On Complex Analytic tools and the Holomorphic Rotation methods Ronald R. Coifman Jacques Peyri ere and Guido Weiss

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On Complex Analytic tools, and the

Holomorphic Rotation methods

Ronald R. Coifman, Jacques Peyri`ere, and Guido Weiss

1 introduction

This paper in honor of Guido Weiss was written posthumously, jointly with

him, as we had, all of his initial notes and ideas related to the program

described below.

Our task, here, is to recount ideas, explorations, and visions that Guido his

collaborators and students, developed over the last 60 years. To point out the

connection of ideas between the original views of the interplay between com-

plex and real analysis as envisioned by Zygmund and his students Calder´on,

Guido Weiss, Eli Stein,and many others, 70 years ago, and the current ap-

proaches introducing nonlinear multi layered analysis for the organization

and processing of complicated oscillatory functions.

It was Zygmund’s view that harmonic analysis provides the infrastructure

linking most areas of analysis, from complex analysis to partial diﬀerential

equations, to probability, number theory, and geometry.

In particular he pushed forward the idea that the remarkable tools of

complex analysis, which include; contour integration, conformal mappings,

factorization, tools which were used to provide miraculous proofs in real anal-

ysis, should be deciphered and converted to real variable tools. Together with

Calder´on, they bucked the trend for abstraction, prevalent at the time, and

formed a school pushing forward this interplay between real and complex

analysis. A principal bridge was provided by real variable methods, multi-

scale analysis, Littlewood Paley theory, and related Calderon representation

Ronald R. Coifman

Department of Mathematics, Program in Applied Mathematics, Yale University, New

Haven, CT 06510, USA, e-mail: coifman-ronald@yale.edu

Jacques Peyri`ere

Institut de Math´ematiques d’Orsay, CNRS, Universit´e Paris-Saclay, 91405 Orsay, France,

e-mail: jacques.peyriere@universite-paris-saclay.fr

arXiv:2210.01949v1 [math.CA] 4 Oct 2022

2 Ronald R. Coifman, Jacques Peyri`ere, and Guido Weiss

formulas. Our aim, here, is to elaborate on the ”magic” of complex analy-

sis and indicate potential applications in Higher dimensions. An old idea of

Calder´on and Zygmund, the so called ”rotation method”, enabled the reduc-

tions of the study of Lpestimates for multi dimensional singular integrals

to a superposition,over all directions,of Hilbert transforms. Thereby allowing

the use of one complex variable methods. A related idea was the invention of

systems of Harmonic functions satisfying generalised Cauchy Riemann equa-

tions, such as the Riesz systems, exploiting their special properties. [6]

Our goal is to extend these ideas to enable remarkable nonlinear complex

analytic tools for the adapted analysis of functions in one variable, to apply

in higher dimensions.

Guido has been pushing the idea that factorization theorems like Blaschke

products are a key to a variety of nonlinear analytic methods [3]. Our goal

here is to demonstrate this point, deriving amazing approximation theorems,

in one variable, and opening doors to higher dimensional applications. Appli-

cation in which each harmonic function is the average of special holomorphic

functions in planes and constant in orthogonal directions.

We start by describing recent developments in nonlinear complex analy-

sis, exploiting the tools of factorization and composition. In particular we

will sketch methods extending conventional Fourier analysis, exploiting both

phase and amplitudes of holomorphic functions. The ”miracles of nonlinear

complex analysis”, such as factorization and composition of functions lead to

new versions of holomorphic wavelets, and relate them to multiscale dynam-

ical systems.

Our story interlaces the role of the phase of signals with their ana-

lytic/geometric properties. The Blaschke factors are a key ingredient, in

building analytic tools, starting with the Malmquist-Takenaka orthonormal

bases of the Hardy space H2(T), continuing with ”best” adapted bases ob-

tained through phase unwinding, and describing relations to composition of

Blaschke products and their dynamics (on the disc and upper half plane).

Speciﬁcally we construct multiscale orthonormal holomorphic wavelet bases,

generalized scaled holomorphic orthogonal bases, to dynamical systems, ob-

tained by composing Blaschke products.

We also, remark, that the phase of a Blaschke product is a one layer

neural net with (arctan as an activation sigmoid) and that the composition

is a ”Deep Neural Net” whose ”depth” is the number of compositions. Our

results provide a wealth of related libraries of orthogonal bases.

