Fourier transform in bounded domains

The functional analysis, the concept of distributions u ∈ D′ in the sense of Schwartz and their extension given by Gel'fand and Shilov, to ultradistributions u ∈ Z′, enables us to find by the means of the Fourier transform a second 'language' to characterise physical behaviour. Almost any expression with physical meaning can be transformed, even if it is formulated in domains with complicated boundaries and even if it is not integrable. Numerical procedures in the transformed space can be developed in analogy to those well-known in engineering mechanics like the methods of Finite or Boundary Elements (FEM or BEM). Basis of the approaches presented here is the analytical representation of a characteristic distribution of a domain and the theorem of Parseval, which states the invariance of energy in respect to the transformation. In addition, the concept of the characteristic distribution leads to a very simple derivation of the Green-Gauss formulas fundamental for the Boundary or Finite Elements.     «

The functional analysis, the concept of distributions u ∈ D′ in the sense of Schwartz and their extension given by Gel'fand and Shilov, to ultradistributions u ∈ Z′, enables us to find by the means of the Fourier transform a second 'language' to characterise physical behaviour. Almost any expression with physical meaning can be transformed, even if it is formulated in domains with complicated boundaries and even if it is not integrable. Numerical procedures in the transformed space can be develo...     »