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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)
Abstract:
In 1991, Borwein and Borwein proved [1] 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 [2] on the proof, showing how to extract a uniform and computable rate of metastability for the above family of sequences.
References:
[1] D. Borwein, J. Borwein, Fixed point iterations for real functions. J. Math. Anal. Appl. 157, no. 1, 112–126, 1991.
[2] A. Sipoș, Rates of metastability for iterations on the unit interval. arXiv:2008.03934 [math.CA], 2020.