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# Guido L. Weiss (1928–2021)

Communicated by *Notices* Associate Editor Daniela De Silva

Guido Weiss was born in Trieste, Italy, on December 29, 1928, and died in St. Louis, Missouri, on December 24, 2021. His family emigrated to the USA in 1939 and settled initially in Topeka, Kansas, moving two years later to Chicago. He obtained two degrees in mathematics from the University of Chicago, a bachelor’s degree and a PhD under the guidance of Antoni Zygmund.

In 1961, Guido moved from DePaul University to Washington University in St. Louis where he later became the Elinor Anheuser Professor of Mathematics. He married Barbara Gibgot, then a doctoral student in molecular biology, and they had two sons, Paul and Michael.

Guido played a huge role in the development of harmonic analysis at Washington University. In particular, he entertained a host of students and collaborators from around the world. He met R. Raphael (Raphy) Coifman in Geneva in 1964 and started a long-term friendship and collaboration. Coifman moved to Washington University, where he and Guido collaborated on many results, including the beautiful theory of atomic decompositions of Hardy spaces. He met Yves Meyer at Oberwolfach in 1965, and beginning in the 1970s Guido, Meyer, and Coifman developed a broader understanding of Calderón–Zygmund operators. Simultaneously, Guido and Eli Stein were writing the book (SW71), which served as an invaluable reference for generations of harmonic analysis students. He and Stein also developed a program studying spaces of several variables.

Interpolation of operators was the subject of Guido’s PhD dissertation. He returned to this subject in the 1980s with the theory of interpolation of families of Banach spaces (CCR 82). After reading M. Frazier and B. Jawerth’s work on phi-transforms and wavelets, Guido became interested in reproducing formulae. From the 1990s until his retirement in 2018, his papers with postdocs and other collaborators were strongly influenced by these developments.

The above paragraphs give only a quick sketch of some of the highlights of Guido’s mathematical career. The reader is referred to the article by Susan Kelly and Rodolfo Torres (KT21) for additional details. The tributes that follow offer a sample of the teaching style and mathematical achievements of Guido from the viewpoint of some of his former students and collaborators. This is done with the intention of keeping the memory of Guido Weiss alive.

Guido Weiss was an extraordinary human being and mathematician; he has been my mentor and friend for over 50 years.

We met in 1964 in Geneva, Switzerland. Guido and Barbara came to spend a sabbatical at the university of Geneva. I was finishing my thesis under J. Karamata, and as his assistant, I was expected to welcome them and facilitate their arrival. After various misadventures they settled in, and in his usual way Guido initiated both a friendship and mentorship with me. I was going to teach him skiing, and he would teach me Harmonic Analysis.

During that year, he was writing his book with Stein on Harmonic Analysis in Euclidean spaces, which I helped to proofread. This exercise allowed Guido to communicate his long-term program in analysis to me, expanding on the vision of his own mentor Antoni Zygmund. This interaction opened a whole new world of mathematics to me, as well as mathematical friendships. We traveled to Paris to meet with Antoni Zygmund and Jean Pierre Kahane, and we went skiing in Zermatt. Guido and Barbara welcomed Lucienne and me to their home. The informality and friendliness of these meetings was quite astounding to us “kids” (coming from a senior professor). Guido would lounge on the floor with a glass of Dole wine in his hand discussing everything from Italian Pizza that he would bake to music, and to mathematics (when we were alone).

While in Geneva, Guido convinced me to go to Chicago for a special year in Harmonic Analysis, while he and Barbara returned to Washington University. We had already started on several papers, on basic analysis ranging from group representations to Blaschke products in complex analysis. These were part of his program to extend real and complex analysis beyond their classical context. Three years later Guido organized a special year on analysis on symmetric spaces with a vision of building interactions between different groups of analysts and geometers.

Lucienne and I move to Washington University in St Louis continuing our friendship with Guido and Barbara and our collaboration; it was our best decision. For the next fifteen years we continued mathematics and “skiing” together.

