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Second kind integral equations on the real line [Elektronische Ressource] : solvability and numerical analysis in weighted spaces / von Kai Haseloh

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Second Kind Integral Equations on the Real Line:Solvability and Numerical Analysisin Weighted SpacesVomFachbereich Mathematikder Universit at Hannoverzur Erlangung des GradesDoktor der NaturwissenschaftenDr. rer. nat.genehmigte DissertationvonDipl.-Math. Kai Haselohgeboren am 23. September 1974 in Bad Oeynhausen2004Referent: Prof. Dr. Schmidt-Westphal, Universit at HannoverKorreferent: Prof. Dr. S. N. Chandler-Wilde, University of Reading, UKTag der Promotion: 26. Mai 2004AbstractThe exact and numerical solution of integral equations taking the formR∞ x (s) v(s,t)x(t)dt=y(s)incertainweightedsubspacesX ofthew ∞spaceX :=BC(R)(ofboundedandcontinuousfunctionsoverR)isstud-ied. Here, X denotes the weighted space of all functions x∈X satisfy-wing|w(s)x(s)|=O(1) as|s|→∞, for some weight function w1. Thekernel v is assumed to satisfy the simple condition|v(s,t)||(s t)|,1for some ∈L (R).Conditions on v and w are obtained, which ensure that the integral op-erator K in above equation is bounded on X and X . These conditionsware then strengthened to imply the equivalence of the spectrum (and es-sential spectrum) of K on X and X as well as several other statementswabout the solvability of the above integral equation.NSimilarboundednessandspectralresultsareshownfortheoperatorsKarising from suitable quadrature approximations of the integral op-erator K.

