Multivariate prediction, de Branges spaces, and related extension and inverse problems

416 Pages·2018·2.342 MB·English

by  Arov, Damir Z., Dym, Harry

Operator Theory
Advances and Applications
266
Damir Z. Arov
Harry Dym
Multivariate Prediction,
de Branges Spaces, and
Related Extension and
Inverse Problems

Operator Theory: Advances and Applications
Volume 266
Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Albrecht Böttcher (Chemnitz, Germany)
Harry Dym (Rehovot, Israel)
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More information about this series at http://www.springer.com/series/4850
Damir Z. Arov • Harry Dym
Multivariate Prediction, de
Branges Spaces, and Related
Extension and Inverse
Problems
Damir Z. Arov Harry Dym
Institute of Physics and Mathematics Department of Mathematics
South Ukrainian National Pedagogical University Weizmann Institute
Odessa, Ukraine Rehovot, Israel
ISSN 0255-0156 ISSN 2296-4878 (electronic)
Operator Theory: Advances and Applications
ISBN 978-3-319-70261-2 ISBN 978-3-319-70262-9 (eBook)
https://doi.org/10.1007/978-3-319-70262-9

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This monograph is dedicated to our principal teachers
Vladimir Petrovich Potapov, Mark Gregorovich Krein;
Edward Arthurs, Henry McKean and Israel Gohberg;
and to our students and collaborators,
who were also our teachers.
Preface
Thismonographextendsanumberoftheresultsfortheproblemofpredictingthe
future based on a finite segment of the past that were presented in [Kr54] and
[DMc76] for scalar weakly stationary processes and scalar processes with weakly
stationary increments to a class of p-dimensional weakly stationary processes and
a class of p-dimensional processes with weakly stationary increments.
The solution of prediction problems by identifying the spectral function of
the process with the spectral function of a canonical of integral or differential
system and solving the inverse spectral problem for that system is in the spirit of
anapproachinitiatedbyM.G.Kreinforscalarprocesses.InashortDokladynote
[Kr54] without proofs, Krein proposed a strategy for computing the orthogonal
projectionoff ∈L2(dσ)(i.e.,Lp2(dσ)withwithp=1)ontosubspacesoftheform
eitλ−1
Z[−a,a](dσ)=cls{ft : −a≤t≤a} in L2(dσ) for ft(λ)= λ
and nondecreasing functions σ on R that are subject to the constraint
(cid:2)
∞
(1+μ2)−1dσ(μ)<∞.
−∞
Krein identified σ as the spectral function of a generalized string equation (that
later became known as the Feller–Krein string equation). In particular, subspaces
ofL2(dσ)wereidentifiedintermsofapairoftransformsthatareanaloguesofthe
classicalcosineandsinetransforms.Thisleadstoaclassificationofsubspacesthat
is finer than that which is obtained from the set of subspaces Z[−a,a](dσ). Krein’s
program was completed some twenty years later in the monograph [DMc76] for
absolutely continuous spectral functions σ with even density σ(cid:3)(μ) = Δ(μ) =
Δ(−μ). In that monograph extensive use was made of scalar de Branges spaces
andafundamentalorderingtheoremduetodeBranges.Thesamegeneralstrategy
(based on de Branges spaces and the identification of the spectral density of a
process with the spectral density of a differential system) was considered earlier
for a restricted version of this problem in [DMc70a], before Loren Pitt called
the authors attention to the existence of Krein’s Doklady note; see, e.g., [Dy00]
for historical remarks. Additional information related to the underlying theory of
vii
viii Preface
strings may be found in the papers [KaKr74a] and [KaKr74b] of I.S. Kac and
M.G. Krein. The strategy proposed by Krein was also recently completed in the
paper [KrL14] by M.G. Krein and H. Langer that also made use of de Branges’
uniqueness theorem.
The classical problem of predicting the future based on the full past for
vector-valued weakly stationary processes will also be discussed.
The classes of vector processes under consideration in this monograph are
restricted to have locally absolutely continuous spectral functions with spectral
densities Δ(μ) that are subject to the Szeg˝o condition
(cid:2)
∞ |lndet{Δ(μ)}|
dμ<∞.
1+μ2
−∞
The main focus is on the analytic counterpart of these problems, which
amountstocomputingprojectionsontosubspacesofHilbertspacesH=Lp(Δ)of
