Kernel theorems in spaces of generalized functions

Antoine Delcroix. "Kernel theorems in spaces of generalized functions." Banach Center Publications 88.1 (2010): 77-89. <http://eudml.org/doc/282250>.

@article{AntoineDelcroix2010,
abstract = {In analogy to the classical isomorphism between ((ℝⁿ), $ ^\{\prime \}(ℝ^\{m\}))$ and $ ^\{\prime \}(ℝ^\{m+n\})$ (resp. $((ℝⁿ),^\{\prime \}(ℝ^\{m\}))$ and $^\{\prime \}(ℝ^\{m+n\})$), we show that a large class of moderate linear mappings acting between the space $_\{C\}(ℝⁿ) $ of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space $_\{\}(ℝⁿ)$ of Colombeau rapidly decreasing generalized functions and the space $_\{τ\}(ℝⁿ)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $(ℝ^\{m+n\})$ (resp. $_\{τ\}(ℝ^\{m+n\})$). 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.},
author = {Antoine Delcroix},
journal = {Banach Center Publications},
keywords = {kernel theorems; nuclear spaces; Colombeau generalized functions; Colombeau temperate generalized functions; integral operator; Schwartz distributions; tempered distributions},
language = {eng},
number = {1},
pages = {77-89},
title = {Kernel theorems in spaces of generalized functions},
url = {http://eudml.org/doc/282250},
volume = {88},
year = {2010},
}

TY - JOUR
AU - Antoine Delcroix
TI - Kernel theorems in spaces of generalized functions
JO - Banach Center Publications
PY - 2010
VL - 88
IS - 1
SP - 77
EP - 89
AB - In analogy to the classical isomorphism between ((ℝⁿ), $ ^{\prime }(ℝ^{m}))$ and $ ^{\prime }(ℝ^{m+n})$ (resp. $((ℝⁿ),^{\prime }(ℝ^{m}))$ and $^{\prime }(ℝ^{m+n})$), we show that a large class of moderate linear mappings acting between the space $_{C}(ℝⁿ) $ of compactly supported generalized functions and (ℝⁿ) of generalized functions (resp. the space $_{}(ℝⁿ)$ of Colombeau rapidly decreasing generalized functions and the space $_{τ}(ℝⁿ)$ of temperate ones) admits generalized integral representations, with kernels belonging to specific regular subspaces of $(ℝ^{m+n})$ (resp. $_{τ}(ℝ^{m+n})$). 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.
LA - eng
KW - kernel theorems; nuclear spaces; Colombeau generalized functions; Colombeau temperate generalized functions; integral operator; Schwartz distributions; tempered distributions
UR - http://eudml.org/doc/282250
ER -