Distributions and Their Fourier transforms

Distributions are linear continuous functionals over the space of so-called basic functions. Space of basic functions we choose as the Schwartz space S(ℝ m ) (ℝ m is m — dimensional Euclidean space) of infinitely differentiable on ℝ m functions decreasing under |x| → ∞ more rapidly than any power of |x|−1, x = (x 1, ..., x m ), $$|x| = \sqrt {{x_1}^2 + \ldots + {x_m}^2} .$$ We determine the counting number of norms in S(ℝ m ) by the formula 1.1.1 $$||\varphi |{|_p} = \sup {(1 + |x{|^2})^{p/1}}|{D^a}\varphi (x)|,\varphi \in S({\mathbb{R}^m}),p = 0,1 \ldots ,|a| \leqslant p$$ where $${D^a}\varphi = \frac{{{\partial ^{|\alpha |}}\varphi }}{{\partial {x_1}^{{\alpha _1}} \ldots \partial {x_m}^{{\alpha _m}}}}$$ , α is multi index, |α| = α 1 + ⋯ + α m ; with the help of these norms we define the convergence concept in S(ℝ m ). Namely we say the sequence φ 1, ..., φ k , ... of functions from S(ℝ m ) converges to function φ ∈ S(ℝ m ) iff ∥φ k − φ∥ p → 0, k → ∞ for all ρ = 0, 1, ... The last statement, by virtue of (1.1.1) is equivalent to saying that x α D β φ k (x) uniformly tends to zero under k → ∞ for arbitrary multiindex $$\alpha ,\beta ,{x^\alpha } \equiv {x_1}^{{\alpha _1}} \ldots {x_m}^{{\alpha _m}}$$ .