European Mathematical Society - 30e05
https://euro-math-soc.eu/msc-full/30e05
enThe moment problem
https://euro-math-soc.eu/review/moment-problem
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The moment problem, or one should say moment problems (plural) because there are several different classical moment problems. Some ideas can be found in work of Chebyshev and Markov, but Stieltjes at the end of the nineteenth century was one of the first to formally consider the moment problem named after him. Given a sequence of numbers ($m_k$), is there a positive measure $\mu$ such that $m_k=\int x^k \mu(dx), k=0,1,2,\ldots$? In the case of Stieltjes, the measure was supposed to have a support on the positive real line. First of all one wants to find out under what conditions such a measure exists, then when the solution is unique, and when it is not unique to characterize all possible solutions. Soon (around 1920) other versions were formulated by Hamburger (when the support of the measure is the whole real line) and Hausdorff (when the support is a finite interval) and some ten years later the trigonometric moment problem was tackled by Verblunsky, Akhiezer and Krein where the support is the complex unit circle. There is a basic difference between the trigonometric moment problem and the other classical moment problems on (parts of) the real line. In the latter situation, the existence of a solution is guaranteed by requiring the positivity of Hankel matrices whose entries are the moments. In the trigonometric case, the Hankel matrices are replaced by Toeplitz matrices. The latter involve also the moments $m_{-k}=\overline{m}_k, k=1,2,\ldots$ which are automatically matched as well. Not so for the other moment problems. When also imposing moments with a negative index in those cases, this is called a <em>strong</em> moment problem. When only a finite number of moments are prescribed, this is called a <em>truncated</em> moment problem.</p>
<p>
The importance of the moment problem is a consequence of the fact that it is at the crossroads of several branches and applications of mathematics. It relates to linear algebra, functional analysis and operator theory, stochastic processes, approximation theory, optimization, orthogonal polynomials, systems theory, scattering theory, signal processing, probability, and many more. No wonder that the greatest names in mathematics have contributed to the problem with papers and monographs. Because of the many connections to different fields also many approaches and many generalizations have been considered. The previously described moments are called power moments because of the $x^k$, but one could also prescribe moments based on a set of other functions $M_k(x)$. Traditionally, the Hausdorff moment problem is formulated for the interval [0,1], but one may consider any finite interval $[a,b]$ just like the Stieltjes moment problem could be formulated for any half line $[\alpha,\infty)$. Other generalizations lifts these problems to a block version, by assuming that the moments are matrices and the measure is matrix-valued, or the variable $x$ can have several components, resulting in a multivariate moment problem.</p>
<p>
The fact that today, 100 years after Hamburger and Hausdorff, this is still an active research field is another proof of the importance of moment problems. Many books did appear already that were devoted to moment problems or where moment problems played an essential role. Some classics are Shohat and Tamarkin <em>The Problem of Moments</em> (1943), Akhiezer <em>The classical moment problem and some related questions in analysis</em> (1965), Krein and Nudelman <em>The Markov moment problem and extremal problems</em> (1977). The present book is a modern update of the situation. It gives an operator theoretic approach to moment problems, leaving aside the applications. The univariate classical problems of Hamburger ($\mathbb{R}$), Stieltjes ($[0,\infty)$) and Hausdorff ($[a,b]$), appear both in their full and their truncated version. Also the trigonometric moment problem is represented but by only one chapter.<br />
The introduction to these problems is quite general. It is showing how integral representations for linear functionals can be obtained, and in particular how this works for finite dimensional spaces, and for truncated moment problems. Another essential tool is giving some examples of how moment problems can be defined on a commutative *-semigroup. Indeed, all what is needed is a structure with an involution (which could be the identity) and it should allow the definition of a positive definite linear functional so that it can give rise to an inner product on the space of polynomials (and its completion). With gross oversimplification one could say that a sequence is a moment sequence if the associated linear functional is positive and the solution corresponds to the measure that appears in an integral representation of the functional. For real problems, the involution is the identity: $x^*=x$, for complex problems, the involution $x^*=1/\overline{x}$ allows to treat the trigonometric moment problem at the same level as the real moment problems.<br />
