Remembering Leon Ehrenpreis (1930–2010)

Daniele C. Struppa

Leon Ehrenpreis: A Note on His Mathemat- ical Work Leon Ehrenpreis passed away on August 16, 2010, at the age of eighty, after a life enriched by his passions for mathematics, music, running, and the scriptures. I first met Leon in 1978, when I was a doctoral student under Carlos Berenstein (himself a student of Leon’s), at a time when Carlos had invited him to lecture at the complex analysis seminar at the University of Maryland. I had just begun to attempt my first (of many) readings of his work on the fundamental principle [11] and I was in awe of his presence. And yet, I discovered a gentle human being, with a passion for mathematics and a genuine desire to help the newcomer (me, in that case) to understand its mysteries. Photograph by François Rouvière. What follows in this section is a quick overview Leon explaining his AU-spaces at the 2008 of Ehrenpreis’s work, mostly biased by my own conference held in Stockholm in honor of Jan interest and work on the fundamental principle. Boman’s seventy-fifth birthday. It is of course impossible to pay full tribute to the complexity of Ehrenpreis’s work in such a short note. The subsequent sections are short skill in his willingness to tackle bold generaliza- remembrances, both mathematical and personal, tions, as he extends distribution theory to the of Leon’s life, from some of his friends and case in which Euclidean spaces are replaced by collaborators. locally compact Hausdorff spaces denumerable at Leon Ehrenpreis’s mathematical career started infinity and the derivations ∂/∂x are replaced by with his 1953 dissertation [4], which he wrote while i a countable family of local, closed operators. doing his doctoral work at He then plunged immediately into the topics under the guidance of Chevalley. The dissertation, that would become the main drivers for his funda- whose main results later appeared in [8], already mental principle. In an impressive series of papers shows some of Ehrenpreis’s most characteristic from 1954 to 1960, Ehrenpreis addressed the prob- Daniele C. Struppa is chancellor at Chapman University, lem of division in a variety of contexts. Specifically, Orange, CA. His email address is [email protected]. he investigated in [5] the issue of surjectivity of

674 Notices of the AMS Volume 58, Number 5 May space coef- the constant in with ficients operators differential partial Carlos’s of honor birthday. in sixtieth Fairfax 2004 in the held at conference Sabadini Irene student and Daniele’s Struppa, Daniele Berenstein, student Carlos Carlos’s student his with Leon re n ntespace the in and order a-eciggnrlzto ftewl-nw rep- well-known the of generalization far-reaching also in [11]. appeared fact monograph proof important most complete discovered his in whose references and Russian concurrently therein) the was and [28] Ehrenpreis- see Palamodov; and by result of principle independently (this known fundamental became Palamodov what the of 1960 as in of [9] nouncement class larger a considered he where spaces. [7], later such in continued to he on that solutions analysis an exponential [30], on equations results Schwartz the L. extended special of some and theory. of equations case this the convolution of tackled form Ehrenpreis modern [6] In the develop help [27] to Malgrange and [23] Hörmander of those joined results par- Ehrenpreis’s and coefficients equations, differential constant tial linear of of study solvability the the to distributions of theory the appli- of the cation of years heroic the were These functions. e scnie h space the on consider class us let introductory any in equations. teach differential we ordinary, of which equations, differential solutions linear coefficients for constant theorem resentation ibefntoson functions tiable hs ybl r h polynomials the are symbols whose r o let Now eetal ucin htaesltoso the of solutions are that functions ferentiable Photo by Fabrizio Colombo. ieeta prtr ihcntn coefficients, constant with operators differential h udmna rnil a ese sa as seen be can principle fundamental The an- striking his with culminated this of All nodrt tt t(tlati atclrcase), particular a in least (at it state to order In 2011 E P eoetesaeo nntl dif- infinitely of space the denote R n D E n let and , F ′ ′ fifiieydifferentiable infinitely of fdsrbtoso finite of distributions of E fifiieydifferen- infinitely of P 1 D,...,P , . . . (D), P 1 , r (D) P , be oie fteAMS the of Notices n . odtosta utb aifidb holomor- variety by the satisfied on