In 1970, our two families spent the year in Orsay. Guido and I were teaching a Harmonic Analysis course on spaces of homogeneous type. Our goal was to develop a general setting on which many classical tools and methods could be extended and applied, thereby providing a bridge between geometry and multiscale analysis, as it applies to the structural understanding of operators which are not convolutions.

As usual Guido befriended many in our audience: Jacques Peyriere and Yves Meyer, as well as Aline Bonami and Jean Louis Clerc, who participated in writing up our lecture notes. I should note that much current activity in computer science and data geometry on networks is directly related to the vision of blending algorithms (real variable methods) with the geometries of nature as developed in our lecture notes.

At the time, we were challenged by proving estimates; this was always understood as a test of understanding, and our goal as formulated by Zygmund and his school was to develop “methods” of analysis. Guido always stressed this point; it has been the challenge all along, in particular, understanding the role of complex methods to prove inequalities, and the role of geometry to understand complex methods and the power of generalized analytic functions to control singular integrals.

I should add that the combination of an extended view of analysis in mathematics, together with Guido’s extraordinary ability to assemble communities of mathematical friends in France, Italy, Spain, China, and more, and collaborating and exploring the world around his vision, led to a remarkable broadening of the Calderon–Zygmund school, and our mathematical horizons.

Guido liked to quote A. Zygmund, who said that you assess the mathematical contributions of a person by “integrating his positive part.” This generous philosophy helped generate many friendly, collaborative teams.

Ronald Coifman is the Sterling Professor of Mathematics and a professor of computer science at Yale University. His email address is ronald.coifman@yale.edu.

Two people have profoundly influenced my career in mathematics: Miguel de Guzmán (1936–2004) while I was an undergraduate student and Guido Weiss when I was a graduate student and later as a collaborator and friend.

I arrived at Saint Louis in August 1977, with a backpack and a TWAFootnote^{1} bag of toiletries, because my two suitcases were lost on the trip from Spain. After Guido learned of my situation, he wrote a letter to TWA officials. Two weeks later I had a check for $400 to compensate for the loss. This little story sheds light on the kind of care Guido took of his students.

While I was a second-year graduate student, Guido suggested I prove an interpolation theorem for operators acting on spaces using the atomic decomposition developed by Raphy Coifman. I am sure that as an expert on complex interpolation and spaces, Guido could have proved the result in a couple of hours. It took me several months, during which he patiently guided me towards the correct proof, which resulted in my first math paper.

Guido always tried to share his math projects with others. After my four-year term as Vice Rector of the Universidad Autónoma de Madrid, I sought his advice on how I should spend a 1994–95 sabbatical year retooling in mathematical research. He soon sorted out the finances and offered me an invitation to visit Washington University and to collaborate with him in “organizing” the mathematical theory of wavelets. It gave me the opportunity to learn this theory and the trials and tribulations of writing a book with the mind always set on finding the best and simplest ways to explain mathematics. Our collaboration produced not only HW96 but also many papers over the following years together with several of his collaborators, of which I would like to mention D. Labate, H. Šikić, and E. Wilson.

I am thankful to Guido for all the mathematics he taught me, his lessons of life, the tennis matches we played, as partners and as opponents, and for his friendship.

The contributions of Guido L. Weiss to mathematical analysis, and harmonic analysis in particular, have spanned multiple decades and multiple research directions—from Hardy spaces to wavelets—often tying together different areas of analysis. Guido’s work has influenced the work of many mathematicians, and often it was the fruit of his collaborations with researchers from all over the world.

I was very fortunate to be welcomed by Guido at WashU, and become one of his students—at the time some conjectured I would be the last one, but luckily that turned out to be far from being the case—not only because of his scholarship in mathematics and the endless amount of advice I would receive from him, but also because of the many lessons I received, in mathematics and in life. Guido made me feel welcome from the very beginning. He was always generous with his time, with meetings that never felt hurried and in which working on a problem on the blackboard felt natural. He was always available to answer questions and provide interesting research directions together with references to beautiful papers and books to enrich my very limited knowledge of mathematics.