This general approach is not really needed for the classical one dimensional moment problems that are treated in part I and the truncated version in part II, but the generality of the introduction allows more easy generalizations to the multivariate case and its truncated version that are discussed in parts III and IV respectively. What is treated in the first two parts are the classical results: the representation of positive polynomials, conditions for the existence of a solution of the moment problem, Hankel matrices, orthogonal polynomials and the Jacobi operator, determinacy (i.e. uniqueness) of the solution, the characterization of all solutions in the indeterminate case, and the relation with complex interpolation problems for Pick functions. For truncated moment problems one may look for some special, so called N-extremal, solutions which lie on the boundary of the solution set, or a canonical solution or solutions that maximize the mass in a particular point of an atomic solution.</p>
<p>
For the multivariate case, it takes some more work and we do not have the classical cases where the measure should be supported and generalizations can go in many different directions. Nevertheless, the corresponding chapters in parts III and IV go through the same steps as in the univariate case as much as possible. What are representing measures and when are polynomials positive? By defining the moment problem for a finitely generated abelian unital algebra, and using a fiber theorem that characterizes moment functionals, some generalizations of the one-dimensional case can be obtained (like for example a rational moment problem) or moment problems on some cubics. Determinacy of the multivariate moment problem is given in the form of a generalized Carleman condition, moments for the Gaussian measure on the unit sphere, and complex one- and two-sided moment problems are all discussed. Characterizing a canonical or extreme solution(s) is not as simple as in the one-dimensional case. Only for the truncated multivariate problems Hankel matrices are introduced and atomic solutions with maximization of a point mass can be characterized.</p>
<p>
The book appears in the series <em>Graduate Texts in Mathematics</em> which means that it is conceived as a as a text that could be used for lecturing with proofs fully included and extra exercises after every chapter as well as notes the refer to the history and the related literature. It is however marvellously capturing the present state of the art of the topic. So it will be also a reference work for researchers. It captures a survey of the univariate case and indicates research directions for the multivariate problem. The list of references at the end of the book has both historical as recent publications, but it is restricted to what has been discussed in the present book. Schmüdgen has published two books before on operator theory, so he knows how to write a book on a difficult subject and still keep it accessible for the audience that he is addressing (graduate students and researchers). Lists of symbols are really helpful to remember notation. The fact that on page 4 Chebyshev and Markov are situated in 1974 and 1984 respectively is just a glitch in an otherwise carefully edited text.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a modern operator approach surveying classical one-dimensional moment problems, but the setting is general by formulating the problem on an abelian *-semigroups. This allows to also capture an introduction to multivariate moment problems which is much more recent and a subject that is still in evolution. The characterization of moment sequences, associated linear moment functionals, and determinate as well as indeterminate problems for the full or the truncated problems are discussed. Particular canonical and N-extremal solutions or solutions with a maximal mass point are discussed.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/konrad-schm%C3%BCdgen" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Konrad Schmüdgen</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/springer-internationa" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Internationa</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-64545-2 (hbk); 978-3-319-64546-9 (ebk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">84,79 € (hbk); 67,82 € (ebk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">535</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319645452" title="Link to web page">http://www.springer.com/gp/book/9783319645452</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/42a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">42A70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/44a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">44A60</a></li></ul></span>Tue, 13 Mar 2018 07:38:36 +0000Adhemar Bultheel48323 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/moment-problem#commentsLarge Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics
https://euro-math-soc.eu/review/large-truncated-toeplitz-matrices-toeplitz-operators-and-related-topics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This book is a volume 259 in the Bikhäuser OT Series <em>Operator Theory Advances and Applications</em>. It contains 30 contributions celebrating Albert Böttcher's 60th birthday.</p>
<p>
Albert Böttcher is a professor of mathematics at the TU Chemnitz in Germany. His main research topic is functional analysis. At his 18th he won the silver medal at the International Math Olympiad in Moscow. He studied mathematics at the TU Karl-Marx-Stadt (now TU Chemnitz) and finished a PhD in 1984 entitled <em>The finite section method for the Wiener-Hopf integral operator</em> under supervision of V.B. Dybin at Rostov-on-Don State University in Russia (this was all while the Berlin wall was still up). Since then he stayed at the TU Chemnitz. At the time of writing he has (co)authored 9 books and over 220 papers. The complete list is in the beginning of the book but one may also consult his <a href="https://www-user.tu-chemnitz.de/~aboettch/" target="_blank">website</a> where he keeps his list of publications up to date.</p>