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Second Kind Integral Equations on the Real Line:
Solvability and Numerical Analysis
in Weighted Spaces
Vom
Fachbereich Mathematik
der Universit at Hannover
zur Erlangung des Grades
Doktor der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
von
Dipl.-Math. Kai Haseloh
geboren am 23. September 1974 in Bad Oeynhausen
2004Referent: Prof. Dr. Schmidt-Westphal, Universit at Hannover
Korreferent: Prof. Dr. S. N. Chandler-Wilde, University of Reading, UK
Tag der Promotion: 26. Mai 2004Abstract
The exact and numerical solution of integral equations taking the form
R∞
x (s) v(s,t)x(t)dt=y(s)incertainweightedsubspacesX ofthew ∞
spaceX :=BC(R)(ofboundedandcontinuousfunctionsoverR)isstud-
ied. Here, X denotes the weighted space of all functions x∈X satisfy-w
ing|w(s)x(s)|=O(1) as|s|→∞, for some weight function w1. The
kernel v is assumed to satisfy the simple condition|v(s,t)||(s t)|,
1for some ∈L (R).
Conditions on v and w are obtained, which ensure that the integral op-
erator K in above equation is bounded on X and X . These conditionsw
are then strengthened to imply the equivalence of the spectrum (and es-
sential spectrum) of K on X and X as well as several other statementsw
about the solvability of the above integral equation.
NSimilarboundednessandspectralresultsareshownfortheoperatorsK
arising from suitable quadrature approximations of the integral op-
erator K. Nystr om/product integration and nite section methods are
studied and it is shown that, under certain conditions, whenever a
method is stable on X it is also stable on X , with equivalence holdingw
in many cases. Error estimates in the norm of X are given.w
The class of kernels considered is large and contains, in particular, all
1kernels of the form v(s,t) =(s t), ∈L (R), leading to convolution
orWiener-Hopfequations. Specialemphasisislaidonfamiliesofkernels
of the form v(s,t)k(s,t) or (s t)k(s,t), with k varying in a bounded
2and equicontinuous subset W of BC(R ), for which the stability results
hold uniformly in k∈W.
Asanapplication,numericalmethodsforafamilyofkernelsv withweak
logarithmic singularity are analysed. For these methods, stability and
convergence results in certain weighted spaces are obtained. The kernels
considered arise, e.g., in boundary integral equations for rough surface
scatteringproblemsoverunboundeddomains,whicharestudiedasprac-
ticalexamples. ForacombinedNystrom andnitesectionmethod,novel
error estimates are obtained for the case of a point source.
Keywords: integral equations, Nystr om method, weighted spacesZusammenfassung
Dieexakte undnumerischeL osbarkeitvonIntegralgleichungenderForm
R∞
x (s) v(s,t)x(t)dt = y(s) in gewichteten Unterr aumen X desw ∞
Raumes X := BC(R) (der stetigen und beschankr ten Funktionen ub er
R) wird untersucht. Dabei ist X X der Gewichtsraum aller Funk-w
tionen x∈ X, die der Bedingung |w(s)x(s)| = O(1), |s|→∞, fur eine
Gewichtsfunktion w 1 genugen. Ferner wird angenommen, dass der
1Kern v die einfache Bedingung |v(s,t)| |(s t)| fur ein ∈ L (R)
erfullt.
Es werden Anforderungen an v und w formuliert, die hinreichend dafur
sind, dass der Integraloperator K in obiger Gleichung beschrankt auf
X und X ist. Diese Bedingungen werden so verst arkt, dass sie diew
Ubereinstimmung des Spektrums (wesentlichen Spektrums) von K auf
X und X implizieren und weitere Aussagen ub er die L osbarkeit obigerw
Integralgleichung angegeben werden konn en.
Ahnliche Beschanr ktheits- und Spektralergebnisse ergeben sich fur die
NOperatoren K , die durch geeignete Quadraturapproximation des Inte-
graloperators K entstehen. Nystrom-/Produktintegrations- und Reduk-
tionsverfahren ( nite section methods ) werden untersucht und es wird
gezeigt, dass unter bestimmten Bedingungen die Stabilit at auf X die
Stabilit at auf X impliziert und in vielen F allen sogar Aquivalenz gilt.w
Fehlerschranken in den gewichteten Normen werden angegeben.
Die Klasse der betrachteten Kerne ist umfangreich und beinhaltet ins-
1besondereams tlicheKernederForm v(s,t)=(s t),∈L (R),dieauf
Faltungsgleichungen bzw. Wiener-Hopf-Gleichungen fuhr en. Ein beson-
derer Schwerpunkt liegt auf Familien von Kernen der Form v(s,t)k(s,t)
oder (s t)k(s,t), wobei k in einer beschankr ten und gleichstetigen
2Teilmenge W von BC(R ) variiert; die Stabiliatst resultate gelten dann
gleichm a ig fur k∈W.
Als Anwendung werden numerische Verfahren fur eine spezielle Familie
von Kernen v mit schwacher logarithmischer Singularitat analysiert, fur
die Stabilit at in bestimmten Gewichtsr aumen gezeigt wird. Die betrach-
teten Kerne ergeben sich u.a. bei der Randintegralmethode fur Streu-
probleme mit unbeschr ankten Oberachen, auf die besonders eingegan-
gen wird. Fur ein kombiniertes Nystrom- und Reduktionsverfahren wer-
den im Falle einer punktformigen Quelle neuartige Fehlerabsch atzungen
angegeben.
Schlagw orter: Integralgleichungen,Nystr omverfahren,Gewichtsr aumeContents
1 Introduction 1
2 Background material 11
2.1 Normal solvability and Fredholm operators . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Weighted spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 The strict topology and equicontinuous sets . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Integral equations over unbounded domains 17
4 Spectral properties of integral operators in weighted spaces 21
4.1 Boundedness in weighted spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Solvability in weighted spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Sharpness of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.4 Su cient conditions on kernels and examples . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 The real line case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Numerical methods in weighted spaces 47
5.1 Nystr om and product integration methods . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Boundedness and spectral properties in weighted spaces . . . . . . . . . . . . . . . . . . 52
5.3 Stability and uniform stability in weighted spaces . . . . . . . . . . . . . . . . . . . . . . 57
5.3.1 Su cient conditions for stability and uniform stability on X . . . . . . . . . . . 60
5.4 The nite section method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 Sums of integral operators and their approximation . . . . . . . . . . . . . . . . . . . . . 72
6 Applications 75
6.1 A few technical prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Kernels with polynomial decay and error estimates for smooth inhomogeneities . . . . . 78
6.2.1 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2.2 Kernels with logarithmic singularities . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 A problem in rough surface scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Bibliography 100Chapter 1
Introduction
The focus of this thesis is on the theoretical and numerical solution of Fredholm integral equations of
the second kind over unbounded domains, which take the following general form:
Z
x (s) v(s,t)x(t)dt=y(s), s∈
, (1.1)

