The Discrete Calder´on Reproducing Formula of Frazier and Jawerth ´ Arp´ ad B´ enyi and Rodolfo H. Torres Abstract. We present a brief recount of the discrete version of Calder´on’s reproducing formula as developed by M. Frazier and B. Jawerth, starting from the historical result of Calder´ on and leading to some of the motivation and applications of the discrete version. Alberto P. Calder´on’s genius has produced a plethora of deep and highly in- fluential results in analysis [6], and his reproducing formula certainly qualifies as a gem among them. Also commonly referred to as Calder´on’s resolution of identity, Calder´on’s Reproducing Formula (CRF) is a strikingly elegant relation, inspired by the simple idea of breaking down a function as a sum of appropriate convolutions or wave like functions (see (1.1) below). Moreover, mathematicians working on wavelets consider nowadays Calder´ on as one of the forefathers of the theory. The purpose of this expository note is to honor the memory of Bj¨orn Jawerth by presenting a brief account of some aspects of one of his major contributions to mathematics: the development in collaboration with Michael Frazier of what they called “the φ-transform” and how it relates to Calder´ on’s original formula. None of what is presented here is new and we will repeat some arguments commonly found in the literature; but some others that we shall present seem to be part of the folklore of the subject or are hard to locate in the references. We will try to follow a partially historical and partially formal approach to this beautiful formula as we learned it from the horse’s mouth, in particular from Bj¨orn Jawerth himself, but also from Michael Frazier, Richard Rochberg, Michael (Mitch) Taibleson, and Guido Weiss. 1 The goal is to summarize here in a succinct way some simple motivations while pointing to both classical as well as some not so well-known references, including only some technical details for completion purposes or to illustrate some concepts. Our hope is to convey, perhaps not to the experts but rather to a broader uninitiated audience, some of the profound contributions of Jawerth and collaborators, which are sometimes overlooked in the continuing proliferation and rediscovery of results in the area of function space decompositions. We also want to insight the curiosity of those who may have not read numerous original works we shall mention, and motivate them to further explore the reach existing literature. 2010 Mathematics Subject Classification. Primary: 42B20; Secondary: 42B15, 47G99. 1 In particular, in the development of some topics in this survey we have benefited a lot from unpublished lectures notes of courses taught by Frazier and Jawerth at Washington University in the 80’s. 1