Reference letter for Professor Paul Koosis

Transcription

1 V.P.Havin Professor of St. Petersburg State University, Department of Mathematics and Mechanics, St. Petersburg State University,Stary Peterhof, ,St. Petersburg, RUSSIA Professor J.F.Jardine, Chair of the CMS Research Committee for the Jeffery-Williams Prize Lectureship 2009 Reference letter for Professor Paul Koosis The aim of this letter is to corroborate the following assertion: Paul Koosis is an excellent candidate for the Jeffery-Williams Prize Lectureship The work of P.Koosis embraces a wide field of classical analysis and includes numerous and various concrete topics and results which it would be a nonrealistic task to review in a reasonably short and readable text. I prefer to concentrate on some crucial points, gravity centers inspiring and organizing a large part of work to be reviewed. These themes of my choice are three: 1)weighted approximation by polynomials,trigonometric sums, and entire functions of finite degree; 2) the Beurling - Malliavin multiplier teorem; 3)normal families of functions determined by logarithmic sums. These themes are very classical. Their roots can be traced to the second decade of the twentieth century. They were started by great mathematicians and still remain topical, and their present state is essentially influenced by Paul Koosis. His contribution to the first theme is (partly) summarized in his 1966 Acta paper and in his book on the logarithmic integral (to be discussed at the end of this letter). In this subject a prominent role is played 1

2 by the logarithmic integral L(w), L(w) = 1 π log + w(x) 1 + x 2 dx, w being a weight on the real line (=a non-negative measurable function). The log. integral (=the title of the two-volume treatise by Koosis) plays an outstanding role in many parts of analysis and its applications (complex analysis,harmonic analysis,operator theory,probability). And our present understanding of this object which appeared first in the works of classics (Szegö, Nevanlinna, V.I. Smirnov) is now in many respects due to the results and approaches by Koosis. The emergence of the logarithmic integral in weigted approximation as Koosis turned to this theme was quite natural and fully prepared by his predecessors; the main interest of what Koosis has done in this direcction was a passage to discrete approximation (on the set of integers instead of the whole line). But the second theme (maybe the main theme of the whole mathematical life of Paul Koosis) is different. The log. integral plays a crucial role in this theme too, but this is not so easy to explain. Let me quote the beginning of an article by Paul Koosis in Duke Mathematical Journal, 1971,v3,N3: Let F(z) be an entire function of exponential type. A beautiful and important theorem of Beurling and Malliavin... states that if L( F ) < + then there are non-zero entire functions of arbitrarily small exponential type whose products with F(z) are bounded on the real axis. The condition on F(x) is in form identical to one figuring in the statement of Szegö s theorem and various other results in analysis, all arising ultimately from Jensen s formula. It enters, however, into the work of Beurling and Malliavin in entirely different fashion, and there seems to be no connection between its occurrence there and in the other more classical results. In spite of this, it is natural to believe that the appearances of the same criterion in the context of these different problems must be somehow related. In other words,the role of the log.integral in the Beurling-Malliavin theorem (=BM theorem,or just BM) was not as clear in 1961 when BM was published, as it is now, after many illuminating contributions by Koosis. The modest outlook of BM in the last quotation is deceptive. This short statement conceals a deep and multifaceted content admitting many different interpretations and generating a list of impressive connections (quite a few are due to Koosis). The original proof of BM contained some mysterious energy techniques from potential theory which look alien to the problem in question. The BM theorem attracted analysts since its first appearance (and still does), hundreds of pages are devoted to it. This important theme 2

3 is essentially marked by many original contributions by Koosis, impressive in their scope, volume, diversity and richness of new turns of the initial theme. P.Koosis gave four new proofs of BM: one is the theme of his long Acta paper of 1979, the second is in Annales de l institut Fourier of 1983,the third is in his green book on BM (see below), and the fourth in his publications on logarithmic sums (to be briefly discussed below). My superficial way to enlist these proofs is slightly misleading: it suggests the wrong impression that all this is just a repetition of one and the same result,the BM theorem. But in fact every time we have an essentially new result, enriching analysis with something fresh. The first proof is essentially about the Green function of a slit region (=the complement of a union of segments of the real axis), the second is a development of the method of least superharmonic majorants etc. All these subjects deserve attention in their own right. I will dwell on the fourth proof, a true jewel of analysis in my very personal feeling. Passing to the fourth proof we actually turn to the third theme (see the beginning of the letter). This time the BM theorem is reduced to a (seemingly quite different) problem on normal families: let a be a positive number and denote by F(a) the family of all polynomials p satisfying n log + p(n) 1 + n 2 < a, the sum being taken over all integers n. Is (F(a) normal (i.e. uniformly bounded on every disc)? Replacing the sum by the corresponding log.integral we simplify our question considerably and in this case the answer is known to be yes FOR ANY a. Koosis discovered that the answer to HIS question is yes provided a is sufficiently small! And this was only a start: in a joint work with his Danish student H.Pedersen they contrived to replace polynomials p by entire functions of a finite fixed and small degree - with the same conclusion, F(a) remaining normal if a is majorized by a small number (depending on the above degree). Later on Koosis extended the result to entire functions of any fixed degree strictly less than π. This result is sharp. These delicate phenomena admit further development (in particular a passage to some non-integer values of n in the logarithmic sums, as shown recently by Koosis and Pedersen). And let me emphasize that these results on normal families yield a new proof of BM, thus exhibiting an interesting new aspect of that remarkable theorem. 3

