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A proof mining case study on the unit interval
March 18, 2021 @ 10:00 am - 12:00 pm
Title: A proof mining case study on the unit interval
Speaker: Andrei Sipoș (University of Bucharest & IMAR)
In 1991, Borwein and Borwein proved  the following: if L>0, f : [0,1] → [0,1] is L-Lipschitz, (xn), (tn) ⊆ [0,1] are such that for all n, xn+1=(1-tn)xn+tnf(xn) and there is a δ>0 such that for all n, tn≤(2-δ)/(L+1), then the sequence (xn) converges.
The relevant fact here is that the main argument used in their proof is of a kind that hasn’t been analyzed yet from the point of view of proof mining, and thus it may serve as an illustrative new case study. We shall present our work  on the proof, showing how to extract a uniform and computable rate of metastability for the above family of sequences.
 D. Borwein, J. Borwein, Fixed point iterations for real functions. J. Math. Anal. Appl. 157, no. 1, 112–126, 1991.
 A. Sipoș, Rates of metastability for iterations on the unit interval. arXiv:2008.03934 [math.CA], 2020.