- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 09:56:53
12
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=14&iss=2&update_since=2024-03-28
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
14
1995
2
On Continuous Capacities
M.
Brzezina
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Capacities, continuous capacities, semipolar sets, essential bases
Let $(X, W)$ be a balayage space, $\gamma$ a Choquet capacity on $X$, $\beta(E)$ the essential base of $E \subset X$ and, for a compact set $K \subset X, \alpha (K) = \gamma (\beta(K))$. Then some properties of the set function $\alpha$ are investigated. In particular, it is shown when $\alpha$ is the Choquet capacity. Further, some relation a to the so-called continuous capacity deduced from a kernel on $X$ is given. At last, some open problems from the book [1] by G. Anger are solved.
Potential theory
General
213
224
10.4171/ZAA/671
http://www.ems-ph.org/doi/10.4171/ZAA/671
Loewner Interpolation in Matrix Hardy Classes
Daniel
Alpay
Ben Gurion University of the Negev, BEER SHEVA, ISRAEL
J.
Leblond
INRIA, SOPHIA ANTIPOLIS CEDEX, FRANCE
Loewner interpolation, reproducing kernel spaces, Hardy spaces
We study a Loewner type interpolation problem in the framework of matrix-valued $\mathbb H_2$ functions. We establish a necessary and sufficient condition for a matrix-valued function given on an arc of circle to be the trace of such an $\mathbb H_2$ function subject to some constraint.
Operator theory
Functions of a complex variable
General
225
233
10.4171/ZAA/672
http://www.ems-ph.org/doi/10.4171/ZAA/672
On the Decomposition of Unitary Operators into a Product of Finitely Many Positive Operators
G.
Peltri
Universität Leipzig, LEIPZIG, GERMANY
Operator theory, von Neumann algebras, non-commutative geometry
We will show that in an infinite-dimensional separable Hilbert space $\mathcal H$, there exist constants $N \in \mathbb N$ and $c,d \in \mathbb R$ such that every unitary operator can be written as the product of at most $N$ positive invertible operators $\{a_k\} \subseteq B(\mathcal H)$ with $\| a_k \| ≤ c$ and $\|a^{–1}_k \| ≤ d$ for all $k$. Some consequences of this result in the context of von Neumann algebras are discussed.
Functional analysis
General
235
248
10.4171/ZAA/673
http://www.ems-ph.org/doi/10.4171/ZAA/673
On Associated and Co-Associated Complex Differential Operators
R.
Heersink
Technische Universität Graz, GRAZ, AUSTRIA
Wolfgang
Tutschke
Technische Universität Graz, GRAZ, AUSTRIA
Initial value problems, interior estimates, Cauchy-Kovalevskaya theorem
The paper deals with initial value problems of the form $$\frac{\partial u}{\partial t} = \mathcal L u, \ \ \ u = u_0 \ \mathrm {for} \ t = 0$$ in $[0,T] \times G \subset \mathbb R^+_0 \times \mathbb R^n$ where $\mathcal L$ is a linear first order differential operator. The desired solutions will be sought in function spaces defined as kernel of a linear differential operator $l$ being associated to $\mathcal L$. Mainly two assumptions are required for such initial value problems to be solvable: Firstly, the operators have to be associated, i.e. $lu = 0$ implies $l(\mathcal L u) = 0$. Secondly, an interior estimate $\| \mathcal L u \|_{G'} ≤ c(G,G’) \| u \|_G$ (with $G' \subset G$) must be true. Moreover, operators $\mathcal L$ are investigated possessing a family of associated operators $l_k$ (which then are said to be co-associated). The present paper surveys the use of associated and co-associated differential operators for solving initial value problems of the above (Cauchy-Kovalevskaya) type. Discussing interior estimates as starting point for the construction of related scales of Banach spaces, the paper sets up a possible framework for further generalizations. E.g., that way a theorem of Cauchy-Kovalevskaya type with initial functions satisfying a differential equation of an arbitrary order k (with not necessarily analytic coefficients) is obtained.
Partial differential equations
General
249
257
10.4171/ZAA/674
http://www.ems-ph.org/doi/10.4171/ZAA/674
Non-Symmetric Matrix Riccati Equations
Gerhard
Freiling
Universität Duisburg-Essen, DUISBURG, GERMANY
G.
Jank
RWTH Aachen, AACHEN, GERMANY
Matrix Riccati differential equation, algebraic Riccati equation, asymptotic properties
We prove a fundamental representation formula for all solutions of the matrix Riccati differential equation and of the corresponding algebraic Riccati equation. This formula contains the complete information on the phase portrait of the matrix equation and on the structure of the set F of all solutions of the corresponding algebraic equation. In particular we describe all constant, periodic and almost-periodic solutions of the matrix Riccati differential equation. Further we give an application of the fundamental representation formula to the investigation of non-autonomous Riccati equations.
Ordinary differential equations
Calculus of variations and optimal control; optimization
Differential geometry
General
259
284
10.4171/ZAA/675
http://www.ems-ph.org/doi/10.4171/ZAA/675
Some Results on Non-Coercive Variational Problems and Applications
F.