be functions growth the phic must however, of that nature proof, the the conditions of of understanding the crux is complex The several [25]). edition on third variables monograph the fun- in Hörmander’s the and of to [2] in dedicated principle chapters damental the (see, example, reinterpretations and for analysis of subject decades still the is even it announcement, and original Ehrenpreis’s complicated, after of quite subvarieties is algebraic theorem of sequence V finite a is hr the where hteeyfunction states every which that theorem, representation the is result ooti oooia smrhs ewe the between isomorphism space possible topological is a it obtain that to states roughly principle mental nteuino eti varieties certain of conditions union growth the on suitable satisfying functions ovlto qain) aifigsm specific some satisfying solu- even equations), (or are equations convolution differential that of functions systems holomorphic of of tions for of spaces holds general fact peculiarity in more but variables a several in not functions problems, phe- is division Hartogs’ of nomenon that study [10] his demonstrated developed of Ehrenpreis had course es- he the arguments in compact By the functions. of using sentially holomorphic removability for the Har- classical singularities on the theorem of proof togs’ elegant and beautiful of systems equations conditions. suitable of differential satisfying coefficient solutions constant of linear spaces to holomorphic philo- for functions of hold that sort extend results could a structural one most that was held years that principle those sophical in work preis’s be to the has course). functions, of above interpreted, generalized suitably stated objects are theorem the spaces representation (when these etc. in functions, holomorphic spaces, Beurling distributions, Schwartz’s includes moe o uto h ucin,btas on polynomials also the to but related be multiplicities the functions, incorporate somehow must the that derivatives on conditions suitable just growth not Such imposed place. take to one esrsspotdby supported measures bounded system eylrecaso pcs(htEhrenpreis (what spaces for of holds class it large that called very and equations a differential systems rectangular of to extends actually principle { = ab h otipratcneuneo this of consequence important most the Maybe ab h otsrkn uhrsl a his was result such striking most the Maybe Ehren- of most in themes common the of One ti motn opitotta h fundamental the that out point to important is It z (x) f E nltclyuiomspaces uniform analytically : P P P 1 1 n h ulo pc fholomorphic of space a of dual the and (D)f (z) = Q k X = = = k t r oyoil and polynomials are s 1 = = Z V k f Q in k (x) P r E (z) P P exp r (D)f a erpeetdas represented be can = i,x d x) (iz, 0 V } . = ,actgr that category a ), V h ro fthe of proof The o h duality the for V k .Tefunda- The 0. k . where , k (z), P j . k are s { V k } 675 algebraic properties. The paper is beautiful and fifty years. We began as teacher-student when I elegant and shows Ehrenpreis’s talent for simplic- was still doing my undergraduate degree. I heard ity and anticipates the kind of algebraic treatment about a famous, brilliant young mathematician of systems of differential equations that became who was going to give a course entitled Mathe- so important in years to come. matics and the Talmud and I decided to register The interest of Ehrenpreis in removability of for it. After the semester I asked him whether he singularity phenomena is also apparent in another would be willing to write a letter supporting my beautiful series of papers ([12], [13], [14], [15]) application to graduate school, and he told me dealing with elegant and surprising variations on that I would be better off asking a person more the edge-of-the-wedge theorem. familiar with my mathematical abilities. After that I am obviously unable to touch upon all the I did not have much personal contact with Leon important work of Ehrenpreis, but it would be until after I finished my graduate work and our impossible not to mention his interest in the paths crossed again. I was working on problems Radon transform, which absorbed much of his related to compact Riemann surfaces, and we mathematical efforts in the last ten or so years. would meet now and then. Leon was also very His work on this topic culminated in his second much interested in this subject (Ehrenpreis con- monograph [16], in which Ehrenpreis introduces a jecture, Schottky problem) and always expressed generalized definition of the