Unicity in piecewise polynomial -approximation via an algorithm

Authors:
R. C. Gayle and J. M. Wolfe

Journal:
Math. Comp. **65** (1996), 647-660

MSC (1991):
Primary 41A15, 41A52; Secondary 41A05

MathSciNet review:
1333314

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Abstract | References | Similar Articles | Additional Information

Abstract: Our main result shows that certain generalized convex functions on a real interval possess a unique best approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [11]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the case in [5]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [10]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the case.

**1.**J. Al-Siwan,*Best segmented -approximation with free knots*, Dissertation, University of Oregon, Eugene, March 1986.**2.**D. L. Barrow, C. K. Chui, P. W. Smith, and J. D. Ward,*Unicity of best mean approximation by second order splines with variable knots*, Math. Comp.**32**(1978), no. 144, 1131–1143. MR**0481754**, 10.1090/S0025-5718-1978-0481754-1**3.**Dietrich Braess,*Nonlinear approximation theory*, Springer Series in Computational Mathematics, vol. 7, Springer-Verlag, Berlin, 1986. MR**866667****4.**H. G. Burchard and D. F. Hale,*Piecewise polynomial approximation on optimal meshes*, J. Approximation Theory**14**(1975), no. 2, 128–147. MR**0374761****5.**Jeff Chow,*On the uniqueness of best 𝐿₂[0,1] approximation by piecewise polynomials with variable breakpoints*, Math. Comp.**39**(1982), no. 160, 571–585. MR**669650**, 10.1090/S0025-5718-1982-0669650-9**6.**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****7.**R. L. Eubank,*On the relationship between functions with the same optimal knots in spline and piecewise polynomial approximation*, J. Approx. Theory**40**(1984), no. 4, 327–332. MR**740644**, 10.1016/0021-9045(84)90006-6**8.**Kurt Jetter,*𝐿₁-Approximation verallgemeinerter konvexer Funktionen durch Splines mit freien Knoten*, Math. Z.**164**(1978), no. 1, 53–66 (German). MR**514607**, 10.1007/BF01214789**9.**J. B. Kioustelidis,*Optimal segmented approximations*, Computing**24**(1980), 1-8. MR**82i:**

41032**10.**J. B. Kioustelidis,*Optimal segmented polynomial 𝐿_{𝑠}-approximations*, Computing**26**(1981), no. 3, 239–246 (English, with German summary). MR**619794**, 10.1007/BF02243481**11.**John B. Kioustelidis,*Uniqueness of optimal piecewise polynomial 𝐿₁ approximations for generalized convex functions*, Functional analysis and approximation (Oberwolfach, 1980) Internat. Ser. Numer. Math., vol. 60, Birkhäuser, Basel-Boston, Mass., 1981, pp. 421–432. MR**650294****12.**James M. Ortega,*Numerical analysis. A second course*, Academic Press, New York-London, 1972. Computer Science and Applied Mathematics. MR**0403154****13.**Allan M. Pinkus,*On 𝐿¹-approximation*, Cambridge Tracts in Mathematics, vol. 93, Cambridge University Press, Cambridge, 1989. MR**1020300****14.**M. J. D. Powell,*Approximation theory and methods*, Cambridge University Press, Cambridge-New York, 1981. MR**604014****15.**Jerry M. Wolfe,*On the convergence of an algorithm for discrete 𝐿_{𝑝} approximation*, Numer. Math.**32**(1979), no. 4, 439–459. MR**542206**, 10.1007/BF01401047

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Additional Information

**R. C. Gayle**

Affiliation:
Department of Science and Mathematics, Montana State University-Northern, P. O. Box 7751, Havre, Montana 59501

Email:
gayle@nmc1.nmclites.edu

**J. M. Wolfe**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
wolfe@bright.uoregon.edu

DOI:
https://doi.org/10.1090/S0025-5718-96-00709-0

Keywords:
Polynomial approximation,
Lagrange interpolation,
$L^{1}$ approximation

Received by editor(s):
April 13, 1994

Received by editor(s) in revised form:
January 10, 1995

Article copyright:
© Copyright 1996
American Mathematical Society