An integral transform of generalized functions. II (original) (raw)

Some extensions of a certain integral transform to a quotient space of generalized functions

In this paper, we establish certain spaces of generalized functions for a class of " s 2;1 transforms. We give the definition and derive certain properties of the extended " e 2;1 transform in a context of Boehmian spaces. The extended " e 2;1 transform is therefore well defined, linear and consistent with the classical " s 2;1 transforms. Certain results are also established in some detail.

Some classes of integral transforms on distribution spaces and generalized asymptotics

2014

In this doctoral dissertation several integral transforms are discussed.The first one is the Short time Fourier transform (STFT). We present continuity theorems for the STFT and its adjoint on the test function space K1(ℝn) and the topological tensor product K1(ℝn) ⊗ U(ℂn), where U(ℂn) is the space of entirerapidly decreasing functions in any horizontal band of ℂn. We then use such continuity results to develop a framework for the STFT on K'1(ℝn). Also, we devote one section to the characterization of K’1(ℝn) and related spaces via modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform. Part of the thesis is dedicated to the ridgelet and the Radon transform. We define and study the ridgelet transform of (Lizorkin) distributions and we show that the ridgelet transform and the ridgelet synthesis operator can be extended as continuous mappings Rψ : S’0(ℝn) → S’(Yn+1) and Rtψ: S’(Yn+1) → S’0(ℝn). We then use our results to develop a distrib...

Kernel theorems in spaces of generalized functions

Linear and Non-Linear Theory of Generalized Functions and its Applications, 2010

In analogy to the classical isomorphism between L(D(R n), D (R m)) and D (R m+n) (resp. L(S(R n), S (R m)) and S (R m+n)), we show that a large class of moderate linear mappings acting between the space GC (R n) of compactly supported generalized functions and G(R n) of generalized functions (resp. the space GS (R n) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n) of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of G(R m+n) (resp. Gτ (R m+n)). The main novelty is to use accelerated δ-nets, which are unit elements for the convolution product in these regular subspaces, to construct the kernels. Finally, we establish a strong relationship between these results and the classical ones.

Kernel theorems in spaces of tempered generalized functions

Mathematical Proceedings of The Cambridge Philosophical Society, 2007

In analogy to the classical isomorphism between L (S (R n ) , S ′ (R m )) and S ′ R n+m , we show that a large class of moderate linear mappings acting between the space GS (R n ) of Colombeau rapidly decreasing generalized functions and the space Gτ (R n ) of temperate ones admits generalized integral representations, with kernels belonging to Gτ R n+m . Furthermore, this result contains the classical one in the sense of the generalized distribution equality. : 45P05, 46F05, 46F30, 47G10

Generalized functions beyond distributions

Arabian Journal of Mathematics, 2014

Ultrafunctions are a particular class of functions defined on a non-Archimedean field R * ⊃ R. They have been introduced and studied in some previous works (Benci, Adv Nonlinear Stud 13:461-486, 2013; Benci and Luperi Baglini, EJDE, Conf 21:11-21, 2014; Benci, Basic Properties of ultrafunctions, to appear in the WNDE2012 Conference Proceedings, arXiv:1302.7156, 2014. In this paper we introduce a modified notion of ultrafunction and discuss systematically the properties that this modification allows. In particular, we will concentrate on the definition and the properties of the operators of derivation and integration of ultrafunctions.

Generalized integral operators and Schwartz kernel type theorems

Journal of Mathematical Analysis and Applications, 2005

In analogy to the classical Schwartz kernel theorem, we show that a large class of linear mappings admits integral kernels in the framework of Colombeau generalized functions. To do this, we introduce new spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that, in some sense, Schwartz' result is contained in our main theorem.