<p>
The contributions start with reminiscences and best wishes by friends, colleagues and students of Albrecht Böttcher. Besides personal recollections, there is some discussion of his work, some photographs and reproductions of slides he used in presentations to illustrate that he is not only an excellent mathematician but also a passionate teacher and lecturer.</p>
<p>
That leaves about 700 pages of original research papers all of which relate from far or near to subjects that Böttcher has worked on. The Toeplitz operators and Toeplitz matrices of the title are indeed well represented, but there are all the other "Related Topics" which are close to his work too. About fifty renowned authors are involved.</p>
<p>
The Toeplitz operator (and hence also its spectrum) is characterized by a function, which is called its symbol. It features in a multiplication or convolution in the definition of the operator. With respect to a standard monomial basis, Toeplitz operators are represented by (infinite) Toeplitz matrices that have constant entries along diagonals. Of course the spectral and other properties of truncations of the infinite matrices to large finite ones relate to corresponding properties of Toeplitz operators, and similarly it can be related to other operators such as convolution and Wiener-Hopf operators. These matrices and operators have applications in differential and integral equations, systems and control, signal processing, and many more. Depending on the application the symbol may get an interpretation of transfer function of a system, power spectrum or autocorrelation of a signal, the kernel of an integral equation, or just a weight function in a Hilbert space. So, Toeplitz matrices and operators are also related to numerical methods for solving functional equations after discretization. Or to orthogonal polynomials (on the unit circle), which then in turn links to (trigonometric) moment problems, quadrature, and approximation theory (on the unit circle, but in a similar way also to analogs on the real line).</p>
<p>
Obviously this is not the place to discuss every paper in detail. The table of contents is available on the publisher's website and for convenience the research papers are also listed below. From the titles you will recognize the papers on determinants and eigenvalues for Toeplitz matrices, in particular their asymptotic behaviour as their size goes to infinity. Of course circulant and Hankel operators and combinations of these as operators or matrices are not far off the central theme and they are thus also treated in some of the chapters. The majority of the papers present new results. Note that most of them are (functional) analysis. Only a few exceptions are more linear algebra or make a link to physics or explicitly discuss numerical aspects (see [14, 16, 18, 23, 25, 27] below).</p>
<p>
Some of the papers are quite long (more than 30 pages and some even up to 50 pages). They are basically true research papers, sometimes a bit more expository, but they are not of the introductory broad survey type. So this is not the book you should read to be introduced to the subject, but is is more a sketch of the state-of-the-art for who is already famiiar. The style of course depends on the authors, but the book is homogeneous because of the subjects that all somehow relate to Böttcher's work. These topics discussed here are also close to the core idea of this book series <em>Operators Theory Advances and Applications</em>, founded by Israel Gohberg as a complement to the journal <em>Integral Equations and Operator Theory</em>. Only one of Böttcher's books appeared in this series though (<em>Convolution Operators and Factorization of Almost Periodic Matrix Functions </em> (2002) authored with Yu. I. Karlovich, and I. M. Spitkovsky appeared as volume 131) but several of his books are with Springer / Birkhäuser. That these topics are still a main focus of research is illustrated by the successful annual IWOTA conferences (<em>International Workshop on Operator Theory and its Applications</em>), the proceedings of which are also published in this OT series. The IWOTA 2017 is organized by A. Böttcher, D. Potts and P. Stollmann at the TU Chemnitz.</p>
<p>
Thus for anyone interested in the general topics of this book series, this collection will be a worthy addition. For those who are more selective, there is of course still the possibility to get some separate chapters, which is the advantage of having it also available as an ebook.</p>
<p>
Here are the titles and authors of the research papers in this volume:</p>
<p>
<br />
7. <em>Asymptotics of Eigenvalues for Pentadiagonal Symmetric Toeplitz Matrices, </em> Barrera, M. (et al.), Pages 51-77<br />
8. <em>Echelon Type Canonical Forms in Upper Triangular Matrix Algebras, </em> Bart, H. (et al.), Pages 79-124<br />
9. <em>Asymptotic Formulas for Determinants of a Special Class of Toeplitz + Hankel Matrices, </em> Basor, E. (et al.), Pages 125-154<br />
10. <em>Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series, </em> Bini, D.A. (et al.), Pages 155-178<br />
