p|.

It follows that if x [member of] K and x ≠ 0 then |x| = qk for some k [member of] Z.

Notation. D* = D ~ B = (x [member of] K: |x| - 1). D* is the group of units K*. If x ≠ 0 we may write x = pkx' with x' [member of] D*.

Example. For K the 2-adic integers, D* = {[summation]∞k=0 ak2k: a0 = 1}.

Notation. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is called a fractional ideal.

Each Bk is compact and open and is a subgroup of K+. If k ≥ 0 then Bk is a subring of K. The collection {Bk}k [member of] ZZ~ is a fundamental system of neighborhoods of 0.

Notation. Let A0 = D*, Ak = 1 + Bk [subset] D*, k = 1, 2, ....

The collection {Ak}∞k=0 is a fundamental system of neighborhoods of 1 in D*.

A few miscellaneous facts about K.

A. If a [member of] K then limn - > ∞ an = 0 iff |a| < 1.

B. If L is a discrete subfield of K then L is finite.

From A we see that if x [member of] L then |x| = 1 or |x| = 0. Thus, L [subset] D* and being a discrete subset of a compact set, it is finite.

C. Let V be a topological vector space over K. Let V' be a finite dimensional subspace of V with basis (v1, ..., vn). Then the map (x1, ..., xn) -> [summation] xkvk of Kn to V' is an isomorphism of Kn and V' as topological vector spaces. where Kn is given the topological structure induced by the norm |(x1, ..., xn)| = supk|xk|.

D. Every finite dimensional vector space over K can be given only one structure as a topological vector space.

E. If V is a locally compact topological vector space over K then V has finite dimension d, and modV(x) ... |x|d for all x [member of] K.

F. Let A be an endomorphism of a vector space, V, of finite dimension over K. Then modV(A) ... |det A|.

Theorem (A summary).

Let K be a local field, D = {x [member of] K: |x| ≤ 1}, D* = {x [member of] K: |x| = 1}, and B = {x [member of] K: |x| < 1}. Then K is ultrametric, D is the unique maximal compact subring in K, D* is the group of units in K*, and B is the unique maximal ideal in D. There is a p [member of] B such that B = p D. The residue space | = D/B is a finite field of characteristic p. If q is the number of elements in | then the image of K* in (0, ∞) under the valuation |·| is the subgroup of (0, ∞) generated by q. |p| = q-1. A Haaz measure on K* is given by dx/|x|.

6. More facts about K

A. Let {ak}∞k=0 be any sequence in K with limit 0 in K. Then, [summation]∞k=0 ak converges commutatively.

An easy consequence of the ultrametric inequality.

B. Let U = [ai]qi=1 be any fixed full set of coset representatives of B in D. Then if x [member of] Bk, k [member of] ZZ, x can be expressed uniquely as x = [summation]∞l=k cl pl, cl [member of] U.

Proof. We may assume k = 0. The cl are defined inductively by the relations x [equivalent to] [summation]Nl=0 cl pl, mod(BN+1).

C. Let A be an automorphism of K (as a topological field). Then A maps D onto D, B onto B and has module 1 as an automorphism of K+.

Proof. Clearly the image of D is a maximal compact subring of K and so AD = D. Similarly AB = B. Now note that modK(A)|D| = |AD| = |D| so modk(A) = 1.

D. Let A be as in C. Then for all x [member of] K, |Ax| = |x|.

Proof. From C it follows that A maps D* = D ~ B onto itself, so (|x| = 1) -> (|Ax| = 1). Since Ap generates AB = B we must have |Ap| = q-1. If x ≠ 0 write x = pk x', x' [member of] D*. Then |Ax| = |Ap|k · |Ax'| = q-k · 1 = |x|. If x = 0 then Ax = 0 and again |Ax| = |x|.

E. Let K be a local field. Then K* contains a subgroup of order q-1 which is cyclic and unigue. Let M* be that subgroup. M* is the group of roots of 1 that are of order prime to p, which is to say the roots of xq-1 = 1. M = M* [union] {0} is a full set coset representatives of B in D. If K is of characteristic p then M = GF(q).

7. The dual of K*

Let [member of] be a generator of M*. Then, since {[member of]l}q-2l=0 U {0} is a complete set of coset representatives of B in D we have that if x [member of] K* then x can be written uniguely: x = pk [member of]l a, where a [member of] A = 1 + B, k [member of] ZZ, l = 0,1, ..., q-2.

k is determined by |x| = q-k, l by p-k x [equivalent to] [member of]l (mod B), and it is obvious that p-k [member of]-l x = a [member of] A.

It is now simple to see that D* = ZZq-1 x A and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and A = 1 + B.

*(Continues...)*

Excerpted fromFourier Analysis on Local FieldsbyM. H. Taibleson. Copyright © 1975 Princeton University Press. Excerpted by permission of Princeton University Press and University of Tokyo Press.

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## Table of Contents

- Frontmatter, pg. i
- Preface, pg. v
- Introduction, pg. vii
- Table of Contents, pg. xi
- Chapter I: Introduction to local fields, pg. 1
- Chapter II: Fourier analysis on K, the one-dimension case, pg. 20
- Chapter III. Fourier analysis on Kn., pg. 115
- Chapter IV. Regularization and the theory of regular and sub-regular functions, pg. 168
- Chapter V. The Littlewood-Paley function and some applications, pg. 195
- Chapter VI. Multipliers and singular integral operators, pg. 217
- Chapter VII. Conjugate systems of regular functions and an F. and M. Riesz theorem, pg. 241
- Chapter VIII. Almost everywhere convergence of Fourier series, pg. 262
- Bibliography, pg. 286