We sketch these ideas in various ”vignette” subsections and refer for more

details on analytic methods [2], related to the Blaschke based nonlinear phase

unwinding decompositions [4, 5, 11]. We also consider orthogonal decomposi-

tions of invariant subspaces of Hardy spaces. In particular we constructed a

multiscale decomposition, described below, of the Hardy space of the upper

half-plane.

Such a decomposition can be carried in the unit disk by conformal map-

ping. A somewhat diﬀerent multiscale decomposition of the space H2(T)

On Complex Analytic tools, and the Holomorphic Rotation methods 3

has been constructed by using Malmquist-Takenaka bases associated with

Blaschke products whose zeroes are (1 −2−n)e2iπj/2nwhere n≥1 and

0≤j < 2n[8]. Here we provide a variety of multiscale decompositions by

considering iterations of Blaschke products.

In the next chapter we will show how with help of an extended Radon

transform we can introduce a method of rotations to enable us to lift the one

dimensional tools to higher dimensions. In particular the various orthogonal

bases of holomorphic functions in one dimension, give rise to orthogonal bases

of Harmonic functions in the higher dimensional upper half space.

2 Preliminaries and notation

For p≥1, Hp(T) stands for the space of analytic functions fon the unit

disk Dsuch that

sup

0<r<1Z2π

0|f(reiθ)|pdθ

2π<+∞.

Such functions have boundary values almost everywhere, and the Hardy space

Hp(T) can be identiﬁed with the set of Lpfunctions on the torus T=∂D

whose Fourier coeﬃcients of negative order vanish. We will alternate between

analysis on the disk, and the parallel theory for analytic functions on the

upper half plane H={x+ iy:y > 0}. The space of analytic functions fon

Hsuch that

sup

y>0kf(·+ iy)kLp(R)<+∞

is denoted by Hp(R). These functions have boundary values in Lp(R) when

p≥1. The space Hp(R) is identiﬁed to the space of Lpfunctions whose

Fourier transform vanishes on the negative half line (−∞,0).

3 Analysis on The upper half plane

We present some known results [2], without proof. In this section one simply

writes H2instead of H2(R).

Malmquist-Takenaka bases

Let (aj)1≤jbe a sequence (ﬁnite or not)) of complex numbers with positive

imaginary parts and such that

4 Ronald R. Coifman, Jacques Peyri`ere, and Guido Weiss

j≥0

=aj

1 + |aj|2<+∞.(1)

The corresponding Blaschke product is

B(x) = Y

j≥01 + a2

j

1 + a2

x−aj

where, 0/0, which appears if aj= i, should be understood as 1. The factors

1 + a2

j

1 + a2

insure the convergence of this product when there are inﬁnitely many

zeroes. But, in some situations, it is more convenient to use other convergence

factors as we shall see below.

Whether the series (1) is convergent or not, one deﬁnes (for n≥0) the

functions

φn(x) = 1

√π

Y

0≤j<n

x−aj

x−aj

1

x−an

Then these functions form an orthonormal system in H2. If the series (1)

diverges, it is a Malmquist-Takenaka orthonormal basis of H2, otherwise it is

a basis of the orthogonal complement of B H2in H2.

We remark that roughly a hundred years ago these bases were con-

structed [13, 9] through a Gram Schmidt orthogonalization of the list of

rational functions with poles in the lower half plane .

Observe that for a rational function with a pole of order M at athe cor-

responding M basis functions have the form

φn(x)=einθ(x)1

x−an

(n= 1..M).

These are localized ”Fourier like” basis functions around the real part of

ascaled by the imaginary part.

Example of a multiscale Wavelet decomposition

The inﬁnite Blaschke products

Gn(x) = Y

j≤n

j−i

j+ i

x−j−i

x−j+ i and G(x) = Y

j∈Z

j−i

j+ i

x−j−i

x−j+ i

can be expressed in terms of known functions:

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OnComplexAnalytictools,andtheHolomorphicRotationmethodsRonaldR.Coifman,JacquesPeyriere,andGuidoWeiss1introductionThispaperinhonorofGuidoWeisswaswrittenposthumously,jointlywithhim,aswehad,allofhisinitialnotesandideasrelatedtotheprogramdescribedbelow.Ourtask,here,istorecountideas,explorations,andvisi...

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On Complex Analytic tools and the Holomorphic Rotation methods Ronald R. Coifman Jacques Peyri ere and Guido Weiss

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