11. <em>Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness, </em> Bogoya, J.M. (et al.), Pages 179-212<br />
12. <em>On the Asymptotic Behavior of a Log Gas in the Bulk Scaling Limit in the Presence of a Varying External Potential II, </em> Bothner, T. (et al.), Pages 213-234<br />
13. <em>Useful Bounds on the Extreme Eigenvalues and Vectors of Matrices for Harper's Operators, </em> Bump, D. (et al.), Pages 235-265<br />
14. <em>Fast Inversion of Centrosymmetric Toeplitz-plus-Hankel Bezoutians, </em> Ehrhardt, T. (et al.), Pages 267-300<br />
15. <em>On Matrix-valued Stieltjes Functions with an Emphasis on Particular Subclasses, </em> Fritzsche, B. (et al.), Pages 301-352<br />
16. <em>The Theory of Generalized Locally Toeplitz Sequences: a Review, an Extension, and a Few Representative Applications, </em> Garoni, C. (et al.), Pages 353-394<br />
17. <em>The Bézout Equation on the Right Half-plane in a Wiener Space Setting, </em> Groenewald, G.J. (et al.), Pages 395-411<br />
18. <em>On a Collocation-quadrature Method for the Singular Integral Equation of the Notched Half-plane Problem, </em> Junghanns, P. (et al.), Pages 413-462<br />
19. <em>The Haseman Boundary Value Problem with Slowly Oscillating Coefficients and Shifts, </em> Karlovich, Yu.I., Pages 463-500<br />
20. <em>On the Norm of Linear Combinations of Projections and Some Characterizations of Hilbert Spaces, </em> Krupnik, N. (et al.), Pages 501-510<br />
21. <em>Pseudodifferential Operators in Weighted Hölder-Zygmund Spaces of Variable Smoothness, </em> Kryakvin, V. (et al.), Pages 511-531<br />
22. <em>Commutator Estimates Comprising the Frobenius Norm - Looking Back and Forth, </em> Lu, Zhiqin (et al.), Pages 533-559<br />
23. <em>Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary, </em> Militzer, E. (et al.), Pages 561-591<br />
24. <em>Traces on Operator Ideals and Related Linear Forms on Sequence Ideals (Part IV), </em> Pietsch, A., Pages 593-619<br />
25. <em>Error Estimates for the ESPRIT Algorithm, </em> Potts, D. (et al.), Pages 621-648<br />
26. <em>The Universal Algebra Generated by a Power Partial Isometry, </em> Roch, S., Pages 649-662<br />
27. <em>Norms, Condition Numbers and Pseudospectra of Convolution Type Operators on Intervals, </em> Seidel, M., Pages 663-680<br />
28. <em>Paired Operators in Asymmetric Space Setting, </em> Speck, F.-O., Pages 681-702<br />
29. <em>Natural Boundary for a Sum Involving Toeplitz Determinants, </em> Tracy, C.A. (et al.), Pages 703-718<br />
30. <em>A Riemann-Hilbert Approach to Filter Design, </em> Wegert, E., Pages 719-740</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of papers dedicated to Albrecht Böttcher's 60th birthday. The contributions are by friends, colleagues and students. After the impressive list of his publications, many of which dealing with asymptotics of Toeplitz and related operators, the book has some birthday addresses sketching Böttcher as a person and some of his work. The major part however consists of research papers written on invitation by specialists on topics related by far or near to the work of Böttcher. <br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/dario-bini" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dario A. Bini</a></li><li class="vocabulary-links field-item odd"><a href="/author/torsten-ehrhardt" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Torsten Ehrhardt</a></li><li class="vocabulary-links field-item even"><a href="/author/alexei-yu-karlovich" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alexei Yu. Karlovich</a></li><li class="vocabulary-links field-item odd"><a href="/author/ilya-matvey-spitkovsky" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ilya Matvey Spitkovsky</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-49180-6 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">174,89 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">766</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319491806" title="Link to web page">http://www.springer.com/gp/book/9783319491806</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/47b35" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47B35</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/45e10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">45E10</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/47a57" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47A57</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/15b05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">15B05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65d15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65D15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/65g50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65G50</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/65j10" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">65J10</a></li></ul></span>Sat, 20 May 2017 12:07:31 +0000Adhemar Bultheel47678 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/large-truncated-toeplitz-matrices-toeplitz-operators-and-related-topics#commentsRecent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes
https://euro-math-soc.eu/review/recent-advances-inverse-scattering-schur-analysis-and-stochastic-processes
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Lev Aronovich Saknovich was born on February 24, 1932 in Lugansk, Ukraine. Almost agains all odds as a Jew in the communist Russia of those days, be became a mathematician and had teachers such as D.P. Milman, V.P. Potapov, M.S. Brodskij, and M.S. Livšic and his PhD was supported a.o. by I.M. Gelfand, M.G. Krein, and M.A. Naimark.</p>