He fostered interactions with a wide variety of colleagues, short- and long-term visitors to the department, and among students. This led to invaluable opportunities to learn mathematics in other areas and to forge new collaborations. These interactions often moved from the department to his home, where he and his wife Barbara offered the kindest and warmest hospitality, to guests from all over the world, speaking many different language— a surprisingly large subset of which Guido could speak proficiently. He consistently refused to speak anything but my mother language with me, which was truly helpful to me especially at my arrival in the United States.

Guido’s lectures, always at the chalkboard, had utmost clarity and perfect pacing, only matched by that of his handwriting. His papers and books were superbly written, with a terse prose and a natural organization of the materials. They have influenced generations of mathematicians, researchers, and students alike, and will continue to do so. In fact, I remember that I had to quickly acquaint myself with one of Guido’s books: at our first meeting in his office, after about a week at WashU, and after explaining the story behind a small fraction of the memorabilia there, Guido asked me if I knew measure theory and some functional analysis, to gauge the possibility of waiving the first-year analysis course. I answered in the affirmative, quite possibly with an excess of self-assurance. He suggested that I study his book with Eli Stein, *Introduction to Fourier Analysis on Euclidean Spaces*, reconvene in a week to discuss it, and then see if I could indeed waive the Analysis course. As soon as I started reading, I realized I knew close to none of the material in the book, so I stocked up on food for the week, and started studying. While I was far from having understood the book in that one week, Guido generously considered what I had learned enough for a waiver,Footnote^{2} and we went on discussing the book in greater depth in our subsequent meetings. This book started opening my mind to what the next level of Analysis looked like, and I loved it. Under Guido’s direction, I then learned about spaces of homogeneous type, Calderón–Zygmund theory, representation theory, and wavelets; research in these areas inspired me and propelled me to a career as a researcher.

^{2}

I only learned much later that the first-year graduate course in Analysis was at a much more basic level than the material in the book; there were multiple lessons that I learned in that first week.

I always remember Guido as setting an example by showing the importance of hard work and humility in tackling difficult mathematical problems, and of kindness and openness in welcoming others in mathematics and life. He will be missed by many in the mathematical community, who were influenced by him personally or through his work.

Mauro Maggioni is the Bloomberg Professor of Mathematics, and of Applied Mathematics and Statistics, at Johns Hopkins University. His email address is mauromaggionijhu@icloud.com.

## Guido Weiss at Oberwolfach 1965

I met Guido Weiss in August 1965, at an Oberwolfach conference organized by Paul L. Butzer. I was still a graduate student. Guido was ten years older than me and already a famous mathematician. There were only nineteen participants at this meeting. The old castle still existed, and we were accommodated there. It was deliciously obsolete. I became fascinated by Guido, by his mathematics and by his extraordinary personality. Discussions with Guido were great. He was at the same time a deep mathematician and an accomplished humanist. For Guido, mathematics was a part of human culture. He thought that mathematics should be shared by everyone and should contribute to happiness. Doing mathematics should be as pleasant as playing tennis. This idea was completely new to me. To me doing mathematics was a difficult experience. It was a frustrating confrontation with a few great mathematicians. Guido changed my mind. He was my mentor and often corrected my naive beliefs in politics. He had deep and original views on many issues.

Barbara Weiss and my wife Anne were allowed to participate in the conference without being mathematicians. They did not listen to the mathematical talks. Instead they violated the established rules by arriving ten minutes before lunches and dinners and changing the seating arrangement. As a result, the four of us ate together almost every day. This was forbidden by the organizers since we were supposed to interact every day with new participants. At our table, we spoke French. Guido was fluent in many languages. He was fond of playing with words, which he did in a mixture of English, French, and Italian. His jokes were absurd and hilarious.

I spent 1970 with Guido at Orsay. Then in 1974, he invited me to work with him at Washington University. But while I was there, he was busy with another program. Then Raphy Coifman convinced me to attack Calderón’s conjectures. That is another story.

Yves Meyer is a member of the Académie des Sciences, Paris, France. His email address is yves.meyer305@orange.fr.

Many aspects of Guido Weiss’s personality are fixed in my mind, which I often recall with nostalgia, and whose impact I recognize at various moments of my professional life.

