Extremal L1 problem for entire functions and spectral theory

In the late 80`s almost periodic operators were one of the centers of attention of the mathematical community. Those operators serve in solid state physics as model of disordered systems, such as alloys, glasses and amorphous materials. One of the key results in the subject is the Kotani Theorem. It claims quite a specific spectral property of almost periodic Jacobi matrices: on the support of the absolutely continuous spectrum the operator should be reflectionless. Note that Sodin--Yuditskii under the assumption of regularity of the spectral set E with respect to the Lebesgue measure (the set should be homogeneous) proved the opposite statement: every Jacobi matrix, which is reflectionless on E, is almost periodic. A recent break-through result of Remling generated a new wave of interest in reflectionless operators. His theorem claims that each right limit of a Jacobi matrix is reflectionless on the support of the absolutely continuous spectrum of the initial operator. To indicate the importance of this result it is enough to say that in the simplest case when E is a single interval this theorem becomes the celebrated Rakhmanov theorem. In combination with our result it shows that if E is homo-geneous then each limit point is almost periodic. Let us point out that Remling`s theorem does not require any assumption on E. Thus the following fundamental question arises: investigate properties of the right limits depending on the spectral set E beyond the homogeneous case. For instance Poltoratski and Remling recently proved that every reflectionless Jacobi matrix has purely absolutely continuous spectrum if E is weakly homogeneous. Our previous result stated that such matrices have absolutely continuous spectrum if the resolvent domain is of Widom type and the Direct Cauchy Theorem (DCT) holds on it. Now we constructed a Widom type domain such that a reflectionless matrix may have a singular continuous spectral component. We want to start our studies with the case when the set E consists of a family of intervals accumulating to the lower bound of E and with the question: find a geometric or analytic characterization of the set E such that the domain is of Widom type and the DCT holds on it. We are able to show that the DCT in such a case is intimately related with an L1 extremal problem for entire functions with respect to the Lebesgue measure restricted to E. We give a precise setting of this problem on a modern level, which generalizes the classical one for polynomials and entire functions of a given exponential type. Similar to the polynomial case we plan to reduce the L1 extremal problem to the question on orthogonalization of entire functions with respect to a special family of (reflectionless) weights. All these studies continue a very classical line of investigations in Analysis (Chebyshev problems, weighted polynomial approximations, spectral theory of canonical systems) and, we believe, are of high independent interest.

Classical extremal problems for polynomials and entire functions form an essential core of approximation theory and have numerous connections with diverse problems in analysis, and mathematical physics (in particular, with spectral theory). The earliest version of the results we were interested in is the famous Korkin-Zolotarev theorem. Its various generalizations were given by Stieltjes, Markov, Posse, Bernstein, Akhiezer and Krein. Having this classical background, we dramatically extended the setting of the problem. The main point deals with the involving of the theory of Hardy spaces in infinitely connected domains in the studies, especially the concept of Widom domains. This problem was solved completely and explicit formulas for the solutions were given if the domain had a certain additional specific property: the so-called Direct Cauchy Theorem (DCT) holds. Moreover, we obtained a completely new characterization of Widom domains with DCT in terms of related canonical systems. If DCT failed we found two kinds of Hardy spaces, which form natural upper and lower boundaries for all possible character automorphic Hardy spaces on surfaces of Widom type with a given character. This technique was developed for a further application in the spectral theory of ergodic and almost periodic Jacobi matrices and 1-D Schrödinger operators. Those operators serve in solid-state physics as model of disordered systems, such as alloys, glasses and amorphous materials. In particular, we clarified relations between the following three important isospectral classes of reflectionless Jacobi matrices: a) its common spectral set is homogeneous in the Carleson sense b) all operators have purely absolutely continuous spectrum, c) DCT holds in the common resolvent domain. We developed a general theory of comb functions on a modern level. On its base we obtained precise asymptotics of the error of the best polynomial approximation of a piecewise constant function on two arbitrary intervals (the problem was posed independently by W. Hayman and H. Stahl); solved some problems on a complex non-analytic best uniform polynomial approximation in the 2-D ball; gave a parametric description of the spectral sets of periodic 5-diagonal matrices related to the strong moment problem. Some other problems in approximation theory and spectral theory were studied in the project, including the disproval of a Widom conjecture. The project covers basically purely mathematical aspects, nevertheless one can mention that our results in uniform approximation had certain impact e.g. in computer science: the finding of an efficient approximation scheme in the so-called approximate inclusion-exclusion problem. Probably, the main current application of the project results inside the field deals with solving the Kotani-Last problem, as well as an essential progress in solving the Killip-Simon problem. General significance of the area was confirmed by the recent decision of the International Mathematical Union: the Fields Medal was awarded to A. Avila in particular for his contribution in the spectral theory of 1-D Schrödinger operators.