<p>
This book is compiled on the occasion of his 80th birthday. It starts with a short biography and a list of his publications. Lev Saknovich himself gives an account of his studies and teachers. The main part of the book consists of 10 scientific papers that are of course related to the work of Sakhnovich. They fit very well in the Birkhäuser series on <em>Operator Theory Advances and Applications</em> founded by I. Gohberg in 1979 and more particulary in the subseries <em>Linear Operators and Linear Systems</em>. Contributors to this book are well known is this domain. Several of them published books in this series before like for example D. Alpay, V. Dubovoy, A. Kheifets, A.E. Frazho, M.A. Kaashoek, B. Fritzsche, B. Kirstein, J. Rovnyak, and by Lev Sakhnovich himself. Volume 84 in the OTAA series <a href="/review/integral-equations-difference-kernels-finite-intervals-2nd-ed" target="_blank"><em>Integral equations with difference kernels on finite intervals</em></a> by L.A. Sakhnovich got a second revised edition in 2015 along with the present book.</p>
<p>
The papers are listed alphabetically, but they can be organized in 4 groups. Three papers deal with interpolation and moment problems. Infinite product representations of reproducing kernels are used to study iterated function systems, harmonic analysis, and stochastic processes (Alpay et al). Commutant lifting and solution of Riccati equations are used to generate all rational solutions to a Leech problem in state space form (Frazho et al). It is the continuation of a previously published paper generating minimum entropy solutions. The solutions for the truncated matrix Hamburger moment problem are traditionally treated separately for the even and the odd case. Here Schur analysis is used to treat both simultaneously via a Schur-type algorithm (Fritzsche et al).</p>
<p>
Two papers fall under the flag of indefinite inner product spaces. The paper on quaternionic Krein spaces is a continuation of a published paper that treated quaternionic Pontryagin spaces (Alpay et al). In a joint paper Rovnyak and Sakhnovich continue exploring the relation between interpolation problems, operator identities and Krein-Langer representation of Carathéodory functions. In their previous paper they had treated the case of Nevanlinna functions.</p>
<p>
Different aspects of operator-valued functions are treated in four papers. One deals with operator-valued Q-functions with positive definite boundary conditions (Arlinskii et al). Another paper gives proofs for the Radon-Nikodym theorem for measures that are vector- or operator-valued (Boiko et al). Semi-separable iintegral kernels in infinite dimensional spaces, and in particular their Fredholm determinants are analyzed using a Jost-Pais type reduction (Gesztesy et al). The relation between certain analytic function classes, their closedness under addition and multiplication, and other properties is the subject of another paper (Makarov et al).</p>
<p>
Finally a paper by Lev Sakhnovich and his son discusses stability of nonlinear Fokker-Planck equations in inhomogeneous space.</p>
<p>
From this short sketch of the contents, it is clear that, besides the brief biographic part, the essence is a collection of research papers that will be of interest to the mathematcians and engineers that feel at home in this Birkhäuser series. Most of the papers could as well have been published in journals, and are not of introductory of survey type</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of research papers dedicated to Lev A. Sakhnoviches 80th birthday. It also has a short biography and a list of publications and some reminiscences of Lev Sakhnovich on his teachers and studies. The research papers cover (matrix valued) moment problems and interpolation, indefinite inner product spaces, operatr valued functions, abd nonlinear Fokker-Planck equations.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/daniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">daniel alpay</a></li><li class="vocabulary-links field-item odd"><a href="/author/bernd-kirstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Kirstein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-10334-1 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">100,69 €</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">394</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/analysis-and-its-applications" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Analysis and its Applications</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319103341" title="Link to web page">http://www.springer.com/gp/book/9783319103341</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00b30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00B30</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-06</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/46c20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46C20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/46e40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">46E40</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/35q84" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35Q84</a></li></ul></span>Mon, 20 Jul 2015 15:10:38 +0000Adhemar Bultheel46311 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/recent-advances-inverse-scattering-schur-analysis-and-stochastic-processes#commentsInterpolation, Schur Functions and Moment Problems II
