ABSTRACT
In finite dimensional distribution theory, there are several important continuous linear operators acting on the space of test functions; e.g., differentiation, translation, dilation, multiplication, convolution, and the Fourier transform. We can use duality to define adjoint operators. But often they turn out to be the extensions of continuous operators on the space of test functions to the space of generalized functions because of the nature of the Lebesgue measure. For instance, the adjoint D x * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203733813/e9279b26-91e9-40dd-8029-b908c0f36f6c/content/eq1463.tif"/> of the differential operator Dx is related to the extension D ˜ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203733813/e9279b26-91e9-40dd-8029-b908c0f36f6c/content/eq1464.tif"/> of Dx to the space of generalized functions by D x * = − D ˜ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203733813/e9279b26-91e9-40dd-8029-b908c0f36f6c/content/eq1465.tif"/> . However the situation is quite different in white noise distribution theory since the Gaussian measure is used. We will see that adjoint operators play an important role.
