Bøger af R. E. Edwards

appear in Volume 1, a Roman numeral "e;I"e; has been prefixed as a reminder to the reader; thus, for example, "e;I,B.2.1 "e; refers to Appendix B.2.1 in Volume 1. An understanding of the main topics discussed in this book does not, I hope, hinge upon repeated consultation of the items listed in the bibli- ography. Readers with a limited aim should find strictly necessary only an occasional reference to a few of the book listed. The remaining items, and especially the numerous research papers mentioned, are listed as an aid to those readers who wish to pursue the subject beyond the limits reached in this book; such readers must be prepared to make the very considerable effort called for in making an acquaintance with current research literature. A few of the research papers listed cover devel- opments that came to my notice too late for mention in the main text. For this reason, any attempted summary in the main text of the current standing of a research problem should be supplemented by an examin- ation of the bibliography and by scrutiny of the usual review literature.

These notes provide a reasonably self-contained introductory survey of certain aspects of harmonic analysis on compact groups. The first part of the book seeks to give a brief account of integration theory on compact Hausdorff spaces. The second, larger part starts from the existence and essential uniqueness of an invariant integral on every compact Hausdorff group. Topics subsequently outlined include representations, the Peter-Weyl theory, positive definite functions, summability and convergence, spans of translates, closed ideals and invariant subspaces, spectral synthesis problems, the Hausdorff-Young theorem, and lacunarity.

§1 Faced by the questions mentioned in the Preface I was prompted to write this book on the assumption that a typical reader will have certain characteristics. He will presumably be familiar with conventional accounts of certain portions of mathematics and with many so-called mathematical statements, some of which (the theorems) he will know (either because he has himself studied and digested a proof or because he accepts the authority of others) to be true, and others of which he will know (by the same token) to be false. He will nevertheless be conscious of and perturbed by a lack of clarity in his own mind concerning the concepts of proof and truth in mathematics, though he will almost certainly feel that in mathematics these concepts have special meanings broadly similar in outward features to, yet different from, those in everyday life; and also that they are based on criteria different from the experimental ones used in science. He will be aware of statements which are as yet not known to be either true or false (unsolved problems). Quite possibly he will be surprised and dismayed by the possibility that there are statements which are "definite" (in the sense of involving no free variables) and which nevertheless can never (strictly on the basis of an agreed collection of axioms and an agreed concept of proof) be either proved or disproved (refuted).