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Taut Strings and Real Interpolation
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The taut string problem concerns finding the function with the shortest graph length, i.e. the taut string, in a certain set  of continuous piecewise linear functions. It has appeared in a broad range of applications including statistics, image processing and economics. As it turns out, the taut string has besides minimal graph length also minimal energy and minimal total variation among the functions in Ω.

The theory of real interpolation is based on Peetre’s K-functional. In terms of the K-functional, we introduce invariant K-minimal sets and show a close connection between taut strings and invariant K-minimal sets.

This insight leads to new problems of interpolation theory, gives possibility to generalize the notion of taut strings and provides new applications.

The thesis consists of four papers. In paper I, connections between invariant K-minimal sets and various forms of taut strings are investigated. It is shown that the set Ω′ of the derivatives of the functions in  can be interpreted as an invariant K-minimal set for the Banach couple (ℓ1, ℓ) on Rn. In particular, the derivative of the taut string has minimal K-functional in Ω′. A characterization of all bounded, closed and convex sets in Rn that are invariant K-minimal for (ℓ1, ℓ) is established.

Paper II presents examples of invariant K-minimal sets in Rn for (ℓ1, ℓ). A convergent algorithm for computing the element with minimal K-functional in such sets is given. In the infinite-dimensional setting, a sufficient condition for a set to be invariant K-minimal with respect to the Banach couple L1 ([0,1]m) ,L ([0,1]m) is established. With this condition at hand, different examples of invariant K-minimal sets for this couple are constructed.

Paper III considers an application of taut strings to buffered real-time communication systems. The optimal buffer management strategy, with respect to minimization of a class of convex distortion functions, is characterized in terms of a taut string. Further, an algorithm for computing the optimal buffer management strategy is provided.

In paper IV, infinite-dimensional taut strings are investigated in connection with the Wiener process. It is shown that the average energy per unit of time of the taut string in the long run converges, if it is constrained to stay within the distance r > 0 from the trajectory of a Wiener process, to a constant C2/r2 where C ∈ (0,∞). While the exact value of C is unknown, the numerical estimate C ≈ 0.63 is obtained through simulations on a super computer. These simulations are based on a certain algorithm for constructing finite-dimensional taut strings.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. , 24 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1801
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-132421DOI: 10.3384/diss.diva-132421ISBN: 9789176856499 (Print)OAI: oai:DiVA.org:liu-132421DiVA: diva2:1045576
Public defence
2016-12-02, Nobel BL32, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2016-11-10 Created: 2016-11-10 Last updated: 2016-11-10Bibliographically approved
List of papers
1. Discrete taut strings and real interpolation
Open this publication in new window or tab >>Discrete taut strings and real interpolation
2016 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 270, no 2, 671-704 p.Article in journal (Refereed) Published
Abstract [en]

Classical taut strings and their multidimensional generalizations appear in a broad range of applications. In this paper we suggest a general approach based on the K-functional of real interpolation that provides a unifying framework of existing theories and extend the range of applications of taut strings. More exactly, we introduce the notion of invariant K-minimal sets, explain their connection to taut strings and characterize all bounded, closed and convex sets in R-n that are invariant K-minimal with respect to the couple (l(1), l(infinity)). (C) 2015 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
ACADEMIC PRESS INC ELSEVIER SCIENCE, 2016
Keyword
Taut strings; Real interpolation; Invariant K-minimal sets
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-124087 (URN)10.1016/j.jfa.2015.10.012 (DOI)000366144300007 ()
Available from: 2016-01-25 Created: 2016-01-19 Last updated: 2016-11-10
2. Invariant K-minimal Sets in the Discrete and Continuous Settings
Open this publication in new window or tab >>Invariant K-minimal Sets in the Discrete and Continuous Settings
2016 (English)In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, 1-40 p.Article in journal (Refereed) Epub ahead of print
Abstract [en]

A sufficient condition for a set Ω ⊂ L1([0,1]m to be invariant K-minimal with respect to the couple (L1([0,1]m)), L([0,1]m) is established. Through this condition, different examples of invariant K-minimal sets are constructed. In particular, it is shown that the L1-closure of the image of the L-ball of smooth vector fields with support in (0,1)m, under the divergence operator is an invariant K-minimal set. The constructed examples have finite-dimensional analogues in terms of invariant K-minimal sets with respect to the couple (ℓ1, ℓ) on Rn . These finite-dimensional analogues are interesting in themselves and connected to applications where the element with minimal K-functional is important. We provide a convergent algorithm for computing the element with minimal K-functional in these and other finite-dimensional invariant K-minimal sets.

Place, publisher, year, edition, pages
Springer, 2016
Keyword
Invariant K-minimal sets, Taut strings, Real interpolation
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-132425 (URN)10.1007/s00041-016-9479-5 (DOI)
Available from: 2016-11-10 Created: 2016-11-10 Last updated: 2016-12-01Bibliographically approved
3. Energy of taut strings accompanying Wiener process
Open this publication in new window or tab >>Energy of taut strings accompanying Wiener process
2015 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, Vol. 125, no 2, 401-427 p.Article in journal (Refereed) Published
Abstract [en]

Let W be a Wiener process. For r greater than 0 and T greater than 0 let I-W (T, r)(2) denote the minimal value of the energy integral(T)(0) h(t)(2)dt taken among all absolutely continuous functions h(.) defined on [0, T], starting at zero and satisfying W(t) - r less than= h(t) less than= W(t) + r, 0 less than= t less than= T. The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant C E (0, infinity) such that for any q greater than 0 r/T-1/2 I-W (T, r) -greater than(Lq) C, as r/T-1/2 -greater than 0, and for any fixed r greater than 0, r/(TIW)-I-1/2 (T, r)-greater than(a.s.) C, as T -greater than infinity. Although precise value of C remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of W, we also consider an adaptive version of the problem by giving a construction (called Markovian pursuit) of a random function h(t) based only on the values W(s), s less than= t, and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns out to be related with a classical minimization problem for Fisher information on the bounded interval.

Place, publisher, year, edition, pages
Elsevier, 2015
Keyword
Gaussian processes; Markovian pursuit; Taut string; Wiener process
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-115334 (URN)10.1016/j.spa.2014.09.020 (DOI)000349501200001 ()
Note

Funding Agencies| [RFBR 13-01-00172]; [SPbSU 6.38.672.2013]

Available from: 2015-03-13 Created: 2015-03-13 Last updated: 2016-11-10

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