Linear ordinary differential equations with boundary conditions on arbitrary point sets
Michael Golomb and Joseph Jerome
Trans. Amer. Math. Soc. 153 (1971), 235-264
Full-text PDF Free Access
Similar Articles |
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 .
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.
Halperin, Introduction to the theory of distributions. Based on the
lectures given by Laurent Schwartz, University of Toronto Press,
Toronto, 1952. MR 0045933
Goldberg, Unbounded linear operators: Theory and applications,
McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692
Riesz and Béla
Sz.-Nagy, Leçons d’analyse fonctionnelle,
Akadémiai Kiadó, Budapest, 1953 (French). 2ème
- 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)
- F. Riesz and B. Sz. Nagy, Leçons d'analyse fonctionnelle, 2nd ed., Akad. Kiadó, Budapest, 1953; English transl., Ungar, New York, 1955. MR 15, 132; MR 17, 175. MR 0056821 (15:132d)
Retrieve articles in Transactions of the American Mathematical Society
Retrieve articles in all journals
-splines with arbitrary sets of knots,
lower degree at infinity
© Copyright 1971
American Mathematical Society