https://euro-math-soc.eu/review/interpolation-schur-functions-and-moment-problems-ii
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
<em>Interpolation, Schur Functions and Moment Problems</em> (Part I) was edited by D. Alpay and I. Gohberg and published by Birkhäuser in 2006 as volume 165 in the same series on <em>Operator Theory: Advances and Applications</em>. This book is volume 226 in the same series. Many (in fact all) other volumes in this series are also related to topics in operator theory and analysis that play a role in the interaction between pure mathematics and applications in systems theory, signal processing, linear algebra, etc. By the number of volumes in the series, founded in 1979 by I. Gohberg, it is seen that this is a very productive and active research area. That is why the series has now two subseries: <em>Linear Operators and Linear Systems</em> (to which the present book belongs) and <em>Advances in partial Differential Equations</em>.</p>
<p>
The topics of this volume, as well as the ones in volume 1 dwell in the realm of <em>Schur analysis</em>. The name stems from I. Schur who published two papers in 1917/18 where he proposes an algorithm to solve a coefficient problem (i.e. the trigonometric moment problem), which is a kind of continued fraction-like decomposition of a function analytic in the complex unit disk that is bounded by 1 (now called a function in the Schur class). This involves a recurrence that coincides with the recurrence relation of polynomials orthogonal on the unit circle (studied by Ya. Geronimus and G. Szegő) and can be interpreted as a discretized transmission line in circuit theory (V. Belevich) or a digital prediction filter in signal processing (N. Wiener and P. Masani). It were the applications that revived the interest in Schur analysis in the 1960s and it hasn't stopped since. This has been generalized much further in many different directions and linked to other work by some great mathematicians like R. Nevanlinna, G. Pick, G. Herglotz, T. Stieltjes, H. Weyl, etc. A translation of Schur's original papers into English can be found e.g. in volume 18 of the OT series (<em> I. Schur methods in operator theory and signal processing</em> (I. Gohberg (ed.), 1986) and many other of the basic papers in Schur analysis are collected in <em>Ausgewählte Arbeiten zu den Urspüngen de Schur-Analysis</em> Band 16 of Teubner Archiv zur Mathematik, (B. Fritzsche, B. Kirstein (eds.), 1991).</p>
<p>
The six papers of this volume are mainly focussing on block generalizations of moment problems and Nevanlinna's theory. Moment theory asks for the existence and characterization of a measure when a sequence of moments are given. Nevanlinna's analogue is a generalization when the moments are not given at one point, but at more than one point, i.e. instead of Taylor series coefficients at just one point, a number of Taylor coefficients are given at different points, leading to a form of Hermite interpolation at these points. The block generalization refers to the fact that the functions and the moments are matrix valued, and hence also the measure one is looking for. As a consequence, the theory does not involve (matrix valued) orthogonal polynomials but more general matrix valued orthogonal rational functions.</p>
<p>
Bernd Fritzsche and Bernd Kirstein from Leibniz University have been passionate ambassadors of Schur analysis. The papers in this volume are all written by them and their coworkers at their institute (C. Mädler, T. Schwartz, A. Lasarow, A. Rahn) with one exception, the paper by A.E. Choque Rivero (Mexico) is a continuation of his earlier collaboration with the Leipzig team. Here is a short summary. Papers 1-3 involve reciprocal sequences and their applications, papers 4-5 discuss power moment problems on the real line and the last papers is about orthogonal rational functions.</p>
<ol>
<li>
<em>B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz.</em> On the Concept of Invertibility for Sequences of Complex <em>p × q</em>-matrices Application to Holomorphic <em>p×q</em>-matrix-valued Functions
<p>
This involves the construction the a <em>q× p</em>-sequence which is the reciprocal of a <em>p×q</em>-matrix sequence <em>f(z)</em>. This is a matrix version of constructing the scalar coefficients in a series <em>1/f(z)</em>, given the coefficients of the series <em>f(z)</em>. It applies to investigating the holomorphicity of the Moore-Penrose inverse of a matrix valued holomorphic function.</p>
</li>
<li>
<em>B. Fritzsche, B. Kirstein, A. Lasarow and A. Rahn.</em> On Reciprocal Sequences of Matricial Carathéodory Sequences and Associated Matrix Functions
<p>
This studies a particular case of the previous paper. It concentrates on the case <em>p = q</em> for Carathéodory functions (functions holomorphic is the disc and whose Hermitian part is nonnegative definite). The reciprocal of such Carathéodory function is again a Carathéodory function.</p>
</li>
<li>
<em>B. Fritzsche, B. Kirstein, C. Mädler and T. Schwarz.</em> On a Schur-type Algorithm for Sequences of Complex <em>p×q</em>-matrices and its Interrelations with the Canonical Hankel Parametrization
<p>