where the domain of integration
is one of the setsR orR . In operator notation, (1.1) is written as+
x Kx=y. (1.2)
Our study is centered around the investigation of (1.2) in the following class of weighted subspaces
of X :=BC( ) (the space of all bounded and continuous functions over , equipped with the uniform
normkzk:=sup |z(s)|): For an even weight function w :R→R satisfyings∈

w(0)=1, w(s)w(t) for st0, lim w(s)=∞, (1.3)
s→∞
we let X denote the weighted subspace {x∈ X :kxk :=kxwk <∞} of X. X is a Banach spacew w w
when equipped with the normkk .w
We will show that many spectral properties of the operatorK and suitable discretizations, obtained
by quadrature methods, are essentially the same on X and X . Moreover, we will prove that, for aw
wide range of Nystrom / nite section methods, stability on X is su cient for stability to hold also on
X , with equivalence holding for many Nystomr methods. If stability holds, we provide estimates forw
the resulting error when y is contained in X .w
Integral equations in weighted spaces
Since the ground-breaking work of Fredholm and Riesz at the beginning of the 20th century, second-
kind Fredholm integral equations over nite and in nite intervals have been of continuing interest to
mathematicians (see [11] for a historical account of the theory). A part of their importance stems from
the fact that many problems in physics, electromagnetics and mechanics lead to such equations; the
list of examples is long, and we refer the reader to [39, 41, 54, 29] and the references therein. Indeed,
the reformulation of partial dierential equations as boundary integral equations (see, e.g. [28]), in
particular for the Helmholtz equation in two dimensions, is a rich source of practical applications for
the results we present in this thesis (see, e.g., [25, 6, 23]; [27, 53] provide an introduction).
For second-kind integral equations over a bounded interval [a,b] of the real-line, with the integral
operator being compact on a suitable function space, the theory is mostly complete, with the Fredholm
alternativetheoremandtheRiesz-Schaudertheorybeingthemostprominentandusefultheoreticaltools
(see,e.g.[40]). From1950onwards,muche ortwasdevotedtothedevelopmentandanalysisofsuitable
numerical methods for solving such equations, some of the most outstanding examples being Galerkin,
1Introduction 2
Nystr om, product integration, projection and degenerate kernel methods. An excellent overview of
these can be found in the monograph [10] of Atkinson. In the preface of this book we are also informed
that “this work is nearing a stage in which there will be few major additions to the theory”.
However, if, as in (1.1), the domain of the integral equation is unbounded, we usually lose compact-
ness of the integral operator K and the analysis of the numerical methods mentioned above requires
much more subtlety than in the case of a nite interval.
The most prominent and well-studied examples of integral equations with a non-compact K are
those in which the integral operator is a convolution operator, i.e.
v(s,t)=(s t), s∈
, a.e. t∈
, (1.4)
1holds, for some ∈L (R). In this case (1.1) is called a convolution ( = R) or Wiener-Hopf ( = R )+
integral equation: Z
x (s) (s t)x(t)dt=y(s), s∈
. (1.5)

The solvability of equation (1.5) is usually studied by Fourier transform methods, which give explicit
pexpressions for the spectrum and essential spectrum of the operator K. The essential results for L -
spaces, 1 p ∞, and BC( ) can be found in [40] (we also mention the recent survey [12], an
pup-to-date overview of the L -theory for 1 < p < ∞ for Wiener-Hopf operators with discontinuous
1symbols, arising when is not in L (R)).
Throughout this thesis, we will consider a much more general class of kernels. Let us make this
precise: Our