Radon transform in an interest in what I was doing. We even wrote terms of very general geometric objects defined two papers together ([20], [17]), but this is not the on a manifold (what he calls spreads). Just like measure of the influence he had on me. In fact, his previous monograph [11], this more recent my book with Irwin Kra, Theta Constants, Riemann work brings a new and invigorating perspective Surfaces and the Modular Group, [21] grew out of to his subject matter. And as in the case of a conversation with Leon. A visit to New York [11], this work will provide mathematicians with would never be complete without meeting Leon. In ideas and challenges that should keep them busy fact I arrived in New York on August 15 planning for a long time to come. As Ehrenpreis’s former to meet with Leon in the middle of the week to student Carlos Berenstein states in his review [1] discuss my current work, but he passed away on of [16] “[this is] a book that is worth studying, Monday, August 16. Mathematics has lost one of although mining may be a more appropriate word, its finest practitioners. He will be sorely missed as the reader may find the clues to the keys by the mathematical community as both a scholar he’s searching for to open up subjects that are and a gentleman. seemingly unrelated to this book. Thus, one finds at the end that the title is justified.” Takahiro Kawai Hershel Farkas The Fundamental Principle, Hamburger’s Theorem and Date Line A Remembrance When I was a graduate student (late sixties), the Leon Ehrenpreis was one of the leading mathemat- “fundamental principle of Ehrenpreis” [9] was ical figures of the past century. In many areas of really a guiding principle for many young ana- mathematics, specifically differential equations, lysts, including me. It aimed at, and succeeded Fourier analysis, number theory, and geometric in, analyzing general systems of linear differential analysis, his name is a household word. He was a equations with constant coefficients, in the days person who could discuss mathematical problems when a single equation or a determined system from a very wide variety of subfields and directly was the main target of specialists in differential and indirectly had a great influence and impact on mathematicians and mathematics. He had the abil- equations, and the cohomological machinery was ity of getting to the core of a problem and usually not in the toolboxes of most of them. Thus the the ability to give suggestions for possible solu- publication of his book [11] had been yearned for; tions; sometimes he left these for others, worked actually I asked a bookseller to get it by airmail as jointly with other people toward the solution, or soon as it appeared—it was an exceptional case as just solved the problem himself. Ehrenpreis’s di- one U.S. dollar equaled 360 yen in 1970. Unfortu- versity extended way beyond mathematics. He was nately I could not indulge in reading [11] as I had a pianist, a marathon runner, a Talmudic scholar, hoped; in 1970 and 1971 I was totally absorbed in and above all a fine and gentle soul. My personal the collaboration with Sato and Kashiwara to com- association with Leon Ehrenpreis goes back over plete [29], which was a successor to [9] in the sense that it presented the microlocal structure theorem Hershel Farkas is professor of mathematics at the He- brew University of Jerusalem. His email address is Takahiro Kawai is emeritus professor at the Research [email protected]. Institute for Mathematical Sciences, Kyoto University.

676 Notices of the AMS Volume 58, Number 5 May is address mathematics corrections. and email comments useful of for Struppa His C. D. professor University. 1 A&M is [email protected] Texas Kuchment at Peter prototype endorse the that explained incidents I of When two impression. note the I sincerity. of memory ending man In this now. kind until continued a has impression was The he that impression such of outcomes Leon. of with example chatting chatting. an were and is I [26] if ideas paper the as The feel in the I browse [11], I of of When pages full Ehrenpreis. Leon is of [11] dreams through book browsing the enjoy occasionally; to Still, continued variety. have characteristic its I coefficients, of equa- point constant generic differential a with at linear necessarily of not system tions, general a for Stockholm the at conference. Boman Jan with Leon hepes h huh fhmbiggn has has Leon gone now about being in. one him talking sunk yet that of when not thought believe tense The to past Ehrenpreis. the