Growing specialization and diversification have brought a hor'st of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are sudden ly seen to be related. Further, the kind and level of sophistication of mathematics applied invarious sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This programme, Mathematics and Its Applications, is devoted to such (new) interrelations as exempli gratia: - a central concept which plays an important role in several different mathematical andjor scientific specialized areas; - new applications of the results and ideas from one area of scien tific endeavor into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.

Inhalt
I.- 1. Groups and Homogeneous Spaces.- 2. Representations of Locally Compact Groups.- 3. Decomposition Theory of Unitary Representations.- 4. Representations of Compact Groups.- 5. Theory of Spherical Functions.- II.- 6. The Euler ?- and B-Functions.- 7. Bessel Functions.- 8. Theory of Jacobi and Legendre Polynomials.- 9. Gegenbauer Polynomials.- 10. Jacobi and Legendre Functions.- Chapter11. Harmonic Analysis on the Lobatschevsky space.- 12. The Laguerre Polynomials.- Chapter13. The Hypergeometric Equation.- III.- 14. Affine Transformations.- 15. Symmetric Spaces.- 16. General Harmonic Analysis on a Symmetric Space.- 17. Semisimple Algebras. Semisimple Groups. Symmetric Spaces of the Non-Compact Type.- 18. Harmonic Analysis on Symmetric Spaces of the Non-Compact Type.- Table of Formulas.- References.- List of Symbols.- Author Index.