Geodesic
Approximation and Interpolation by Entire Functions of Several Variables
Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 11-22

Voir la notice de l'article provenant de la source Cambridge University Press

Let $f\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be ${{C}^{\infty }}$ and let $h\,:\,{{\mathbb{R}}^{n}}\,\to \,\mathbb{R}$ be positive and continuous. For any unbounded nondecreasing sequence $\{{{c}_{k}}\}$ of nonnegative real numbers and for any sequence without accumulation points $\{{{x}_{m}}\}$ in ${{\mathbb{R}}^{n}}$ , there exists an entire function $g\,:\,{{\mathbb{C}}^{n}}\,\to \,\mathbb{C}$ taking real values on ${{\mathbb{R}}^{n}}$ such that $$\left| {{g}^{\left( \alpha\right)}}\left( x \right)-{{f}^{\left( \alpha\right)}}\left( x \right) \right|\text{ }<h\left( x \right),\left| x \right|\ge {{c}_{k}},\left| \alpha\right|\le k,k=0,1,2,...,$$ $${{g}^{\left( \alpha\right)}}\left( {{x}_{m}} \right)\,=\,{{f}^{\left( \alpha\right)}}\left( {{x}_{m}} \right),\,\,\,\left| {{x}_{m}} \right|\,\ge \,{{c}_{k}},\,\left| \alpha\right|\,\le \,k,\,m,\,k\,=\,0,\,1,\,2,\,.\,.\,.\,.$$ This is a version for functions of several variables of the case $n\,=\,1$ due to $L$ . Hoischen.
DOI : 10.4153/CMB-2010-006-4
Mots-clés : 32A15, entire function, complex approximation, interpolation, several complex variables 
Burke, Maxim R. Approximation and Interpolation by Entire Functions of Several Variables. Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 11-22. doi: 10.4153/CMB-2010-006-4
@article{10_4153_CMB_2010_006_4,
     author = {Burke, Maxim R.},
     title = {Approximation and {Interpolation} by {Entire} {Functions} of {Several} {Variables}},
     journal = {Canadian mathematical bulletin},
     pages = {11--22},
     year = {2010},
     volume = {53},
     number = {1},
     doi = {10.4153/CMB-2010-006-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2010-006-4/}
}
TY  - JOUR
AU  - Burke, Maxim R.
TI  - Approximation and Interpolation by Entire Functions of Several Variables
JO  - Canadian mathematical bulletin
PY  - 2010
SP  - 11
EP  - 22
VL  - 53
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2010-006-4/
DO  - 10.4153/CMB-2010-006-4
ID  - 10_4153_CMB_2010_006_4
ER  - 
%0 Journal Article
%A Burke, Maxim R.
%T Approximation and Interpolation by Entire Functions of Several Variables
%J Canadian mathematical bulletin
%D 2010
%P 11-22
%V 53
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2010-006-4/
%R 10.4153/CMB-2010-006-4
%F 10_4153_CMB_2010_006_4

[Ca] Carleman, T., Sur un théorĕme de Weierstrass. Ark. Mat. Astronom. Fys. 20B(1927), 1–5. Google Scholar

[En] Engelking, R., General topology. Second edition, Sigma Series in Pure Mathematics 6, Heldermann Verlag, Berlin, 1989. Google Scholar

[FG] Frih, E. M. and Gauthier, P. M., Approximation of a function and its derivatives by entire functions of several variables. Canad. Math. Bull. 31(1988), no. 4, 495–499. Google Scholar

[GP] Gauthier, P. M. and Pouryayevali, M. R., Covering properties of most entire functions on Stein manifolds. Comput. Methods Funct. Theory 5(2005), no. 1, 223–235. Google Scholar

[GF] Grauert, H. and Fritsche, K., Several complex variables. Graduate Texts in Mathematics 38, Springer-Verlag, New York-Heidelberg, 1976. Google Scholar

[H1] Hoischen, L., Eine Verschärfung eines approximationssatzes von Carleman. J. Approximation Theory 9(1973), 272–277. doi:10.1016/0021-9045(73)90093-2 Google Scholar

[H2] Hoischen, L., Approximation und Interpolation durch ganze Funktionen. J. Approximation Theory 15(1975), no. 2, 116–123. doi:10.1016/0021-9045(75)90121-5 Google Scholar

Cité par Sources :