Among these aspects, I just want to mention his rigor in research and teaching, his sense of responsibility toward persons around him and institutions, and, going more closely to scientific issues, his approach to problems exclusively following his own mathematical taste. Scientifically, I owe him for having introduced me to Euclidean harmonic analysis.

I was fortunate to meet him in the early days of atomic theory of Hardy spaces. Classical Hardy spaces with (in the sense of holomorphic functions on the unit disc) had been the subject of my bachelor’s thesis. Attending the intensive course Guido gave in Perugia in 1976 about his AMS Bulletin paper with Raphy Coifman, I was shocked to learn that a revolution on this topic had taken place and to understand the tremendous generalizing power coming from Guido’s contribution.

This event, followed by a semester at Washington University that Guido had arranged for me, determined the direction that my research has taken ever since. The Stein–Weiss book became a basic reference for me and the source of new ideas. The chapter on spherical harmonics combined nicely with what I was learning at the time about analysis on Lie groups and homogeneous spaces. This included the idea of transplanting Calderón–Zygmund theory from to a Lie group, in particular which was the topic of the Springer Lecture Notes volume by Raphy and Guido. Another important source of inspiration for me was the CBMS Lecture Notes on the transference principle by Raphy and Guido. ,

For various years since then, Guido and I have remained in close contact, though we have not worked together on a specific research project. Following his own taste, Guido became more and more interest in other topics, first in the general theory of complex interpolation, then in wavelet theory. As our interests were gradually diverging, the occasions for meeting unfortunately became less and less frequent over time. When I heard the sad news at the end of last year, my first reaction was regret for not having been closer to him over the last several years.

Another, more important, reason for being grateful to Guido is the great effort and dedication he put into supporting harmonic analysis in Italy. He had a large number of Italian graduate students and promoted, or even just facilitated, many joint initiatives. Altogether, he contributed greatly to the formation of a coherent group of mathematicians with common scientific interests.

Fulvio Ricci is a professor emeritus at Escuola Normale Superiore, Pisa, Italy. His email address is fulvio.ricci@sns.it.

Guido Weiss was one of the most exceptional people I ever met. How does one describe such a unique and multifaceted person? Perhaps one could do so through some personal recollections.

It was in February of 1996 that I entered Guido’s office for the first time. It was located centrally on the second floor of the Cupples I Hall at Washington University in St Louis. His office was packed with memorabilia and in front of the blackboard was a wide olive-green sofa. A cute small Shih Tzu, with the impressive name of Thor, was comfortably settled on the sofa. Guido explained that any mathematics discussed was to be approved by Thor and then pointed toward the upper right-hand corner of the blackboard where the word *stupid* was written in several languages. He emphasized that he had already written the word in Croatian, so that I probably could not add much to it. My response was that the word written in Polish indeed meant stupid, but the Croatian one would be better translated as a fool, so hopefully I could be of some help. And it continued like that for the next twenty-five years; mainly mathematics, interrupted by word puns, thumb wrestling, discussions of current events, history, arts, and sports. Guido’s language skills were remarkable. I am not sure how many languages he spoke, but I witnessed conversations of his in most major ones.

Throughout his life, Guido encountered a few historical figures and many people who were in positions of power and influence, but it did not prevent him from treating everybody with respect and open arms. After getting to know him better on a personal level and becoming aware that he had lived through some difficult times in his long life, I asked him how he maintained his optimism. He explained in somewhat mathematical terms that he always tried to focus on the *f-plus* of the person’s character and ignore the *f-minus* for as long as possible.

My wife and I became friends with Barbara and Guido, and we saw both provide help to many people on numerous occasions. Just to give you an illustration of the level of their generosity, when we arrived for one of our longer stays in St Louis, and our rented apartment was not yet ready, Barbara and Guido took us along with our two-year old daughter into their home for three weeks.

