Subspace condition for Bernstein's lethargy theorem

Asuman Güven AKSOY; Monairah AL-ANSARI; Caleb CASE; Qidi PENG

Article Details

Subspace condition for Bernstein's lethargy theorem

Authors:

Asuman Güven AKSOY | Monairah AL-ANSARI | Caleb CASE | Qidi PENG

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2017 Vol.41 Issue 5, pp.1101-1107

Document Type:

Article

Publication Date:

9/2017

Document Language:

Turkish

Document ID (Org No.):

2000745986

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Abstract:

 

In this paper, we consider a condition on subspaces in order to improve bounds given in  Bernstein's lethargy theorem  for Banach spaces.  Let d1≥d2≥…dn≥⋯>0d1≥d2≥…dn≥⋯>0d_1 \geq d_2 \geq \dots d_n \geq \dots > 0 be an infinite sequence of numbers  converging to 000, and let Y1⊂Y2⊂⋯⊂Yn⊂⋯⊂XY1⊂Y2⊂⋯⊂Yn⊂⋯⊂XY_1 \subset Y_2 \subset \dots\subset Y_n \subset \dots \subset X be a sequence of  closed nested subspaces in a Banach space XXX  with the property that  ¯¯¯¯Yn⊂Yn+1Y¯n⊂Yn+1\overline{Y}_{n}\subset Y_{n+1}  for all  n≥1n≥1n\ge1. We prove that  for any cın(0,1]cın(0,1]c ın (0,1] there exists an element xcınXxcınXx_c ın X such thatcdn≤ρ(xc,Yn)≤min(4,~a)cdn.cdn≤ρ(xc,Yn)≤min(4,a~)cdn. c d_n  \leq \rho(x_c, Y_n)  \leq \min (4, \tilde{a}) c\, d_n.Here, ρ(x,Yn)=ınf{||x−y||:yınYn}ρ(x,Yn)=ınf{||x−y||:yınYn}\rho(x, Y_n)= ınf \{ ||x-y||: \,\,yın Y_n\},  ~a=supi≥1sup{qi}{a−3ni+1−1}a~=supi≥1sup{qi}{ani+1−1−3}\tilde{a} =\sup_{i\ge1}\sup_{\left \{ q_{i}  \right \}}\left \{  a_{n_{i+1}-1}^{-3}\right \}  where the sequence {an}{an}\{a_n\}  is defined as: for all n≥1n≥1 n \geq 1 ,an=ınfl≥nınfqın⟨ql,ql+1,…⟩ρ(q,Yl)||q||an=ınfl≥nınfqın⟨ql,ql+1,…⟩ρ(q,Yl)||q||a_n = ınf_{l \geq n} \,  ınf_{q ın \langle q_l, q_{l+1},\dots \rangle} \frac{\rho(q,Y_l)}{||q||}in which each point qnqnq_n is taken from  Yn+1∖YnYn+1∖YnY_{n+1} \setminus Y_{n}, and satisfies  ınf?imitsn≥1an>0ınf?imitsn≥1an>0ınf?imits_{n\ge1} a_n  > 0. The sequence  {ni}i≥1{ni}i≥1\{n_i\}_{i\ge1} is given byn1=1;ni+1=min{n≥1:dna2n?eqdni},i≥1.n1=1;ni+1=min{n≥1:dnan2?eqdni},i≥1.n_1=1; n_{i+1}= \min \left \{ n\ge1 : \frac{d_n}{{a_{n}^{2}}} ?eq d_{n_{i}}\right \}, i\geq 1.

Article Details

Subspace condition for Bernstein's lethargy theorem

Authors: Asuman Güven AKSOY | Monairah AL-ANSARI | Caleb CASE | Qidi PENG

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2017 Vol.41 Issue 5, pp.1101-1107

Document Type: Article

Publication Date: 9/2017

Document Language: Turkish

Document ID (Org No.): 2000745986

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Issues

Authors:

Asuman Güven AKSOY | Monairah AL-ANSARI | Caleb CASE | Qidi PENG

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2017 Vol.41 Issue 5, pp.1101-1107

Document Type:

Article

Publication Date:

9/2017

Document Language:

Turkish

Document ID (Org No.):

2000745986

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098