# Celebratio Mathematica

## Ingrid Daubechies

### Foreword

#### by Sheila NewberyManaging Editor, Celebratio Mathematica

This col­lec­tion is a com­pil­a­tion and tran­scrip­tion of the ex­tant holo­graph let­ters from In­grid Daubech­ies to Yves Mey­er.1 Our aim in of­fer­ing it to the pub­lic is to make its con­tents easy to per­use in their en­tirety: read­ers will find here our type­set ver­sions of the ori­gin­al French texts jux­ta­posed with their Eng­lish trans­la­tions. Yves Mey­er provides an in­tro­duc­tion to ex­plain the math­em­at­ic­al sig­ni­fic­ance of the let­ters and to flesh out the cir­cum­stances of their writ­ing.

The col­lec­tion came to light when I asked Yves (who was then Pro­fess­or Mey­er to me) about a par­tic­u­lar let­ter from In­grid (dated March 12, 1987) which he men­tioned in some re­marks ad­dressed to the ed­it­or of the Amer­ic­an Sci­ent­ist (see “Let­ters to the ed­it­or” in vol. 85, no. 3, 1997, pp. 204–206). There Yves de­scribed the role of Stéphane Mal­lat’s pri­or work and that of his own in what In­grid ac­com­plished in Janu­ary–Feb­ru­ary 1987: the con­struc­tion of smooth or­tho­gon­al wave­lets of com­pact sup­port.

Tak­ing a his­tor­ic­al per­spect­ive, Daubech­ies’s con­struc­tion be­longs […] to Mal­lat’s pro­gram. Al­though this state­ment may be con­strued to mean that Daubech­ies merely treated a spe­cial ex­ample that arose from someone else’s great vis­ion, that was nev­er my opin­ion. I al­ways knew that a large gap ex­is­ted between the form­al­ism that I de­veloped with Mal­lat and what Daubech­ies achieved. When one tries to ap­ply this form­al­ism, many prob­lems oc­cur im­me­di­ately. The first one con­cerns the sta­bil­ity of the con­struc­tion, and the second one is re­lated to the de­sired reg­u­lar­ity. These dif­fi­culties re­quire a subtle and dif­fi­cult ana­lys­is, which was achieved by In­grid and could not be dis­covered by or­din­ary hu­man be­ings.

I kept a beau­ti­ful sev­en-page let­ter that Daubech­ies wrote to me at the time she was work­ing on her con­struc­tion, ex­plain­ing how she dis­covered the bases. Mal­lat’s al­gorithm was not the start­ing point, but it did play a role as an in­gredi­ent.

Yves’ re­marks piqued my curi­os­ity. I wrote to him and asked if he would con­sider let­ting us in­clude In­grid’s let­ter in our Cel­eb­ra­tio volume on her work, as­sum­ing we could get her agree­ment as well. I feared he would say no. Yet he wrote back promptly (and with char­ac­ter­ist­ic gen­er­os­ity, as I would soon learn) to say that he would look for the let­ter in ques­tion, but that he was trav­el­ing and away from his of­fice, and could only do so on his re­turn.

Yves did look, and that’s how this col­lec­tion came to light, for there was not only the lettre sub­lime, as he called it (eight hand­writ­ten pages, in fact), but a folder full of hand­writ­ten cor­res­pond­ence: let­ters and notes from In­grid span­ning the years 1985–2002, the dec­ades of the mo­ment­ous shift from postal mail to email.

I’m grate­ful to Yves and In­grid for their ex­traordin­ary gen­er­os­ity in let­ting us share these let­ters with our read­ers. Yves freely gave of his time to re­view the Eng­lish trans­la­tions and cor­rect our type­set ver­sions (French and Eng­lish), even as he was fully en­gaged in new math­em­at­ic­al re­search and was try­ing to com­plete pub­lish­ing pro­jects of his own. In­grid trus­ted Yves and me both to handle the per­son­al and vis­ion­ary con­tent of these let­ters with care.

I want to thank Silvio Levy, Dir­ect­or of Pub­lish­ing at MSP, who trans­lated the let­ters from the French, and whose rig­or­ous at­ten­tion to de­tail was in­valu­able; Hélène Bar­celo, Deputy Dir­ect­or of the Math­em­at­ic­al Sci­ences Re­search In­sti­tute (MSRI), who helped se­cure crit­ic­al fin­an­cial sup­port for Cel­eb­ra­tio’s Daubech­ies volume as MSRI’s Act­ing Dir­ect­or in 2018–19; Geor­gia Ben­k­art, whose lead­er­ship as the Chair of the Com­mit­tee on Wo­men in Math­em­at­ics at MSRI has helped make pos­sible a series of Cel­eb­ra­tio volumes of wo­men math­em­aticians, in­clud­ing this one; and Eliza­beth Weaver, Seni­or Ed­it­or at MSP, who cata­logued the Daubech­ies let­ters in pre­par­a­tion for this pro­ject.