Acta mathematica scientia,Series A ›› 2009, Vol. 29 ›› Issue (4): 833-842.

• Articles •     Next Articles

Relative Widths of Function Classes of L2(T) Determined by Fractional Order Derivatives in Lq(T)

  

  1. (1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875 2. College of Sciences, North China University of Technology, |Beijing 100144)
  • Received:2007-10-29 Revised:2009-03-25 Online:2009-08-25 Published:2009-08-25
  • Supported by:

    国家自然科学基金(10771016)、北京师范大学``985项目"和北京自然科学基金(1062004)资助

Abstract:

The relative widths Kn(W2α(T), MW2β(T), L2(T)),  T=[0, 2π], is studied and the smallest number M which makes the equality Kn(W2α(T), MW2β(T), L2(T))=dn(W2α(T), L2(T)) valid is obtained, and the asymptotic order of relative widths Kn(W2α(T), W2α(T), Lq(T) ) is obtained, where αβ>0, 1≤q ≤ ∞ , Kn (., ., Lq(T)) and dn(., Lq(T)) denote respectively the relative widths and the widths in the sense of Kolmogorov in Lq(T), and MWpα(T), 1≤ p ≤ ∞ , denotes the collection of 2π-periodic and continuous functions f representable as a convolution f(t)=c+(Bα* g)(t), where Bα* g denotes the convolution of Bα and g, for g ∈ Lp(T) satisfying ∫02πg(τ)dτ=0 and ||g||≤ M. Here Bα is in L1(T) with the Fourier expansion
Bα(t) )=1/2π ∑' k∈ Z(ik)-αeikt, where ∑means that  the term is omitted  when k=0.

Key words: Relative widths, n-K  widths, Derivatives of fractional order

CLC Number: 

  • 41A46
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