ABSTRACT
Spaces of distributions are the duals of spaces of C ∞ $ C^\infty $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math15_1.tif"/> functions on open subsets of R d $ \mathbb R ^{d} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math15_2.tif"/> . The operations of differentiation, convolution, and Fourier transform of functions may be extended by duality to distributions, opening up the possibility of finding non-differentiable solutions, so-called weak solutions, of differential equations that may not have smooth solutions. For example, consider the partial differential equation ∑ α ∈ S ψ α ( x ) ∂ α f ( x ) = g ( x ) $$ \sum _{\alpha \in S} \psi _\alpha ({\boldsymbol{x}})\partial ^\alpha f({\boldsymbol{x}}) = g({\boldsymbol{x}}) $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/um1295.tif"/>
