90 Years of the Reproducing Kernel Property: History Meets Contemporary Research

 

At the beginning of the 20th century two important papers1 appeared in Mathematics and the reproducing kernel property was born in Kraków. Its father was Stanisław Zaremba2, professor at the Jagiellonian University. Stanisław Zaremba (1863-1942) graduated as an engineer in St. Petersburg and went on to study mathematics in Paris, receiving there his Ph.D. His thesis “Sur un probleme concernant l’état calorifique d’un corps homogene indefini” offered a solution to a problem posed by the Paris Academy of Sciences. Among Zaremba’s pupils was Tadeusz Ważewski (1896-1972), Zaremba’s successor, who created the Kraków School of Differential Equations3, as called by specialists in the field. Zaremba’s education determined his scientific interest: widely understood mathematical analysis (traditionally the strength of Cracovian Mathematics) including the theory of partial differential equations, especially those of mathematical physics as well as distinctive applications, in particular in physics. In this context the 1907 paper, still within the theory of partial differential equations, states for the first time what is nowadays called the reproducing kernel property4; the other paper, the 1909 one, is rather orientated towards numerical applications5. In the sizeable list of his scientific achievements one finds also that Zaremba is the author of a method of orthogonal projection in the Dirichlet problem. It is important to mention this because it shows that the discovery of the reproducing kernel property was not made by accident, it is well settled in Zaremba’s mathematical work. These two principles in their abstract form later on have found firm position in the theory of Hilbert space, the vital area which, besides numerous applications, paved the way for mathematical foundations of quantum mechanics.

 

As said before the proper set up for the reproducing kernel property is within the Hilbert space framework. This provides an abstract language as well as allows to derive properties suitable for application. And the applications are diverse, and sometimes contrasting: function theory (mostly thought of as in complex variable), differential equations, functional analysis and operator theory (with application to control theory), harmonic analysis (including magnetic resonance tomography), stationary processes and mathematical physics, as to mention some of them. In all these disciplines the reproducing kernel property has being present as a research tool for over last 60 years. One may say without any hesitation that the aforesaid areas are impacted by the property to the vastest extent. All this gave an impetus to gather mathematicians representing those branches just in Kraków, at the Jagiellonian University.

Thus the conference 90 years of the reproducing kernel property (April, 16-21) was organized by the Chair of Functional Analysis. The conference was attended by over 50 mathematicians from different countries all over the world as well as from major mathematical centers in Poland; each of the participants gave a lecture relevant to her/his personal research interest presenting the updated state of the art (Mathematics is an art.!). In times of over-specialization it was an exceptional event. The program of the conference reflects this truth completely.

 

Professor Franciszek Hugon Szafraniec, Ph.D.

Chair of Functional Analysis, JU

 

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1 Zaremba, L'equation biharmonique etune classe remarquable de fonctions fondamentales harmoniques, Bulletin International de l'Academie des Sciences de Cracovie, Classe des Sciences Mathematiques et Naturelles, 1907 (3), 147-196.

---, Sur le calcul numerique des fonctions demandees dans le probleme de Dirichlet et le probleme hydrodynamique, ibidem, 1909, (2), 125-195.

 

2 For an account on Zaremba's scientific life and his Mathematics with some historical background see Andrzej Pelczar, Stanislaw Zaremba (100th anniversary of taking up a chair at the Jagiellonian University), prepared for the International Confeerence 90 Years of the Reproducing Kernel Property, Kraków, April 16-21, 2000, organized by the Chair of Functional Analysis of the Jagiellonian University (available from the web site http://www.im.uj.edu.pl./~preprints as preprint # 2000/H1).

 

3 The author was fortunate to be a member of this school, in fact he was Wazewski's last Ph.D. student.

 

4 For those who stoll want to learn more let me refer to F.H.Szafraniec, The reproducing kernel Hilbert space and its multiplication operators, Operator Theory: Advances and Applications, 114(2000), 253-263 as a contemporary look at the reproducing kernel story.

 

5 All this is well documented in N.Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68(1950), 337-404.