Ebook: Distributions

This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same “weak” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.

This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.

This book is the third of seven volumes dedicated to solving partial dif-
ferential equations in physics:
Volume 1: Banach, Frechet, Hilbert and Neumann Spaces
Volume 2: Continuous Functions
Volume 3: Distributions
Volume 4: Integration
Volume 5: Sobolev Spaces
Volume 6: Traces
Volume 7: Partial Differential Equations