# Kernel theorems in spaces of generalized functions

Banach Center Publications (2010)

- Volume: 88, Issue: 1, page 77-89
- ISSN: 0137-6934

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topAntoine 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 -

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