The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Extremal values of a class of functionals under restrictions on moments

Translated from Matematicheskie Zametki, Vol. 55, No. 5, pp. 130–141, May, 1994.     

Morse-Bott functions with two critical values on a surface

Czechoslovak Mathematical Journal - We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit...     

Ranges of certain systems of functionals in a class of functions,convex in a certain direction

One finds the range of systems of functionals on classes of univalent functions which are convex in a certain direction.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 51–58, 1981.     

On subclasses of the caratheodory class with matricial extremal property *

The paper can be considered as continuation of the former investigations on inverse problems of entropy optimization (see [FFK 1]. [FFK 21). Two types of a matricial extremal property for matrix-valued Carathéodory functions are studied. The results are interpreted in the context of prediction theory     

Properties of delta functions of a class of observables on white noise functionals

Let δa be the Dirac delta function at a∈R and (E)⊂(L²)⊂^∗(E) the canonical framework of white noise analysis over white noise space (E^∗,μ), where E^∗=S^∗(R). For h∈H=L²(R) with h≠0, denote by Mh the operator of multiplication by Wh=〈⋅,h〉 in (L²). In this paper, we first show that Mh is δa-composable. Thus the delta function δa(Mh) makes sense as a generalized operator, i.e. a continuous linear operator from (E) to ^∗(E). We then establish a formula showing an intimate connection between δa(Mh) as a generalized operator and δa(Wh) as a generalized functional. We also obtain the representation of δa(Mh) as a series of integral kernel operators. Finally we prove that δa(Mh) depends continuously on a∈R.     

Sets of values of equivalent almost periodic functions

The Ramanujan Journal - This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely...     

Integrable Hamiltonian systems with two degrees of freedom associated with holomorphic functions

We focus on integrable systems with two degrees of freedom that are integrable in the Liouville sense and are obtained as real and imaginary parts of a polynomial (or entire) complex function in two complex variables. We propose definitions of the actions for such systems (which are not of the Arnol'd-Liouville type). We show how to compute the actions from a complex Hamilton-Jacobi equation and apply these techniques to several examples including those recently considered in relation to perturbations of the Ruijsenaars-Schneider system. These examples introduce the crucial problem of the semiclassical approach to the corresponding quantum systems. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 204–210, February, 2000.     

Sets of pairs of values omitted by univalent functions

This article proves covering theorems for classes of univalent functions that omit given pairs of values. In particular, a problem posed by D. A. Brannan [R. Zh. Mat. 1978, 2B63] is solved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 140–145, 1990.     

Normal functions and a class of associated boundary functions

Letμ′ be the family of non-empty closed subsets of the Riemann sphere and Λ the family of continuous curves λ with values in the unit disk and lim _t→1 |λ(t)|=1. A meromorphic functionf in |z|<1 induces a mapping\(\hat f\) from Λ intoμ′ by setting\(\hat f\left( \lambda \right)\) equal to the cluster set off on λ. The authors show that if\(\hat f\) is continuous then existence of an asymptotic value ate ^iθ implies the existence of an angular limit. Further if the spherical derivative off iso(1/(1?|z|)) then\(\hat f\) is constant on every open disk in the space Λ.     

On the integrability of the ergodic hilbert transform for a class of functions with equal absolute values

It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable functionf such that the ergodic Hilbert transform of any function equal in absolute values tof is non-integrable.     

Order estimates of the norms of derivatives of functions with zero values on linear functionals and their applications

For a given finite set of linear functionals we construct functions vanishing on them and give order estimates of their derivatives. We also give their different applications.