Calculus of Operators: Covariant Transform and Relative Convolutions

Abstract: This paper outlines a covariant theory of operators defined on groups and homogeneous spaces. A systematic use of groups and their representations allows to obtain results of algebraic and analytical nature. The consideration is systematically illustrated by a representative collection of examples. Keywords: Lie groups and algebras, convolution operator, relative convolution, representation theory, pseudo-differential operators, PDO, singular integral operator, SIO, the Heisenberg group, SL(2,R), induced representation, Fock-Segal-Bargmann (FSB) representation, Shrodinger representation, covariant transform, contravariant transform, wavelet transform, square integrable representations, reproducing kernel, Fourier-Wigner transform, deformation quantization, Bergman space, Berezin's symbol, Wick symbol, Schwartz kernel, Toeplitz operator, operators of local type, Simonenko's localisation, reproducing kernel thesis, Dynin group, twisted convolution, star product, Groenewold-Moyal, composition operator.