As a mathematician, I like to think of Guido as a theory-builder and a storyteller. Think of mathematics as an infinite onion, where you peel off layer after layer. When working with Guido, we did not rush, our pace was moderate and steady; every day we would try to bring some insight, perhaps very small, but an improvement nevertheless. When we managed to peel off a layer, then we stopped to rethink it. Furthermore, we would remember it and very often revisit it in the future. Guido liked to build a solid house, but the level of understanding was never final; a new point of view was not to be neglected. At one point early in our collaboration, Guido, Manos Papadakis, and I worked on a statement for a few months. Eventually, Manos and I brought a seventeen-page proof to Guido. We all checked it and it was correct. But Guido sent us back to work on it further until we had a reasonable proof; the instruction being that “a reasonable proof” would be clear to us once we reached it. And indeed, it was so; the end result was an elegant one-page proof where every step had a clear and simple meaning.

For Guido, exposition in mathematics was not just a matter of style or format. It was the essence of the content and understanding, a guide through a new theory. As a consequence, his writing and his lectures were highly appreciated everywhere in the world and on every level, from undergraduate lectures to research seminars.

An opportunity to work with Guido also meant joining Guido’s academic family, a large and very diverse group of excellent people from every corner of the world. Guido was a natural leader, and the St Louis School of Analysis left its mark in many countries in the world, my own included. Guido visited Croatia on several occasions. One of the visits some twenty years ago was particularly emotional and reminded us again of the horror of the Holocaust. Guido’s mother’s family was from Pakrac, a small town some fifty miles east of the Croatian capital of Zagreb. Just before WWII and their move to the US, Guido and his family visited their relatives in what was then Yugoslavia; as it happens, never to see most of them again. Almost sixty-five years later we brought Guido to Pakrac, where we found his grandfather’s house and managed to trace one of his cousins near Zagreb and reunite them after all these decades. Guido had a long and complex life, and he will remain a symbol of the human spirit and an inspiration for all of us.

Hrvoje Šikić is a professor of mathematics at the University of Zagreb. His email address is hsikic@math.hr.

When I arrived at Washington University, I was not planning to be a student of Guido. But the frequent talks he used to have with us, the Spaniards who came around that time (José Luis Fernández, Juan José Manfredi) and in previous years (Eugenio Hernández, José Dorronsoro, Julián Aguirre, Patricio Cifuentes, just to mention a three-year period), convinced me that indeed it was with him that I wanted to do my doctoral thesis. Thus, after finishing my qualification period, he agreed to be my supervisor jointly with another unforgettable and brilliant mathematician, Mitch Taibleson. Working with them has been the best thing that has happened to me in my entire academic life.

From Guido and his book, written with Eli Stein a few years earlier (SW71), I learned the essence of harmonic analysis. There was something about him that has always captivated me. I am talking about the enthusiasm he showed for mathematics, the cleverness of his arguments, and the care with which he used to write his manuscripts. He would not stop until there was not a single doubt about the validity of the arguments, until they were understood by the most profane of mortals. This was true for the courses he gave, as well as for his research work. I remember the care with which he showed us his own work or the work of others on any subject, whether it was interpolation, Hardy spaces, maximal operators, factorization of weights, atomic decompositions, and the many other fields with which he dealt. I keep some of his handwritten notes which, despite the time that has passed, I still use.

It is also worth mentioning the amount of time he spent with his colleagues and students talking about mathematics. It was incredible to see how a person who was immensely busy with issues not always related to pure academics, could squeeze his time in order to have informal work sessions on almost a daily basis. Every guest who came to WashU to work with him or just for a friendly visit (and Guido had lots of friends!) was immediately invited to one of these sessions in his office in Cupples I Hall. One of his students would begin by describing a problem he or she was working on and then everyone present would give their opinion, relating it to similar problems and suggesting techniques to solve it. Guido, as a connoisseur of the ins and outs of the proposed problem, would take the lead. But he never disdained a suggestion, however banal. A phrase he often repeated after an unexpected suggestion was: *well, what you just said it’s either true or false. We’ll have to see it.*

From a personal point of view, Guido was more than a teacher, a friend, or a colleague. He was the mirror in which I wanted to see myself reflected and from which I learned so many things, professionally and in everyday life. Every time I sit down next to a student to review their master’s or doctoral thesis, his image comes to mind. In particular, I think of the immense patience he always had, sitting next to me, while explaining the reason for this or that correction to my work. It was really an unforgettable, and even priceless experience.

