## Fundamentals of Approximation TheoryThe field of approximation theory has become so vast that it intersects with every other branch of analysis and plays an increasingly important role in applications in the applied sciences and engineering. Fundamentals of Approximation Theory presents a systematic, in-depth treatment of some basic topics in approximation theory designed to emphasize the rich connections of the subject with other areas of study. With an approach that moves smoothly from the very concrete to more and more abstract levels, this text provides an outstanding blend of classical and abstract topics. The first five chapters present the core of information that readers need to begin research in this domain. The final three chapters the authors devote to special topics-splined functions, orthogonal polynomials, and best approximation in normed linear spaces- that illustrate how the core material applies in other contexts and expose readers to the use of complex analytic methods in approximation theory. Each chapter contains problems of varying difficulty, including some drawn from contemporary research. Perfect for an introductory graduate-level class, Fundamentals of Approximation Theory also contains enough advanced material to serve more specialized courses at the doctoral level and to interest scientists and engineers. |

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### Contents

Density Theorems | 1 |

Linear Chebyshev Approximation | 33 |

Degree of Approximation | 101 |

Interpolation | 155 |

Fourier Series | 203 |

Spline Function | 231 |

Orthogonal Polynomials | 321 |

Best Approximation in Normed Linear Spaces | 369 |

515 | |

535 | |

### Common terms and phrases

AK(X algorithm assume B-splines Banach space bounded called Cent v(F Chebyshev polynomials CLB(X CLC(X closed coefficients compact subset completes the proof constant continuous function continuous selection contradicts convergent convex set convex subset Corollary crit data nodes defined Definition denote distinct points distinct zeros equations exists Ext U(X finite dimensional following theorem Fourier series given Haar condition Haar subspace Hausdorff Hausdorff space Hence Hermite interpolation implies inequality integer interval Lemma Let f linear subspace lower semicontinuity matrix metric projection multifunction n-dimensional subspace nonempty normed linear space normed space observe obtain optimal orthogonal polynomials poised proof of Theorem Proposition prove proximinal PV(X Py(X rad v(F reader resp result satisfying Section sequence smooth spline interpolant statements are equivalent Suppose topological trigonometric polynomials uniformly unique element unique solution

### Popular passages

Page 515 - Stosow. 12, 43-63 (Polish) (1978) 453.68003 Bojamc R., An estimate of the rate of convergence for Fourier series of functions of bounded variation. Publ. Inst. Math..