## Lectures on Theory of Approximation |

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### Contents

Introduction | 1 |

Existence of Best Approximants | 8 |

Uniqueness and Characterization of Best Approximants | 20 |

14 other sections not shown

### Common terms and phrases

2n+l algebraic polynomials alternate e-points analytic approximation error assume asymptotic Banach space Bernstein polynomials best approximant changes sign Chapter characterization class ft class of functions coefficients consider constant continuous functions converge convex set COROLLARY D(ft defined denote derivative elements entire function equidistant nodes example exists expansion Fejer formula Fourier series Fourier sums function x function x e hence Hermite interpolation inequality integral interpolating property interpolation operator Lagrange interpolation Lemma linear manifold linear operator lower bound minimal error modulus of continuity n-th obtains orthogonal periodic functions polynomial of degree proof is complete proof of Theorem proved REMARKS S. N. Bernstein satisfies sequence sinh strictly convex subset t-t t-t Tchebycheff polynomials Theorem 9.1 trigonometric interpolation trigonometric polynomial uniformly upper bound x e ft xeft Zygmund