This is another application of reciprocal sequences. The Hamburger moment problem involves power moments for a (matrix-valued) measure on the real line, and the measure has to be recovered. The moments are coefficients in a Taylor series and a second sequence of moments can be defined as the Taylor coefficients in the reciprocal series. On the other hand partially defined (block) Hankel matrices that have a positive definite extension play an important role in the existence of a solution. By recursively constructing Schur complements, a sequence of matrices can be defined characterizing them. This allows to formulate a Schur-type algorithm to find this so called canonical characterization of Hankel matrices and hence finding solutions of truncated Hamburger moment problems.</p>
</li>
<li>
<em>A.E. Choque Rivero.</em> Multiplicative Structure of the Resolvent Matrix for the Truncated Hausdorff Matrix Moment Problem
<p>
The Hausdorff moment problem has to find a measure, supported on a finite interval, given the power moments. This involves two sequences of nested (block) Hankel matrices corresponding to an even or an odd number of prescribed moments. The solution of the moment problem is given in terms of a resolvent matrix. In this paper, the resolvent matrix is constructed and its factorization in elementary factors (Blaschke-Potapov factors). Again this corresponds to some recurrence relation for the (matrix-valued) orthogonal polynomials on a finite interval. This factorization of the resolvent has been treated before in the case of an even number of moments, here the odd case is analysed.</p>
</li>
<li>
<em>B. Fritzsche, B. Kirstein and C. Mädler.</em> On a Special Parametrization of Matricial α-Stieltjes One-sided Non-negative Definite Sequences
<p>
The Stieltjes moment problem looks for a measure supported on the positive real line, given its power moment sequence. Like in the Hamburger case, the existence of a solution involves (block) Hankel matrices containing the moments that have to be in a certain class. Being a member or not can be easily checked in the Hausdorff case by their canonical parametrization that can be constructed by a recursive Schur-like algorithm. Here a similar characterization is derived for the Stieltjes case and the relation between both is established.</p>
</li>
<li>
<em>B. Fritzsche, B. Kirstein and A. Lasarow.</em> On Maximal Weight Solutions of a Moment Problem for Rational Matrix-valued Functions
<p>
Truncated moment problems on the unit circle with moments given at several points involves rational interpolation (rather than matching the initial terms in series expansions). For a finite number of moments, a measure solving the moment problem can be described as a quadrature formula, i.e. it is a discrete measure with mass at a finite number of mass points. In this paper such particular solutions are investigated that have extremal mass at a prescribed point on the unit circle.</p>
</li>
</ol>
<p>
The contributors of this volume are very productive and they have published a large number of papers in many journals and books. Bringing a number of papers together and publish them as a book is a good idea. They have a particular style of writing, involving very precise formulations that also require complex notation. It may take a while for the reader to get used to it, but once familiar with these habits and the constructs involved, it is good reading. It will be of high interest to anyone who is involved from far or near with Schur analysis and the whole universe of related topics that I sketched in the beginning. Given the special character of this book (sub)series, it is clear that whoever was interested in volumes of this OT subseries before will be interested in practically all of them, a fortiori in this one. Be warned though that this book is at an advanced level, treating particular and specialized results in the domain, so this is not the right place to start learning the subject.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Like the previous volume with the same title, this is a collection of papers in the field of Schur analysis, a mathematical research fields involving moment problems and more generally interpolation in the complex plane by bounded analytic (Schur) functions or positive real (Carathéodry) functions. The papers are produced by the team of the U. Leipzig and one coworker who are well known ambassadors of Schur analysis. This volume contains a coherent set of papers related to matrix valued moment problems on the complex unit circle and on real line, a half line or a finite interval.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/baniel-alpay" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Baniel Alpay</a></li><li class="vocabulary-links field-item odd"><a href="/author/bernd-kirstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Kirstein</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2012</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-0348-0427-1</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">94.95 € (net)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">305</div></div></div><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/mathematics/book/978-3-0348-0427-1" title="Link to web page">http://www.springer.com/mathematics/book/978-3-0348-0427-1</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/30-functions-complex-variable" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30 Functions of a complex variable</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/30e05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30e05</a></li></ul></span>Thu, 06 Sep 2012 10:14:39 +0000Adhemar Bultheel45460 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/interpolation-schur-functions-and-moment-problems-ii#comments