2
p×1 vector-valued functions with inner product
(cid:2)
∞
(cid:5)f,g(cid:6)H = g(μ)∗Δ(μ)f(μ)dμ
−∞
based on an appropriately restricted p×p matrix-valued functions Δ(μ).
Extensive use will be made of:
(1) The properties of the resolvent matrices of matrix versions of a number of
extension problems that were first formulated and studied by M.G. Krein.
(2) ThetheoryofreproducingkernelHilbertspacesofentirevector-valuedfunc-
tions that originate with L. de Branges.
(3) TheinversespectralproblemforcanonicalintegralsystemsandKrein–Dirac
differential systems.
Therelevanttheoryissurveyedinthetext.Additionalinformation,references
and proofs of those results that will be presented without proof may, for the most
part, be found in the two monographs [ArD08] and [ArD12]. En route, a number
of results that were (to the best of our knowledge) only known for scalar-valued
functions (i.e., for p = 1) were extended to the setting of p × 1 vector-valued
functions with p > 1. Nevertheless, we have tried to avoid excessive generality in
order to make the text easily accessible.
It is perhaps of minor historical interest to note that the authors began to
consider multivariate prediction problems based on a finite segment of the past
when they first started to work together in the autumn of 1992. However, after
some initial dabbling, they concluded that, in order to make progress, it was
necessary to develop a deeper understanding of the relevant theory of extension
problems, vector-valued de Branges spaces and inverse spectral problems. That
detour took almost twenty years.
Preface ix
The authors gratefully acknowledge and thank the administration of South
Ukranian National Pedagogical University for authorizing extended leaves of ab-
sence to enable the first author to visit the second; and finally, and most impor-
tantly, the Weston and Belkin Visiting Professorship programs of the Weizmann
Institute, and the Belfer and Pnueli funds (administered by the Dean of the Fac-
ultyofMathematicsandComputerScience),forthefinancialsupportwhichmade
these visits possible and enabled the authors to work together under ideal condi-
tions.
The authors also extend their thanks to Dr. Andrei Iacob for a superb job
of copy editing the manuscript, and for his help with the preparation of the index
and in organizing the list of references in a uniform style. The authors thank the
typesetter for his expert work.
Last, but very much not least, special thanks to our very significant others,
NatashaandIrene,fortheircontinuedsupportandencouragementandforputting
up (mostly) with late meals and distracted spouses.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Organization of the monograph . . . . . . . . . . . . . . . . . . 2
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 de Branges matrices E and spaces B(E) . . . . . . . . . . . . . 5
1.4 Some basic identifications . . . . . . . . . . . . . . . . . . . . . 8
1.5 Direct and inverse spectral problems . . . . . . . . . . . . . . . 9
1.6 J -inner mvf’s and de Branges matrices . . . . . . . . . . . . . 12
p
1.7 Helical extension problems . . . . . . . . . . . . . . . . . . . . 14
1.8 Positive extension problems . . . . . . . . . . . . . . . . . . . . 16
1.9 Accelerant extension problems . . . . . . . . . . . . . . . . . . 18
1.10 Inverse spectral problems for Krein systems . . . . . . . . . . . 19
1.11 Prediction for vector processes . . . . . . . . . . . . . . . . . . 20
1.12 Supplementary notes . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Analytic Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Basic classes of mvf’s and related theorems . . . . . . . . . . . 27
2.2 J-inner mvf’s and related transformations . . . . . . . . . . . . 37
2.3 Some subclasses of U(J) . . . . . . . . . . . . . . . . . . . . . . 40
2.4 The Carath´eodory class . . . . . . . . . . . . . . . . . . . . . . 45
2.5 The Stieltjes class . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 The classes Gp×p(0) and Gp×p(0) . . . . . . . . . . . . . . . . . 52
∞ a
2.7 The classes Pp×p and Pp×p . . . . . . . . . . . . . . . . . . . . 55
∞ a
2.8 The classes A˚p×p and A˚p×p . . . . . . . . . . . . . . . . . . . . 58
∞ a
2.9 Supplementary notes . . . . . . . . . . . . . . . . . . . . . . . . 63
xi