me use for to hard is It Colleague. and Friend Ehrenpreis—A Leon Kuchment Peter thanks, many Many, Leon! life. daily in of and mathematics effect the to due Kyoto Line. Date in International day the fast the and about the dubious days of was two he date notice that for was to anything reason happened the taken that Another once not I had [19]. he that paper that is our note intellectual cannot I to of I incident led full Weil. which atmosphere and curiosity, warm Hecke theorem the of the forget works expounded related kindly and he and Hamburger’s theorem to relevance its noticed immediately Photograph by Yael Ehrenpreis Meyer. mgaeu oP ef,M oiesy and Mogilevsky, M. Deift, P. to grateful am I hnIfis e eni 93i yt,Igtthe got I Kyoto, in 1973 in Leon met first I When hsIhv ere uhfo enbt in both Leon from much learned have I Thus R hlnmccmlxst eni 90 he 1980, in Leon to complexes -holonomic 2011 . oie fteAMS the of Notices 1 htnmslk enErnri,PtrLx Louis Lax, Peter Ehrenpreis, was Leon afterward like soon names my emigration that during my did shock and proof truly visit Another constructive first curtain revelation! the iron a being the was all, like outside exist trip of world felt first First the it that very dream. and a my USSR, in was former It the conference outside California. tomography Arcata, and in geometry integral ideas. through wonderful by of rewarded got abundance always one an was one once easy hurdles, minor but an these unusual notations, not preferred and always was Leon terms since It part the detail. in learn in task, to in Zelenko had techniques L. thus We with Leon’s PDEs. work periodic on my 1970s played in late Palamodov, role V. crucial and a and Malgrange fun- papers B. the related by called as books well he the as what since principle, on least damental work at His time, 1970s. long fundamental early very of a for existence solutions) (e.g., results standing oeie ikaro o i nahtlseven hotel a in him visits, for would his room I Kansas, during a Wichita, pick run in sometimes me to visited York liked 1970 he when New in also so the inception He its run 2007. since had till he year and that every known runner marathon well avid is It an some music. was or in Talmud, time the mathematics, the or it all activity—be intellectual engaged of was variant brain his that transform. Radon and the on him [16] chapter book privilege with a his Quinto) the to together Todd had with papers (jointly also contributing joint have writing I of email. by and communicating seeing phone and a basis, conferences, regular quite at a other on met each Texas we and visiting Kansas Leon And in years, me twenty holiday. past the a over times like occasion few every feel made met kind time any we any discuss at to mathematics eagerness of abundant his disposition cheerful and unfailing His encounter. first main the of collaboration one our started. how is is This geometry tools. mathematical integral which in tomography, computed I in and working started geometry, just that integral had on at worked me had to Leon unknown time, that, was discovery happy communicated book his I Analysis translating Geometric when while indirectly, him with earlier me proven Sigurdur been met had to my existence also his was (I but there, kind. Arcata Helgason this in of Leon only and experience all Meeting first at not active. old that very who not were still but mortals, flesh Earth, the in mere this existed referred above to least well at corresponded somewhere or semi–gods covers, book to on obviously only which existed luminaries, other and Nirenberg, e enfrtefis iei 99a the at 1989 in time first the for Leon met I out- Leon’s of some with familiar been have I vroewoke enErnri realized Ehrenpreis Leon knew who Everyone the from Leon loved I else, everyone like Just 2]it usa. Another Russian.) into [22] rusand Groups 677 miles away from the campus, with a sufficiently attractive route to run between the two. So, after his lecture, or just a working day, he would give me his things to take back to the hotel, while he would run. Every time I would meet him after the run, he would have some new ideas (and he had so many great ideas!) about the problem we were working on at the time. Once, when he came back and I was waiting for him in the hotel’s lobby, the receptionist at the front desk asked him: “Did you really run all the way from the campus?” Leon’s reply was: “What else could I do? He refused to give me a ride”—and he pointed at me. I think I lost all the receptionist’s respect at that time.

