Ebook: Theory of Tikhonov Regularization for Fredholm Equations of the First Kind

ft has been two decades since the publication of Tikhonov's groundbreaking
paper on the method of regularization for numerical solution of Fredholm
integral equations of the first kind. The ensuing years have seen an
intensive dcveloprc'nt of the theory of the method as well as its increasing
application to difficult technical problems. A coherent and self-contained
treatment of the general theory of Tikhonov's method for compact operators
in Hilbert space would seem to be a timely and worthwhile undertaking.
These notes represent our own modest attempt at such a treatment.
Our development is approximative rather than numerical, that is, the
approximations themselves lie in the same IFilbert space as the solution and
questions of convergence, rates, etc., are addressed in the general Hilbert
space context. Although most of the results apply to more general
operators, we limit our attention to compact operators, which results in
considerable simplification, because our prine motivating example is a
Fredholm integral equation of the first kind.
A reference in the text of the form "(a.b.c)" refers to theorem number
"c" ,n section `'b" of chapter TTa,'T equations and other important displayed
items are numbered consecutively within each section. The end of a proof
is indicated by the symbol T!
These notes comprise the text of a course of lectures on Tikhonov
regularization which I gave at the University of Kaiserslautern in the
spring of 1983. I wish to thank Professor Eberhard Semtock for inviting me
to lecture in Germany.