Weissbaum
Ecole Polytechnique Federale, LAUSANNE, SWITZERLAND
Calculus of variations, convexity, non-coercive and constraint problems
We give a necessary and sufficient condition to ensure the existence of solutions of three problems of the calculus of variations with non-coercive integrands. The solutions $u$ we consider are lipschitz functions, i.e. $u \in W^{1, \infty} (0,1)$. The three problems depends on the same functionals but are different in the constraints. We consider respectively a problem without constraint, a problem with $u' ≥ 0$ and finally a problem with $u ≥ 0$. These problems can be related to optimal foraging models in behavioural ecology.
Calculus of variations and optimal control; optimization
General
285
325
10.4171/ZAA/676
http://www.ems-ph.org/doi/10.4171/ZAA/676
Variational Bounds to Eigenvalues of Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum
S.
Zimmermann
Technische Universität Clausthal, CLAUSTHAL-ZELLERFELD, GERMANY
U.
Mertins
Technische Universität Clausthal, CLAUSTHAL-ZELLERFELD, GERMANY
Eigenvalue problems, variational methods, upper and lower bounds to eigenvalues
In the present paper a method by Lehmann-Maehly and Goerisch is extended to self-adjoint eigenvalue problems with arbitrary essential spectrum. This extension is obtained by consequently making use of the local character of the method. In this way, upper and lower bounds to all isolated eigenvalues are derived. In our proofs, the close relationship to Wielandt’s inverse iteration becomes quite obvious.
Calculus of variations and optimal control; optimization
Partial differential equations
Numerical analysis
General
327
345
10.4171/ZAA/677
http://www.ems-ph.org/doi/10.4171/ZAA/677
On the Sharpness of Error Bounds for the Numerical Solution of Initial Boundary Value Problems by Finite Difference Schemes
H.
Esser
RWTH Aachen, AACHEN, GERMANY
Steffen
Goebbels
Niederrhein University of Applied Sciences, KREFELD, GERMANY
Rolf Joachim
Nessel
RWTH Aachen, AACHEN, GERMANY
Initial boundary value problems, finite difference methods, sharpness of error bounds
The present paper studies the sharpness of error bounds obtained for approximate solutions of initial boundary value problems by finite difference schemes. Whereas the direct estimates in terms of partial moduli of continuity for partial derivatives of the (exact) solutions follow by standard methods (stability inequality plus Taylor expansion of the truncation error), the sharpness of these bounds is established by an application of a quantitative extension of the uniform boundedness principle. To verify the relevant resonance condition a general procedure is suggested, in contrast to our previous investigations which were based on rather specific properties of the discrete Green’s functions associated. Exemplarily, details are worked out in connection with Crank–Nicolson, Du Fort-Frankel and Saulyev schemes.
Partial differential equations
Numerical analysis
Approximations and expansions
General
347
367
10.4171/ZAA/678
http://www.ems-ph.org/doi/10.4171/ZAA/678
A Convergence Rate Result for a Steepest Descent Method and a Minimal Error Method for the Solution of Nonlinear Ill-Posed Problems
A.
Neubauer
Johannes Kepler Universität Linz, LINZ, AUSTRIA
Otmar
Scherzer
Austrian Academy of Sciences, LINZ, AUSTRIA
Nonlinear ill-posed problems, steepest descent method, minimal error method, regularization methods, discrepancy principle, stopping rule
Recently, convergence and stability of the steepest descent method for the solution of nonlinear ill-posed operator equations have been proven. The same results also hold for the minimal error method. Since for ill-posed problems the convergence of iterative methods may be arbitrarily slow, it is of practical interest to guarantee convergence rates of the iterates under reasonable assumptions. The main emphasis of this paper is to present a convergence rate result in a uniform manner for the steepest descent and the minimal error method for the noise free case.
Numerical analysis
Operator theory
General
369
377
10.4171/ZAA/679
http://www.ems-ph.org/doi/10.4171/ZAA/679
A New Maximal Point Theorem
A.
Göpfert
Universität Halle-Wittenberg, HALLE, GERMANY
Chr.
Tammer
Universität Halle-Wittenberg, HALLE, GERMANY
Maximal point theorem, conical supporting point, variational principle, approximation problem
The aim of our paper is to generalize the maximal point theorem of Bishop and Phelps and to apply this result to derive a new multicriteria Ekeland’s principle in a direct way by induction without making use of Ekeland’s original scalar result.
Operations research, mathematical programming
Calculus of variations and optimal control; optimization
General
379
390
10.4171/ZAA/680
http://www.ems-ph.org/doi/10.4171/ZAA/680
Optimal Transportation Flows
Rolf
Klötzler
, BORSDORF, GERMANY
Transportation flow problems, deposit problems, dual optimization problems, generalized Pontryagin’s maximum principle
This paper deals with a modification of L. C. Young’s flow concept in application to transportation flow problems. There will be proved necessary and sufficient conditions for an optimal transport and its dual deposit problem.
Calculus of variations and optimal control; optimization
General
391
401
10.4171/ZAA/681
http://www.ems-ph.org/doi/10.4171/ZAA/681
Some Properties of the Attainable Set for the Abstract Control Problem with Application to Controllability
B.
Shklyar
Bar-Ilan University, RAMAT GAN, ISRAEL
Attainable sets, controllability, abstract evolution equations, linear hereditary systems
Partial differential equations
Numerical analysis
General
403
412
10.4171/ZAA/682
http://www.ems-ph.org/doi/10.4171/ZAA/682