This was not a one-time event; Leon always liked Photograph by Yael Ehrenpreis Meyer. to crack or to hear a good joke. He was smiling Leon at the Stockholm conference in front of most of the time that I saw him. It was a joy to the Royal Palace. discuss with him not only mathematics, but also religion, music, or anything else. What made this even more enjoyable was that in my experience he he described the big picture, and he illustrated never imposed his opinions, beliefs, or personal his points with examples that brought the ideas problems (and he unfortunately had quite a few) down to earth. Leon’s enthusiasm was contagious, on others. It was relaxing to talk to him. He must and it got even skeptical listeners involved and have been a wonderful rebbe. Over the years, he intrigued. I especially appreciated Leon’s respect became our dear family friend. and concern for the person he was talking with; Leon always liked a good story and had some of he wanted the listener to be as engaged and inter- his own to tell. My favorite, which I tell to students ested as he was. His daughter Yael told me that often, was about him teaching a calculus class he was exactly the same way when he helped her many years ago. As any good teacher would do, he tried to lead his students, whenever possible, with math homework. to the discovery of new things. So, he once said: Leon’s mathematics reflected his emphasis on “Let us think, how could we try to define the unifying principles. In Leon’s book [16] The Uni- slope of a curve?” “What is there to think about?” versality of the Radon Transform, he developed was the reply from one smart student, “it says several overarching ideas and used them to un- on page 52 of our textbook that this is the derstand properties of the transforms, such as derivative.” “Well,” replied Leon, “I haven’t read range theorems and inversion methods. One such till page 52 yet.” The result was that the class overarching idea he defined is the nonparamet- complained to the administration that they were ric Radon transform using spreads (a collection given an unqualified teacher. So much for inspiring of foliations of Euclidean space, the leaves of teaching; it can backfire! which depend on a spread parameter). For the I miss Leon, a great mathematician and human hyperplane transform, the spread is defined by being, and hope that he dwells happily in the a specific hyperplane L0 through the origin (or house of the Lord forever. element of the Grassmannian G(n, n − 1), and the ⊥ spread variable is a point s in L0 : the spread is defined for each L0 by s ֏ s + L0. After ten Eric Todd Quinto pages of the book, he is considering other mani- folds and spread variables, and he is relating the geometric construction to differential equations A Remembrance and groups. The book draws connections between Before I met Leon Ehrenpreis I was a little ap- several fields, including complex variables, PDE, prehensive. He had theorems named after him, harmonic analysis, number theory, and distribu- and I knew he was a brilliant mathematician. His tion theory—all of which have benefited from his work on existence of solutions to PDEs helped contributions over the years. I remember one talk revolutionize the field, and I wondered what its he gave at Harvard while he was writing the book. creator would be like. Leon immediately put me He pointed to about three hundred pages of notes at ease with his warm enthusiasm and love for on the lectern and told us with a broad smile and mathematics. When we talked about mathematics hearty laugh that he was going to talk about all Eric Todd Quinto is Robinson Professor of Mathematics that. Worried that we would be there for a while, at Tufts University. He email address is todd.quinto@ we laughed a little nervously. However, if I am tufts.edu. remembering right, he covered a lot of material

678 Notices of the AMS Volume 58, Number 5 but ended more or less on time. Peter Kuchment his family, the sorrows were of such immensity and I wrote an appendix to the book, and this, too, that would otherwise crush anyone. But Leon bore was a pleasure. his with unimaginable courage and responsibility. Leon was a rabbi, and some of our most lively Courage that, we dare say, none of us could have conversations occurred when I asked him simple possibly comprehended, let alone mustered. But questions about Jewish law. Leon would launch despite it all, and no matter what transpired, Leon into an energetic description of what the rabbis retained every bitof the vitality ofour earlier years. thought about a point and how they resolved His mathematical work