This monograph develops the spectral theory of an $n$th order non-self-adjoint two-point differential operator $L$ in the Hilbert space $L^2[0,1]$. The mathematical foundation is laid in the first part, where the spectral theory is developed for closed linear operators and Fredholm operators. An important completeness theorem is established for the Hilbert-Schmidt discrete operators. The operational calculus plays a major role in this general theory. In the second part, the spectral theory of the differential operator $L$ is developed by expressing $L$ in the form $L = T + S$, where $T$ is the principal part determined by the $n$th order derivative and $S$ is the part determined by the lower-order derivatives. The spectral theory of $T$ is developed first using operator theory, and then the spectral theory of $L$ is developed by treating $L$ as a perturbation of $T$. Regular and irregular boundary values are allowed for $T$, and regular boundary values are considered for $L$. Special features of the spectral theory for $L$ and $T$ include the following: calculation of the eigenvalues, algebraic multiplicities and ascents; calculation of the associated family of projections which project onto the generalized eigenspaces; completeness of the generalized eigenfunctions; uniform bounds on the family of all finite sums of the associated projections; and expansions of functions in series of generalized eigenfunctions of $L$ and $T$.

Spectral Theory of Non-Self-Adjoint Two-Point Differential Operators (Mathematical Surveys & Monographs)

Science & Math

John Locker

Mathematics

American Mathematical Society

Numerical and analytic trigonometry

Davies & Associates

These materials are written primarily for students engaged in independent study in both structured and unstructured environments. Carefully formulated instructional objectives serve as the fundamental organizational unit in Numerical Trigonometry. Each objective is prominently displayed and is followed immediately by focused discussion, summary, examples and practice problems. Various students will wish to use the materials following the objectives differently. Students who work from the general to the specific will want to digest the discussion, verify their understanding by comparing with the summary, work through the examples and, finally, do some practice problems. Other students with a different learning style may prefer to begin with concrete examples and practice problems, then read the summary to verify the insights derived from the examples, and finally, read the discussion to synthesize the information they have gained. Prominent headings invite each reader to use this material in the personally most efficient way.

Numerical Trigonometry

The Davies Group Publishers

The Davies Group

Kenneth F. Klopfenstein

Davies Group Staff

Davies Group Publishers, The

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