Approximation of a function and its derivatives by entire functions (Q2788791)
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scientific article; zbMATH DE number 6543542
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Approximation of a function and its derivatives by entire functions |
scientific article; zbMATH DE number 6543542 |
Statements
22 February 2016
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Weierstrass approximation theorem
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Carleman theorem
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entire function
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Approximation of a function and its derivatives by entire functions (English)
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Let \(m\) be a non-negative integer, \(I=(a,b)\) (\(-\infty\leq a\leq b\leq\infty\)). Denote by \(H(U)\) the family of functions holomorphic on \(U\), and denote by \(C^+(I)\) the positive continuous functions on \(I\). In the paper, the following result is proved. If \(f\in C^{(m)}(I)\), and \(\epsilon\in C^+(I)\), then there exists a function \(g\in H(\mathbb{C}\setminus I^c)\) such that \(|f^{(i)}(x)-g^{(i)}(x)|<\epsilon(x)\), \(x\in I\), \(i=0,1,\dots,m\).
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