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This book provides a modern survey of some basic properties of
Sturm-Liouville problems and to bring the reader to the forefront
of knowledge of some areas of the theory. For example, some special
Sturm-Liouville eigenvalue problems are equivalent to certain
Jacobi and cyclic Jacobi matrix eigenvalue problems. A new approach
to problems with periodic conditions is developed.
Norm inequalities relating (i) a function and two of its
derivatives and (ii) a sequence and two of its differences are
studied. Detailed elementary proofs of basic inequalities are
given. These are accessible to anyone with a background of advanced
calculus and a rudimentary knowledge of the Lp and lp spaces. The
classical inequalities associated with the names of Landau,
Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and
Caravetta, etc., are discussed, as well as their discrete analogues
and weighted versions. Best constants and the existence and nature
of extremals are studied and many open questions raised. An
extensive list of references is provided, including some of the
vast Soviet literature on this subject.
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