continued. His, along with seeming inconsistencies. He always made sure Ahava’s, loving care and unstinting dedication to the conversation was a dialogue and that I was his family continued. His kindness and loyalty to engaged. He cared deeply about Jewish law, and us, his friends, continued. It was who he was. he wanted to share it with me. To think now that this vitality is gone from us I will miss Leon’s theorems and mathematical is very hard to accept. We’ll always cherish our taste, as well as our conversations. His ideas are recollections of him—even as an attempt, however important, but so was his way of communicating illusive, to conjure up a whiff of his vitality. Leon those points and sharing his enjoyment of them. I was one of a kind. We know he is irreplaceable. will miss Leon’s enthusiasm for math and life, his And we miss him. smile, and his menschlichkeit. Alan Taylor Shlomo Sternberg Remembrances of Leon Ehrenpreis A Personal Recollection I first met Leon in the summer of 1966 at an I will leave it to others to describe the extremely AMS summer conference, “Entire Functions and distinguished mathematical career of Leon Ehren- Related Parts of Analysis”, in La Jolla, California. preis. In fact, Leon, Victor Guillemin, and I wrote I had just completed my thesis, which was about one joint paper together [18] of which I am still problems in spaces of entire functions that fit his proud, but this reflects only a very tiny fraction theory of analytically uniform spaces, and I had of his great mathematical achievements. I will studied some of his important papers on division concentrate on the personal. problems and their applications to constant coef- I first met Leon Ehrenpreis in the early 1950s at ficient partial differential operators. So it was very Johns Hopkins University, where I was a student exciting to meet and talk with him. It was also and he an instructor. We became close personal where I had my first encounter with his kindness, friends, and then, when I married Aviva, he became as he suggested that I apply to spend a postdoc- a close friend of ours, a friendship that lasted from toral year at the Courant Institute, where he was a the time that we were teenagers until his death. professor. It was during that year, 1968-1969, that Thinking back through the years, I can’t recall a sin- I really got to spend a lot of time talking with him gle time, no matter how trying the circumstances and met his student at that time, Carlos Beren- may have been, whether casual or serious, that stein. Thinking back on those times is bittersweet; his voice, his eyes, his whole demeanor conveyed it is sad now because of Leon’s death and doubly less than deep warmth, profound generosity, an so because Carlos is too ill to share memories of optimism, a hopefulness that was pure Leon. that special year. But it also reminds me of this When we were young, “pure Leon” might in- most interesting and fun year of my professional clude a dash of madcap charm, a directness, a life. Carlos was finishing his doctoral thesis at boyish whimsy, a ruefulness, that belied his dis- Courant and helping Leon with the final editing of tinguished mathematical achievements. His style his book on Fourier analysis [11]. Leon had just was not professorial. He was not into style or moved to in uptown New York image—then or ever. Leon retained and presented City, where he was giving a course on the book. an honesty, a disarming forthrightness, a genuine- Both Carlos and his wife and I and my family ness, a profound generosity and sheer vitality that lived in -owned housing in he carried with him all of his life. He was certainly Greenwich Village, so each Thursday Carlos and I extremely generous to us at crucial junctures in would take the A train uptown to spend the day our lives. with Leon, attending his class and talking about And it was, and is, this vitality that perhaps mathematics. for us best describes Leon. The years passed; life I really saw Leon’s style of doing mathematics transpired with its joys and sorrows. For Leon and in that class. He was always interested in the

Shlomo Sternberg is professor of mathemat- B. Alan Taylor is professor of mathematics at ics at Harvard University. His email address is the University of Michigan. His email address is [email protected]. [email protected].

May 2011 Notices of the AMS 679 680