As a final thought, I must say that Guido was instrumental in helping mathematical analysis to flourish in Spain, making it easy for almost a generation of Spaniards to go to Washington University, either as doctoral or postdoctoral students and generating collaborations with Spanish institutions. In recognition of his scientific career, the Autonomous University of Madrid organized a conference in his honor in 1993. Undoubtedly, some of the best harmonic analysts of the time came to the congress, many of them personal friends of Guido. I think it was a memorable moment that made him immensely happy.

Fernando Soria is a professor of mathematics at the Universidad Autónoma de Madrid, Spain. His email address is fernando.soria@uam.es.

I met Guido Weiss in October 1982 when I arrived at Washington University to pursue my PhD. I was 22 years old and frightened by having made the decision to leave Italy, where I could have followed a clear and safe road. I was even more frightened by the idea of being evaluated by Professor Weiss, who would immediately notice my knowledge (tiny!) of mathematical analysis.

I still remember my feeling during our first meeting in his office. It seemed that the space was mainly full of books and papers; though it was nothing compared to what I saw the last time I visited Guido in May 2014! Guido was my scientific father and, even more, a very generous mentor. He helped me in all the difficult moments that I had to face as a graduate student. I grew up thanks to him and his example; he had an enormous impact on all the stages of my career.

Guido’s lectures were carefully prepared—always looking for a simple way to explain fundamental points. This was not only the case when he was teaching a course, but also when he wanted to introduce a new research topic. He was attentive to details, but with the general picture very clear in his mind. Guido was a Maestro, and his influence on my way of teaching is undeniable. I keep in full view the notes I took in his courses, his lectures, and in front of a blackboard; they are still valuable to me today.

Studying his book written in collaboration with Elias Stein, *Introduction to Fourier analysis on Euclidean spaces* SW71, and reading the first article he gave me (*Extensions of Hardy spaces and their use in analysis*, coauthored with R. R. Coifman CW77) were fundamental for me. In particular, the paper contained all the ingredients that are still contained in my research today, such as spaces of homogeneous type, Hardy and BMO spaces, interpolation theory, doubling measures, Calderón–Zygmund decomposition.

Guido’s intelligence, pleasure in life and personal relationships, knowledge, and above all generosity, contributed in a fundamental way to the growth of harmonic analysis in Italy. He played a key role in forming a group of friends and colleagues who met and talked while at Washington University.

I want to remember him as he has always been with all of us: available, full of energy, ideas, and great humanity.

I am happy to have met him on my journey.

Anita Tabacco is a professor of mathematics at Politecnico di Torino. Her email address is anita.tabacco@polito.it.

Guido Weiss had a tremendous influence on my life starting when I was a graduate student. Estela Gavosto (now my wife) and I arrived at Washington University in St. Louis from Argentina with the help of Cora Sadosky. I remember Cora encouraging us to go to WashU because it was “a very special place to be a graduate student” and that “Guido will take very good care of you.” Little did we know at the time that in addition to receiving great training and education, we would form many fantastic friendships and relationships with colleagues which continue to this day, due in large part to Guido. This experience has had a huge impact on our professional careers and lives.

Guido and several of our other teachers, in particular Mitch Taibleson and Al Baernstein, were extremely caring to a large group of students coming from China, Italy, Poland, and Spain whose time at WashU overlapped. Guido had established close connections with all these countries and was a magnet for attracting students. In particular, he was one of the first US mathematicians to participate in exchanges with post-Cultural Revolution China, after the reopening of relations with the US.

When we arrived in Saint Louis, Raphy Coifman had moved to Yale from WashU and so the Math Department was eagerly trying to hire other outstanding mathematicians. They succeeded in recruiting Björn Jawerth, Steven Krantz, and later Björn Dahlberg. The atmosphere in the Math Department during this time was incredible. There was a constant stream of visitors from other universities in the US and from all over the world. Practically every week some leading figure in harmonic analysis, complex analysis, several complex variables, or partial differential equations was visiting WashU and giving a lecture. We were encouraged to attend the talks, even if we were not at a point in our education to fully understand everything. The idea was to expose the students to a lot of great math and also to give them a chance to meet people. The accompanying receptions at the homes of faculty, to which graduate students were invited, served the same purpose. The day-long picnics hosted by Guido and his wife Barbara at their farm near St. Louis, were wonderful occasions to build synergy and camaraderie in the department. As were the memorable costume parties that the Italian students used to organize, which several professors would also attend. Guido and his colleagues managed to create a truly unique welcoming environment where we could be both nurtured professionally and also have a good time.

