First works related to the topics covered in this book belong to J. Delsarte and B. M. Le vitan and appeared since 1938. In these works, the families of operators that generalize usual translation operators were investigated and the corresponding harmonic analysis was constructed. Later, starting from 1950, it was noticed that, in such constructions, an important role is played by the fact that the kernels of the corresponding convolutions of functions are nonnegative and by the properties of the normed algebras generated by these convolutions. That was the way the notion of hypercomplex system with continu ous basis appeared. A hypercomplex system is a normed algebra of functions on a locally compact space Q-the "basis" of this hypercomplex system. Later, similar objects, hypergroups, were introduced, which have complex-valued measures on Q as elements and convolution defined to be essentially the convolution of functionals and dual to the original convolution (if measures are regarded as functionals on the space of continuous functions on Q). However, until 1991, the time when this book was written in Russian, there were no monographs containing fundamentals of the theory (with an exception of a short section in the book by Yu. M. Berezansky and Yu. G. Kondratiev BeKo]). The authors wanted to give an introduction to the theory and cover the most important subsequent results and examples."

This is the ?rst of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This?rstvolumeisdevotedtotheoperatortheoryandrelatedtopics. Itopens withthebiographypapersaboutM. G. Kreinandanumberofsurveypapersabout his work. The mainpartof the book consistsof originalresearchpaperspresenting the state of the art in operator theory and its application. The second volume of these proceedings, entitled Di?erential Operators and Mechanics, concerns other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences. The editors are sincerely grateful to the persons who contributed to the preparation of these proceedings: Sergei Marchenko, Myroslav Sushko, Kostyantyn Yusenko and Vladimir Zavalnyuk. Mark Grigorievich Krein, 1907-1989 Operator Theory: Advances and Applications, Vol. 190, xi-xx c 2009 Birkh. auser Verlag Basel/Switzerland Mark Grigorievich Krein (on his 100th birthday anniversary) V. M. Adamyan, D. Z. Arov, Yu. M. Berezansky, V. I. Gorbachuk, M. L. Gorbachuk, V. A. Mikhailets and A. M. Samoilenko April 3, 2007, is the l00th anniversary of the birth of Mark Grigorievich Krein, one of the most celebrated mathematicians of the 20th century, whose whole life was closely connected with Ukraine.

This is the second of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M.G. Krein. This second volume is devoted to the theory of di?erential operators and mechanics. It opens with the description of the conference and a number of survey papers about the work of M.G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of di?erential operators. The ?rst volume of these proceedings, entitled Operator Theory and Related Topics, concerns other aspects of the conference. The two volumes will be of - terest to a wide-rangeof readership in pure and applied mathematics, physics and engineering sciences. OperatorTheory: AdvancesandApplications,Vol.191, xi-xv c 2009Birkh. auserVerlagBasel/Switzerland The World Dimension of the Heritage of a Ukrainian Mathematician International Conference "Modern Analysis and Applications" (MAA - 2007) (April 9-14, 2007, Odessa) Yu. BerezanskyandV.Gorbachuk This forum has been dedicated to the centennial birthday anniversary of one of the most prominent mathematicians of the twentieth century Mark Gr- orievich Krein, a corresponding member of the Academy of Sciences of the Ukr. SSR (1907-1989).

First works related to the topics covered in this book belong to J. Delsarte and B. M. Le vitan and appeared since 1938. In these works, the families of operators that generalize usual translation operators were investigated and the corresponding harmonic analysis was constructed. Later, starting from 1950, it was noticed that, in such constructions, an important role is played by the fact that the kernels of the corresponding convolutions of functions are nonnegative and by the properties of the normed algebras generated by these convolutions. That was the way the notion of hypercomplex system with continu ous basis appeared. A hypercomplex system is a normed algebra of functions on a locally compact space Q-the "basis" of this hypercomplex system. Later, similar objects, hypergroups, were introduced, which have complex-valued measures on Q as elements and convolution defined to be essentially the convolution of functionals and dual to the original convolution (if measures are regarded as functionals on the space of continuous functions on Q). However, until 1991, the time when this book was written in Russian, there were no monographs containing fundamentals of the theory (with an exception of a short section in the book by Yu. M. Berezansky and Yu. G. Kondratiev BeKo]). The authors wanted to give an introduction to the theory and cover the most important subsequent results and examples."