We consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any function can be approximated in [0,1] by a function that is Caputo-stationary in [0,1], with initial point . Otherwise said, Caputo-stationary functions are dense in .
Accepté le :
DOI : 10.1051/cocv/2016056
Mots-clés : Caputo stationary, fractional derivative, nonlocal operators
@article{COCV_2017__23_4_1361_0,
author = {Bucur, Claudia},
title = {Local density of {Caputo-stationary} functions in the space of smooth functions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1361--1380},
publisher = {EDP-Sciences},
volume = {23},
number = {4},
year = {2017},
doi = {10.1051/cocv/2016056},
mrnumber = {3716924},
zbl = {1396.26013},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2016056/}
}
TY - JOUR AU - Bucur, Claudia TI - Local density of Caputo-stationary functions in the space of smooth functions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 1361 EP - 1380 VL - 23 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2016056/ DO - 10.1051/cocv/2016056 LA - en ID - COCV_2017__23_4_1361_0 ER -
%0 Journal Article %A Bucur, Claudia %T Local density of Caputo-stationary functions in the space of smooth functions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 1361-1380 %V 23 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2016056/ %R 10.1051/cocv/2016056 %G en %F COCV_2017__23_4_1361_0
Bucur, Claudia. Local density of Caputo-stationary functions in the space of smooth functions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1361-1380. doi : 10.1051/cocv/2016056. http://www.numdam.org/articles/10.1051/cocv/2016056/
M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables. Reprint of the 1972 ed. A Wiley-Interscience Publication. Selected Government Publications. New York: John Wiley & Sons, Inc; Washington, D.C.: National Bureau of Standards. xiv, 1046 pp. 44.95 (1984). | Zbl
C. Bucur and E. Valdinoci. Nonlocal diffusion and applications. Cham: Springer, Bologna: UMI (2016). | MR
, Linear models of dissipation whose is almost frequency independent. II. Fract. Calc. Appl. Anal. 11 (2008) 4–14. Reprinted from Geophys. J. R. Astr. Soc. 13 (1967) 529–539. | MR | Zbl
, and , Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012) 521–573. | DOI | MR | Zbl
S. Dipierro, O. Savin and E. Valdinoci, All functions are locally -harmonic up to a small error. Preprint (2014). To appear in: J. Eur. Math. Soc. JEMS (2017). | arXiv | MR
A. Anatolii Aleksandrovich Kilbas, Hari M. Srivastava and Juan J. Trujillo. Theory and applications of fractional differential equations. In Vol. 204. Elsevier Science Limited (2006). | MR | Zbl
K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. Wiley New York (1993). | MR | Zbl
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives: Theory and applications. Gordon and Breach, Yverdon (1993). | MR | Zbl
R.L. Wheeden and A. Zygmund, Measure and integral. An introduction to real analysis. CRC Press (1977). | MR | Zbl
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