Let (mathcal {S}=mathcal{S}(mathbb{R}^{n})) be the subspace of (C^{infty}(mathbb {R}^{n})) of Schwartz rapidly decreasing functions and (mathcal {S}'=mathcal{S}'(mathbb{R}^{n})) be the space of all tempered distribution on (mathbb{R}^{n}).
To be precise, we can multiply (widehat{f}) with the Dirac distribution on the circle of radius K, obtaining a tempered distribution on (mathbb {R}^{2}), and define (f_{K}) as the inverse Fourier transform of this multiplication: it is automatically a smooth function.
A "Fourier" transform for tempered distributions on groups of Heisenberg type is introduced.
end{aligned} and (mathcal{S}' (mathbf{R}^{3})), (mathcal{P}(mathbf{R}^{3})) are the spaces of all tempered distributions on (mathbf{R}^{3}) and the set of all scalar polynomials defined on (mathbf{R}^{3}), respectively.
Let S′ be the space of tempered distributions on R and Λ the Lévy noise measure on S′.
We define the atomic Hardy space (H^{p,q}_{L}(mathbb{C}^{n})) to be the set of all tempered distributions of the form (sum_{j} lambda_{j}a_{j}) (the sum converges in the topology of (mathcal{S}'(mathbb{C}^{n}))), where (a_{j}) are (H_{L}^{p,q} -atoms and (sum_{j}|lambda_{j}|^{p}<+infty).
This problem will be analyzed in Section 3.2 where the directional short-time Fourier transform of tempered distributions for (ginmathcal{S}_{0}(mathbb{R})) will be defined.
For example, every f ∈ L p, 1 ≤ p < ∞, defines a tempered distribution by virtue of the relation 〈 f, φ 〉 = ∫ f ( x ) φ ( x ) d x, φ ∈ S ℝ.
Besides, the Fourier transform is extended to infinite dimensions and a new definition of the composition of a tempered distribution with the Brownian motion is given.
In this section, we reformulate and prove the stability theorem of the quartic functional equation (1.4) in the spaces of some generalized functions such as of tempered distributions and of Fourier hyperfunctions.
In this paper, we solve the general solutions and the stability problems of (1.2) in the spaces of generalized functions such as S ′ of tempered distributions, F ′ of Fourier hyperfunctions and D ′ of distributions.