4 One more remification of the BM theorem due to Koosis is a nice and rather striking variant of weighted Helson-Szegö-Muckenhoupt inequalities for the Hibert transform: given a weight w summable on the real line, is there a non-zero weight ω satisfying f 2 ω f 2 w for any f with a spectral gap of a given width ( f stands for the Hilbert transform of f)? We may think of f as of a finite sum [A(λ)cos λx + B(λ)sin λx] λ >a a being a fixed positive number. Koosis proved that the affirmative answer is equivalent to the existence of a non-zero entire function of degree a majorized by w almost as in BM (not pointwise,but in an integral sense). I have to stop and repeat that the above survey is very compressed and does not (by far) represent all that Koosis has done in analysis. I just wanted to give SOME samples of results and approaches clarifying and enriching the most fundamental parts of classical analysis. And I had to select facts and statements which could be reproduced in a reasonably short text. I cannot, however, completely omit an impressive episode which is far away from the preceding material. It is related to a memorable event of function theory of the 20th century, the discovery by Burkholder, Gundy and Silverstein of a real definition of the Hardy H 1 class in the upper halfplane (and its analogs in multidimensional halfspaces). It turned out that the elements of this class (whose origin is purely complex in dimension two) can be characterized by the summability of their maximal functions! The original proof was probabilistic (based on the brownian motion), a challenge to all analysts. Koosis found a very elegant deterministic proof (first in dimension two, then for all dimensions). I should turn now to a very nice treatment of Levinson s theorem on zeros of entire functions of finite degree (a clever reduction to the Kolmogorov theorem on the Hilbert transform), or to Koosis results on the interior compactness in some spaces of functions on the line, the results I really like. But I feel I have to stop, omitting, by the way, two physical articles on the Schroedinger equation, a feast of classical complex analysis, but aimed at properties of concrete physical spectra... A detailed exposition of what Koosis has done would require a book. 4

5 The last phrase opens up another important side of the work by Koosis: he is an outstanding mathematiced writer. He wrote three exellent books, very popular and influential among the analysts. The first is his Introduction to Hardy classes ( the blue Koosis, three editions, one of them in Russian). I witness: three generations of young complex and harmonic analysts in USSR/Russia (not to mention other countries) started their professional lives reading this marvelous book. It is deservedly famous in the whole world of analysts. Then comes the two volume treatise The logarithmic integral (two editions, the gray Koosis ). This is a wonderful encyclopaedic exposition of some parts of classical analysis including moment problems, weighted approximation, a detailed treatment of the BM multiplier theorem and of the so-called Second BM theorem (on completeness of a system of exponentials) and many other topics. The treatise is largely based on the original contributions by the author. Everyone seriously interested in modern function theory and its applications cannot dispense with these two volumes. The third book ( the green Koosis ) is entirely devoted to BM and its (rather vast) neighbourhood. It is an important complement to The logarithmic integral. The predominant style of writing mathematics is nowadays marked by haste and negligence. Authors are not bothered with clarity, their aim is not communication of their results, but only priority and the very fact of being published. The style of Koosis is very different. Once you have opened his text you CAN read and understand it, not struggling with riddles, omissions, numerous it is easy to see etc. His books and articles are masterpieces of clarity and noble simplicity,they are accessible even to a beginner. In conclusion I return to my first claim (see the beginning of the letter): Koosis work as a whole, his results, his ideas, his books are a splendid contribution to modern function theory. He is an influential first-class analyst. His achievements in the complex analysis and approximation theory determine in many important respects the present shape of these disciplines. V.P.Havin, Professor of St.Petersburg State University 5