Guido was always there to help us with anything he could. In fact, it was typical of Guido to grab the phone and call someone and arrange things for any of us, or just “save the world,” as he used to joke while pointing to a big Superman belt buckle he often wore.

When it came time to pick an advisor after the grueling boot camp of first-year qualifying exams, Guido was going to be away for some time, so it was agreed that I would work with Björn Jawerth. Although I was not an official PhD student of Guido, when Björn moved to the University of South Carolina and I decided to stay at WashU, Guido became my closest mentor in the last steps of my thesis work.

In the mid 1980s, harmonic analysis saw the important developments of the T1 and Tb theorems as well as the birth of wavelets. These topics were very central to the focus of much of the research at WashU on the study of singular integrals and functions spaces. Michael Frazier, who was a postdoc at WashU at the time, and Björn developed their theory, producing powerful and unifying decomposition techniques, which include smooth atomic and molecular decompositions for the whole scale of Besov and Triebel–Lizorkin spaces. Many of the applications of these decompositions were then elegantly presented in Guido’s collaborative monograph with Michael and Björn -transformFJW91, which is one of the most cited references on the subject. Björn had suggested to me that I explore a version of the T1 theorem for spaces of smooth functions, in particular those with a lot of regularity, by showing, in the “spirit of St. Louis,” that appropriate Calderón–Zygmund operators map atoms into molecules. This would require improving on and developing arguments to extend previous work by M. Frazier, Y-S. Han, B. Jawerth, and G. Weiss along the same lines, but for small regularity. I had obtained some results when Björn learned that Guido and Michael, who were in a special semester on harmonic analysis at the Mathematical Sciences Research Institute (MSRI), were also working on the problem. I got very worried, but Guido arranged for me to visit MSRI and join forces with them. It was great to write my first math paper with him and Michael. Though I never formally collaborated with Guido again, we discussed a lot of mathematics over the years, and his ideas and mathematical contributions inspired much of my other work.

I continued to be in touch with Guido after leaving St. Louis, visiting him many times, and calling him on the phone. It was always a pleasure to meet him in his legendary office, which was often filled with other colleagues, visitors, and students. Guido liked to include everyone in the discussions and ideas he was investigating. I learned a lot from him and not only in mathematics. He was always ready to provide a letter of recommendation, advice about a difficult situation, or just chat about sports, world politics, or joke about life. When I last saw Guido, in his late eighties during a visit to WashU, he complained that he was no longer the super-strong athlete he used to be and could no longer play tennis, another subject he had taught many of his students, but he was still active mathematically and very engaging. He told me about a wavelets project he had been working on with his lifelong friend Ed Wilson, yet another great professor who was a dean when we were students. Guido took me for dinner at one of his favorite Italian restaurants on The Hill. We had a great evening as usual and, as was his wont, he told the server how to correct the name of the Italian dishes on the menu. The multiple languages that Guido spoke were always a source of conversation, amusement, and play on words.

I only had some brief communications with him after my last visit, but I know through Barbara that Guido enjoyed the biographical article my friend and colleague Susan Kelly and I wrote about him shortly before he passed away. I regard that article as a way to express our admiration for Guido. I miss him and think of him in many of my current roles and activities. When facing a difficult problem or trying to help others, I often ask myself what Guido would have done in a similar case. I am extremely thankful for his mentorship and friendship and feel fortunate he has touched my life. He will be forever remembered.

Rodolfo H. Torres is a Distinguished Professor of Mathematics and the Vice Chancellor for Research and Economic Development at the University of California, Riverside. His email address is rodolfo.h.torres@ucr.edu.

