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.
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.