## Approximation of FunctionsThis is an easily accessible account of the approximation of functions. It is simple and without unnecessary details, but complete enough to include the classical results of the theory. With only a few exceptions, only functions of one real variable are considered. A major theme is the degree of uniform approximation by linear sets of functions. This encompasses approximations by trigonometric polynomials, algebraic polynomials, rational functions, and polynomial operators. The chapter on approximation by operators does not assume extensive knowledge of functional analysis. Two chapters cover the important topics of widths and entropy. The last chapter covers the solution by Kolmogorov and Arnold of Hilbert's 13th problem. There are notes at the end of each chapter that give information about important topics not treated in the main text. Each chapter also has a short set of challenging problems, which serve as illustrations. |

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### Contents

Possibility of Approximation | 1 |

Polynomials of Best Approximation | 16 |

Properties of Polynomials and Moduli of Continuity | 36 |

The Degree of Approximation by Trigonometric Polynomials | 54 |

The Degree of Approximation by Algebraic Polynomials | 65 |

Approximation by Rational Functions Functions of Several Variables | 81 |

Approximation by Linear Polynomial Operators | 92 |

Approximation of Classes of Functions | 111 |

Widths | 132 |

Entropy | 150 |

Representation of Functions of Several Variables by Functions of One Variable | 168 |

179 | |

Index | 187 |

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algebraic polynomials an(f analytic functions arbitrary assume Banach space belongs Bernstein polynomials best approximation bounded Chapter Chebyshev polynomials class Lip coefficients compact Hausdorff complex condition const constant contained continuous derivative continuous function converges convex convex set defined degree of approximation denote differential dimensional dn(A En(f entropy estimate example exceed exists extremal signature finite following theorem formula Fourier series function f given Hausdorff space Hence Ht(A inequality integral interval Kolmogorov Lemma linear combinations linear operators LipM Ln(f metric space modulus of continuity Mp+1 n-tuple norm obtain Pn(x points xk polynomial of best polynomial of degree polynomial operators polynomial Pn positive Problem proof of Theorem prove quasi-smooth rational functions Rn(f rn(x satisfies saturation class sn(f subspace superpositions Theorem 9 tion Tn(x topological space trigonometric polynomial uniformly variables Xn+1