## References

- [CCR 82]
- R. R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher, and G. Weiss,
*A theory of complex interpolation for families of Banach spaces*, Adv. in Math.**43**(1982), no. 3, 203–229, DOI 10.1016/0001-8708(82)90034-2. MR648799Show rawAMSref`\bib{CCR+82}{article}{ author={Coifman, R. R.}, author={Cwikel, M.}, author={Rochberg, R.}, author={Sagher, Y.}, author={Weiss, G.}, title={A theory of complex interpolation for families of Banach spaces}, journal={Adv. in Math.}, volume={43}, date={1982}, number={3}, pages={203--229}, issn={0001-8708}, review={\MR {648799}}, doi={10.1016/0001-8708(82)90034-2}, }`

Close amsref.^{✖} - [CW77]
- Ronald R. Coifman and Guido Weiss,
*Extensions of Hardy spaces and their use in analysis*, Bull. Amer. Math. Soc.**83**(1977), no. 4, 569–645, DOI 10.1090/S0002-9904-1977-14325-5. MR447954Show rawAMSref`\bib{MR447954}{article}{ author={Coifman, Ronald R.}, author={Weiss, Guido}, title={Extensions of Hardy spaces and their use in analysis}, journal={Bull. Amer. Math. Soc.}, volume={83}, date={1977}, number={4}, pages={569--645}, issn={0002-9904}, review={\MR {447954}}, doi={10.1090/S0002-9904-1977-14325-5}, }`

Close amsref.^{✖} - [FJW91]
- Michael Frazier, Björn Jawerth, and Guido Weiss,
*Littlewood-Paley theory and the study of function spaces*, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991, DOI 10.1090/cbms/079. MR1107300Show rawAMSref`\bib{FJW91}{book}{ author={Frazier, Michael}, author={Jawerth, Bj\"{o}rn}, author={Weiss, Guido}, title={Littlewood-Paley theory and the study of function spaces}, series={CBMS Regional Conference Series in Mathematics}, volume={79}, publisher={Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI}, date={1991}, pages={viii+132}, isbn={0-8218-0731-5}, review={\MR {1107300}}, doi={10.1090/cbms/079}, }`

Close amsref.^{✖} - [HW96]
- Eugenio Hernández and Guido Weiss,
*A first course on wavelets*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer, DOI 10.1201/9781420049985. MR1408902Show rawAMSref`\bib{HW96}{book}{ author={Hern\'{a}ndez, Eugenio}, author={Weiss, Guido}, title={A first course on wavelets}, series={Studies in Advanced Mathematics}, note={With a foreword by Yves Meyer}, publisher={CRC Press, Boca Raton, FL}, date={1996}, pages={xx+489}, isbn={0-8493-8274-2}, review={\MR {1408902}}, doi={10.1201/9781420049985}, }`

Close amsref.^{✖} - [KT21]
- Susan E. Kelly and Rodolfo H. Torres,
*Guido Weiss: from immigrant boy to internationally renowned mathematician*, J. Geom. Anal.**31**(2021), no. 9, 9146–9179, DOI 10.1007/s12220-020-00596-8. MR4302216Show rawAMSref`\bib{KT21}{article}{ author={Kelly, Susan E.}, author={Torres, Rodolfo H.}, title={Guido Weiss: from immigrant boy to internationally renowned mathematician}, journal={J. Geom. Anal.}, volume={31}, date={2021}, number={9}, pages={9146--9179}, issn={1050-6926}, review={\MR {4302216}}, doi={10.1007/s12220-020-00596-8}, }`

Close amsref.^{✖} - [SW71]
- Elias M. Stein and Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR0304972Show rawAMSref`\bib{SW71}{book}{ author={Stein, Elias M.}, author={Weiss, Guido}, title={Introduction to Fourier analysis on Euclidean spaces}, series={Princeton Mathematical Series, No. 32}, publisher={Princeton University Press, Princeton, N.J.}, date={1971}, pages={x+297}, review={\MR {0304972}}, }`

Close amsref.^{✖}

## Credits

Figure 1 is courtesy of Washington University in St. Louis.

Figures 2, 3, and 5 are courtesy of Eugenio Hernández.

Figure 4 is courtesy of Fernando Soria.