R.J.Nessel,E.van Wickeren.关于Dini-Lipschitz型逼近定理的统一处理方法方法(英文)[J].数学研究及应用,1984,4(3):137~152
关于Dini-Lipschitz型逼近定理的统一处理方法方法(英文)
An Unified Approach to Approximation Theorems of Dini-Lipschitz-Type
投稿时间:1983-07-12  
DOI:10.3770/j.issn:1000-341X.1984.03.027
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R.J.Nessel Lehrstuhl A für Mathematik RWTH Aachen Templergraben 55 D-5100 Aachen, Germany 
E.van Wickeren Lehrstuhl A für Mathematik RWTH Aachen Templergraben 55 D-5100 Aachen, Germany 
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      This survey paper studies the approximation of (polynomial) processes for which the operator norms do not form a bounded sequence. In view of familiar direct estimates and quantitative uniform boundedness principles, a unified approach is given to results concerning the equivalence of Dini-Lipschitz-type conditions with (strong) convergence on (smoothness) classes. Emphasis is laid upon the necessity of these conditions, essential ingredients of the proofs are suitable modifications of the familiar gliding hump method. Apart from the classical results concerned with Fourier partial sums, explicit applications are treated for (trigonometric as well as algebralc) Lagrange interpolation, interpolatory quadrature rules based upon Jacobl knots, multipliers or strong convergence, and for Bochner-Riesz means of multivariate Fourier series for parameter values below the critical index.
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