Approximation and Interpolation by Entire Functions of Several Variables

Burke, Maxim R.

Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 11-22

Résumé

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.

Détail

DOI : 10.4153/CMB-2010-006-4

Mots-clés : 32A15, entire function, complex approximation, interpolation, several complex variables