MATH 453 Introduction to Generalized Functions

Heaviside function and Delta-sequences. Test functions. Linear functionals and definition of a distribution (generalized function). Regular and singular distributions. Algebraic operations on distributions: linear change of variables, product of a distribution by a function. Analytic operations on distributions: derivative of a distribution. Transformation properties of Dirac-delta distribution. Schwartz space and tempered distributions: definitions and basic properties. Fourier transform of distributions. Convolution. The concept of generalized solution of a differential equation. Applications to Differential equations: Fundamental solutions and Green’s functions.