Non-Differentiability of the Convolution of Differentiable Real Functions

We provide an example of two 2 -periodic everywhere differentiable functions f , g : R → R whose convolution f ∗ g fails to be differentiable at every point of some perfect (thus, uncountable) set P ⊂ R . This shows that the convolution operator can actually destroy the differentiability of these ma...

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Veröffentlicht in:	Real analysis exchange 2020, Vol.45 (2), p.327
Hauptverfasser:	Ciesielski, Krzysztof C., Jiménez-Rodríguez, Pablo, Muñoz-Fernández, Gustavo A., Seoane-Sepúlveda, Juan B.
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Sprache:	eng
Schlagworte:	Convolution Convolutional codes Mathematical functions Mathematics Smoothness Theorems
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description	We provide an example of two 2 -periodic everywhere differentiable functions f , g : R → R whose convolution f ∗ g fails to be differentiable at every point of some perfect (thus, uncountable) set P ⊂ R . This shows that the convolution operator can actually destroy the differentiability of these maps, rather than introducing additional smoothness (as it is usually the case). New directions and open problems are also posed.
doi_str_mv	10.14321/realanalexch.45.2.0327
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subjects	Convolution Convolutional codes Mathematical functions Mathematics Smoothness Theorems
title	Non-Differentiability of the Convolution of Differentiable Real Functions
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