Smoothness of convolutions of zonal measures in symmetric spaces

Fourier and many other mathematicians (e.g. D. Bernoulli, Lagrange and Euler) were motivated by problems in mathematical physics to represent any periodic function f with period 2? on the real line as the sum of the series (the so-called Fourier series of f ) . On the real line, sin nx and cos nx are the simplest possible periodic functions with period 2?. In 1822, Fourier in his book ?Th?orie Analytique de la Chaleur? gave the following formulas to calculate the values of an and bn : The real numbers an, bn are called the Fourier coefficients of f. Throughout the nineteenth century Fourier series was studied and applied to solve problems in natural sciences (see e.g. [20]). Kelvin used Fourier series to predict tides, to estimate the age of earth [20]. Fourier series is used to solve differential equations which occur in the study of natural sciences (e.g. heat equation, wave equation, Laplace?s equation). For a function f on the real line, the analogue of Fourier coefficient is defined as where y is a real number. is called the Fourier transform of f . In the earlier part of twentieth century, it was realized that Fourier series and Fourier transforms can be studied on Lie groups (real line, unit circle and matrix groups are examples of Lie groups). My research proposal is concerned with the study of smoothness of convolutions of zonal measures on symmetric spaces. In the study of these convolutions, we use Fourier transform on Lie groups as a tool.