Applications

Abstract

(a) The translation (τx0Th)(f) = Th(τ−x0f), (b) The modulation (e0Th)(f) = Th(e ix·ξ0f), (c) The rescaling (Th ◦A)(f) = |det(A)|Th(f ◦A−1), (d) The multiplication by xj : (xjTh)(f) = Th(xjf), (e) The multiplication by g : (gTh)(f) = Th(fg), (f) The derivative ∂xj : (∂xjTh)(f) = −Th(∂xjf), (g) The convolution with f : (Th ∗ f)(x) = Th(f(x− ·)) (is a smooth function), (h) The Fourier transform F : F(Th)(f) = Th(F(f)), (i) The inverse Fourier transform F(Th)(f) = Th(F(f)) = Th(FR(f)).