## A descent approach to the Chebyshev approximation problem |

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### Contents

CLASSICAL TECHNIQUES FOR THE CHEBYSHEV | 6 |

DEVELOPMENT OF A DESCENT ALGORITHM | 20 |

ASSESSMENT AND SUGGESTIONS FOR FURTHER | 49 |

2 other sections not shown

### Common terms and phrases

absol absolute value alternation theorem approx approximation to f(t approximation with Chebyshev attained best approximation best Chebyshev approximation Calculate the best changes sign characterization Chebyshev approximation problem Chebyshev norm Chebyshev problem Chebyshev sets continuous function convergence convex functions critical point set decreased defined derivative d/da descent algorithm descent approach descent mapping developed Dg(a direction AA direction of steepest downhill edge direction e(AQ+ e(aQ+Aa,t eCA^t eigenvalues eigenvectors equal error curve error function F(Aa function f(t function sgn g(aQ g(aQ+Aa guaranteed hyperplanes imation initial guess interpolation requirement lemma Let f(t linear programming linearly independent Lp norms magnitude maxima maximum deviation n+1 points negative gradient non-Chebyshev sets parameter space point Aq programming rate of convergence reduce the norm region set of n+1 solution point steepest descent step direction strictly convex subintervals subset subspace Suppose teF(Aa teT r(t three algorithms Tk+1 vertex to vertex whole interval