© C. J. Mozzochi, Princeton, NJ. ai iklti pyis.SvrltmsIheard I was times group Several the (physics). Finkelstein in David Also outstanding mathematicians! an of What group Aumann. Robert Martin and Flatto, Leo Davis, Shapiro, S. Harold Shields, Schwartz, Allen Jacob Rubel, Lee Ehren- Newman, Leon Donald included interacted preis, group who The another. or one mathematicians with year young remarkable a a of was class, there group 1949, that year In the around Michigan. two at for article years colleague his my many in Shields, Allen mentions fact remembering also a [31] York, Shapiro New Harold of under- College that re- remarkable City worth at a class from are Lee graduate came advisor, think Leon my that I from Rubel, learned that I years First, membering. the over Leon Ann visit department. to our Leon and for Arbor visits occasion family family good that a a so always from that suburb, were Detroit was lucky a in Sperka, visit was lived (nee) that I would Ahava Arbor. he wife, Ann when his about in mostly Leon department saw two, our I or was year, year it that and After every think train jogging. I the his and riding in when him especially around always, spent true of I 100% time mathematics the doing some of was to Leon “Edge differential extension equations. partial the coefficient its on constant of and that systems theorem” example, Wedge for much the is work, in view present his is of inquiry of point of style general This this explored. which in functions quasianalytic and series, lacunary boundary problems, general value on of chapters proof contains principle, the fundamental the Fourier presenting contribution, important on most to his view book addition of his in point Indeed, analysis, the analysis. from Fourier analysis of in problem almost at look any the could he in that me interested to seemed It less details. but examples illustrative and in true were theorems that reasons fundamental att asaogtotig ere about learned I things two along pass to want I endsusn ahmtc ihDennis with mathematics discussing Leon ulvnadJh ogna Columbia at Morgan John and Sullivan nvriy 2006. University, oie fteASVolume AMS the of Notices en agag,adHradri h 1950s, the in Hörmander in- and by These answered Malgrange, time. questions, Leon, the fundamental at with the them along cluded about knew operators questions he several differential what of list partial a about with back Schwartz Schwartz written Laurent suggestions. had to thesis-problem write for he asking suggested to had led Cheval- ley that that me problems told He on principle. work fundamental the to come Cheval- Claude had of he student ley, a being how, I of him point space asked some the At on functions. differentiable inverse infinitely right linear continuous had a operators differential partial coefficient with con- stant which talked determining of often problem the I about Leon 1970s, in own my the 1970s, on Throughout influential the so work. was in it because sometime part large Leon with versation youths. stimu- talented very these the between about competition and Shields atmosphere lating and Rubel from stories References he that will and I him. name. miss team, boyhood greatly my baseball “Alan”, me a called enough always form have to to wanted he children tea that drank worked, many always he we remember that when still fact the much I Like knew but things. I little life, that private in his him I about that to say close cannot personally I was friend. great a mathematical and his suggestions with generous was word He unkind him. an heard from never a I and without face, him his on saw smile never I mathematics. about ing case life. the on professional influence indeed entire profound my is a it had Ehrenpreis So Leon years. that twenty me occupied last have the ques- work for and this from 1988, arose by around that solved tions myself was and question Vogt, inverse Meise, right Giorgi in The De the Hörmander 1973.) by of by given characterization being negative a operators surjective with the [3] in Cattabriga analytic and solved func- real was analytic of real tions space on surjectivity the (The on functions. surjectivity operators of such question in- the of Schwartz as from well been as had list we discussing, that the question inverse that right the me cluded linear told of also theory operators. He differential modern partial the coefficient constant underlie now that [1] [3] [2] eodvvdmmr aei facon- a of is have I memory vivid second A enwsakn a,awy neetdi talk- in interested always man, kind a was Leon .A Berenstein A. C. .Cattabriga L. oln,AsedmOfr-e ok 1979. York, Amsterdam-Oxford-New Holland, .E Bj E. J. h ao Transform Radon the islzoiaaiih ieuzoiaderivate a equazioni di analitiche soluzioni di 090 (2007d:58047). 