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Best Uniform Rational Approximations of Functions by Orthoprojections
| Content Provider | Semantic Scholar |
|---|---|
| Author | Pekarskii, Alexandr Antonovich |
| Copyright Year | 2004 |
| Abstract | AbstractLet C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$ |
| Starting Page | 200 |
| Ending Page | 208 |
| Page Count | 9 |
| File Format | PDF HTM / HTML |
| DOI | 10.1023/B:MATN.0000036758.61603.90 |
| Volume Number | 76 |
| Alternate Webpage(s) | https://page-one.springer.com/pdf/preview/10.1023/B:MATN.0000036758.61603.90 |
| Alternate Webpage(s) | https://doi.org/10.1023/B%3AMATN.0000036758.61603.90 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
