Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

The existence, characterization and essential uniqueness of solutions of $ L\sp{\infty }$ extremal problems


Authors: S. D. Fisher and J. W. Jerome
Journal: Trans. Amer. Math. Soc. 187 (1974), 391-404
MSC: Primary 41A65
MathSciNet review: 0364983
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ I = (a,b)$ be an interval in R and let $ {H^{n,\infty }}$ consist of those real-valued functions f such that $ {f^{(n - 1)}}$ is absolutely continuous on I and $ {f^{(n)}} \in {L^\infty }(I)$. Let L be a linear differential operator of order n with leading coefficient $ 1,a = {x_1} < \cdots < {x_m} = b$ be a partition of I and let the linear functionals $ {L_{ij}}$ on $ {H^{n,\infty }}$ be given by

$\displaystyle {L_{ij}}f = \sum\limits_{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,} $

where $ 1 \leq {k_i} \leq n$ and the $ {k_i}$ n-tuples $ (a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)})$ are linearly inde pendent. Let $ {r_{ij}}$ be prescribed real numbers and let $ U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\} $. In this paper we consider the extremal problem

$\displaystyle {\left\Vert {Ls} \right\Vert _{{L^\infty }}} = \alpha = \inf \{ {\left\Vert {Lf} \right\Vert _{{L^\infty }}}:f \in U\} .$ ($ \ast$)

We show that there are, in general, many solutions to $ ( \ast )$ but that there is, under certain consistency assumptions on L and the $ {L_{ij}}$, a fundamental (or core) interval of the form $ ({x_i},{x_{i + {n_0}}})$ on which all solutions to $ ( \ast )$ agree; $ {n_0}$ is determined by the $ {k_i}$ and satisfies $ {n_0} \geq 1$. Further, if s is any solution to $ ( \ast )$ then on $ ({x_i},{x_{i + {n_0}}}),\vert Ls\vert = \alpha$ a.e. Further, we show that there is a uniquely determined solution $ {s_ \ast }$ to $ ( \ast )$, found by minimizing $ {\left\Vert {Lf} \right\Vert _{{L^\infty }}}$ over all subintervals $ ({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1$, with the property that $ \vert L{s_ \ast }\vert$ is constant on each subinterval $ ({x_j},{x_{j + 1}})$ and $ L{s_ \ast }$ is a step function with at most $ n - 1$ discontinuities on $ ({x_j},{x_{j + 1}})$. When $ L = {D^n},{s_ \ast }$ is a piecewise perfect spline. Examples show that the results are essentially best possible.

References [Enhancements On Off] (What's this?)

  • [1] N. I. Achieser and M. Krein, Sur la meilleure approximation des fonctions periodiques au moyen des sommes trigonométriques, Dokl. Akad. Nauk SSSR 15 (1937), 107-111.
  • [2] J. Favard, Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques, Bull. Sci. Math. 61 (1937), 209-224.
  • [3] J. Favard, Sur l’interpolation, J. Math. Pures Appl. (9) 19 (1940), 281–306 (French). MR 0005187 (3,114e)
  • [4] Georges Glaeser, Fonctions composées différentiables, Séminaire d’Analyse, dirigé par P. Lelong, 1962/63, No. 2, Secrétariat mathématique, Paris, 1963, pp. 4 (French). MR 0188382 (32 #5821)
  • [5] Michael Golomb, ℋ^{𝓂,𝓅}-extensions by ℋ^{𝓂,𝓅}-splines, J. Approximation Theory 5 (1972), 238–275. Collection of articles dedicated to J. L. Walsh on his 75th birthday, III (Proc. Internat. Conf. Approximation Theory, Related Topics and their Applications, Univ. Maryland, College Park, Md., 1970). MR 0336161 (49 #937)
  • [6] -, Some extremal problems for differential periodic functions in $ {L_\infty }(R)$, Math. Res. Cent. Tech. Summary Rep. 1069, Madison, Wisconsin, 1970.
  • [7] Philip Hartman, Ordinary differential equations, John Wiley & Sons Inc., New York, 1964. MR 0171038 (30 #1270)
  • [8] Joseph W. Jerome, Minimization problems and linear and nonlinear spline functions. I. Existence, SIAM J. Numer. Anal. 10 (1973), 808–819. MR 0410205 (53 #13955)
  • [9] Joseph W. Jerome, Minimization problems and linear and nonlinear spline functions. II. Convergence, SIAM J. Numer. Anal. 10 (1973), 820–830. MR 0410206 (53 #13956)
  • [10] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. Teil. 1: Gewöhnliche Differentialgleichungen, 3rd ed., Math. und ihre Anwendungen in Physik und Technik, Band 18, Geest & Portig, Leipzig, 1944. MR 9, 33.
  • [11] I. J. Schoenberg and A. Cavaretta, Solution of Landau's problem concerning higher derivatives on the halfline, Math. Res. Cent. Tech. Summary Rep. 1050, Madison, Wisconsin, 1970.
  • [12] P. Smith, $ {W^{r,p}}(R)$-splines, Dissertation, Purdue University, Lafayette, Indiana, June, 1972.
  • [13] Kôsaku Yosida, Functional analysis, Die Grundlehren der Mathematischen Wissenschaften, Band 123, Academic Press Inc., New York, 1965. MR 0180824 (31 #5054)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A65

Retrieve articles in all journals with MSC: 41A65


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0364983-X
PII: S 0002-9947(1974)0364983-X
Keywords: Minimization, interpolation, spline, fundamental interval of uniqueness, Tchebycheff system
Article copyright: © Copyright 1974 American Mathematical Society