Linear ordinary differential equations with boundary conditions on arbitrary point sets
Authors: Michael Golomb and Joseph Jerome
Journal: Trans. Amer. Math. Soc. 153 (1971), 235-264
MSC: Primary 34.36
MathSciNet review: 0269918
Abstract: Boundary-value problems for differential operators of order which are the Euler derivatives of quadratic functionals are considered. The boundary conditions require the solution to coincide with a given function at the points of an arbitrary closed set , to satisfy at the isolated points of the knot conditions of -spline interpolations, and to lie in . Existence of solutions (called ``-splines knotted on ") is proved by consideration of the associated variational problem. The question of uniqueness is treated by decomposing the problem into an equivalent set of problems on the disjoint intervals of the complement of , where denotes the set of limit points of . It is also shown that , considered as an operator from to ), with appropriately restricted domain, has a unique selfadjoint extention if one postulates that the domain of contains only functions of which vanish on has a bounded inverse which serves to solve the inhomogeneous equation with homogeneous boundary conditions. Approximations to the -splines knotted on are constructed, consisting of -splines knotted on finite subsets of , with dense in . These approximations converge to in the sense of .
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-  Michael Golomb, Splines, -widths and optimal approximations, MRC Technical Summary Report #784, 1967.
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-  Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
- Michael Golomb and I. J. Schoenberg, On -extension of functions and spline interpolation, MRC Technical Summary Report #1090, 1970.
- Michael Golomb, Splines, -widths and optimal approximations, MRC Technical Summary Report #784, 1967.
- Israel Halperin, Introduction to the theory of distributions, Univ. of Toronto Press, Toronto, 1952. MR 13, 658. MR 0045933 (13:658b)
- Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill, New York, 1966. MR 34 #580. MR 0200692 (34:580)
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Keywords: Boundary-value problems, quadratic functional, existence, minimization, -splines with arbitrary sets of knots, discrete components, unicity conditions, uniqueness, approximation, selfadjoint extension, Tchebychev set, lower degree at infinity
Article copyright: © Copyright 1971 American Mathematical Society