2019604 ¨o rk , ig fDffrnilOperators Differential of Rings and eiwof Review , .D Giorgi De E. yL Ehrenpreis, L. by , h nvraiyof Universality The 58 Sull’esistenza , Number , ah Rev. Math. North- , 5 parziali a coefficienti costanti in un qualunque nu- ), pp. 125–132, Academic Press, New mero di variabili, Boll. Un. Mat. Ital. (4) 6 (1972), York, 1974. 301–311. [21] H. M. Farkas and I. Kra, Theta Constants, [4] L. Ehrenpreis, Theory of Distributions in Locally Riemann Surfaces and the Modular Group. An Compact Spaces, Part I, Ph.D. thesis, Columbia Introduction with Applications to Uniformization University, 1953, 122 pp, Proquest LLC. Theorems, Partition Identities and Combinatorial [5] L. Ehrenpreis, Solution of some problems of di- Number Theory, Graduate Studies in Mathematics, vision, I, Division by a polynomial of derivation, 37, American Mathematical Society, Providence, Amer. J. Math. 76 (1954), 883–903. RI, 2001. [6] , Solution of some problems of division, II, [22] S. Helgason, Groups and Geometric Analysis, Division by a punctual distribution, Amer. J. Math. Integral Geometry, Invariant Differential Opera- 77 (1955), 286–292. tors, and Spherical Functions, Pure and Applied [7] , Solution of some problems of division, III, Mathematics, 113, Academic Press, Orlando, FL, ′ Division in the spaces D , H , QA, O, Amer. J. Math. 1984. 78 (1956), 685–715. [23] L. Hörmander, On the theory of general partial [8] , Theory of distributions for locally compact differential operators, Acta Math. 94 (1955), 161– spaces, Mem. Amer. Math. Soc. 21 (1956). 248. [9] , The fundamental principle for linear con- [24] , On the division of distributions by stant coefficient partial differential equations, polynomials, Ark. Mat. 3 (1958), 555–568. Proc. Intern. Symp. Linear Spaces, Jerusalem, 1960, [25] , An Introduction to Complex Analysis in Sev- pp. 161–174. eral Variables, 3rd edition, North-Holland Math- [10] , A new proof and an extension of Har- ematical Library, 7, North-Holland, Amsterdam, togs’ theorem, Bull. Amer. Math. Soc. 67 (1961), 1990. 507–509. [26] T. Kawai, The Fabry-Ehrenpreis gap theorem and [11] , Fourier Analysis in Several Complex Vari- systems of linear differential equations of infinite ables, John Wiley & Sons. Inc., New York, 1970, order, Amer. J. Math. 109 (1987), 57–64. Reprinted by Dover, 2006. [27] B. Malgrange. Existence et approximation des so- [12] , Edge of the wedge theorem for partial lutions des équations aux dérivées partielles et differential equations: Harmonic analysis in Eu- des équations de convolution, Ann. Inst. Fourier clidean spaces (Proc. Sympos. Pure Math., Williams Grenoble 6 (1955-56), 271–355. Coll., Williamstown, Mass., 1978), Part 2, pp. 203– [28] V. P. Palamodov, Linear Differential Operators 212, Proc. Sympos. Pure Math. 35, Amer. Math. with Constant Coefficients, Grundl. d. Math. Wiss., Soc., Providence, RI, 1979. Bd. 168, Springer-Verlag, Berlin-Heidelberg, New [13] , Reflection, removable singularities, and York, 1970. approximation for partial differential equations, I, [29] M. Sato, T. Kawai, and M. Kashiwara, Microfunc- Ann. of Math. (2) 112 (1980), no. 1, 1–20. tions and pseudo-differential equations, Lect. Notes [14] , The edge-of-the-wedge theorem for par- in Math., No. 287, Springer, 1973, pp. 265–529. tial differential equations. Recent developments in [30] L. Schwartz, Théorie générale des fonctions several complex variables (Proc. Conf. Princeton moyenne-periodiques, Ann. of Math. 48 (1947), Univ., Princeton, NJ, 1979), pp. 155–169, Ann. of 857–929. Math. Stud. 100, Princeton Univ. Press, Princeton, [31] H. S. Shapiro, Allen Lowell Shields—Some Rem- NJ, 1981. iniscences, Mathematical Intelligencer 12 (1990), [15] , Reflection, removable singularities, and No. 2, pp. 8–10. approximation for partial differential equations, II, Trans. Amer. Math. Soc. 302 (1987), no. 1, 1–45. [16] , The Universality of the Radon Transform, With an appendix by Peter Kuchment and Eric Todd Quinto, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2003. [17] L. Ehrenpreis and H. M. Farkas, Some refinements of the Poincare´ period relation, discontinuous groups and Riemann surfaces (Proc. Conf. Univ. Maryland, College Park, MD, 1973), pp. 105–120, Ann. of Math. Studies, No. 79, Princeton Univ. Press, Princeton, NJ, 1974. [18] L. Ehrenpreis, V. W. Guillemin, and S. Stern- berg, On Spencer’s estimate for δ-Poincaré, Ann. of Math. (2) 82 (1965), 128–138. [19] L. Ehrenpreis and T. Kawai, Poisson’s summation formula and Hamburger’s theorem, Publ. RIMS, Kyoto Univ. 18 (1982), 833–846. [20] L. Ehrenpreis, H. Martens, H. M. Farkas, and H. E. Rauch, On the Poincare´ relation, Contribu- tions to analysis (collection of papers dedicated to

May 2011 Notices of the AMS 681