Asymptotic properties of best $L_{2}[0, 1]$ approximation by splines with variable knots
Authors:
D. L. Barrow and P. W. Smith
Journal:
Quart. Appl. Math. 36 (1978), 293-304
MSC:
Primary 41A15
DOI:
https://doi.org/10.1090/qam/508773
MathSciNet review:
508773
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Let $S_N^k$ be the set of $k$th-order splines on $\left [ {0, 1} \right ]$ having at most $N - 1$ interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best ${L_2}\left [ {0, 1} \right ]$ approximation of a sufficiently smooth function $f$ by the set $S_N^k$: \[ \lim \limits _{N \to \infty } {N^k}dist\left ( {f, S_N^k} \right ) = {\left ( {\left | {{B_{2k}}} \right |/\left ( {2K} \right )!} \right )^{1/2}}{\left ( {{{\int _0^1 {\left | {{f^{\left ( k \right )}}\left ( \tau \right )} \right |} }^\sigma }d\tau } \right )^{1/\sigma }}\], where $\sigma = {\left ( {k + 1/2} \right )^{ - 1}}$ and ${B_{2k}}$ is the $2k$th Bernoulli number. Similar results have previously been obtained for piecewise polynomial (i.e., with no continuity constraints) approximation, but with different constant before the integral term. The approach we use is first to study the asymptotic behavior of dist($f$, $S_N^k\left ( t \right )$), where $S_N^k\left ( t \right )$ is the linear space of $k$th-order splines having simple knots determined from the fixed function $t$ by the rule ${t_i} \\ = t\left ( {i/N} \right ),i = 0,...,N$.
- 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, 1131β1143. MR 481754, DOI https://doi.org/10.1090/S0025-5718-1978-0481754-1
- Carl de Boor, On calculating with $B$-splines, J. Approximation Theory 6 (1972), 50β62. MR 338617, DOI https://doi.org/10.1016/0021-9045%2872%2990080-9
- Carl de Boor, The quasi-interpolant as a tool in elementary polynomial spline theory, Approximation theory (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973) Academic Press, New York, 1973, pp. 269β276. MR 0336159
C. de Boor, Good approximation by splines with variable knots, in Spline functions and approximation theory, A. Meir and A. Sharma, eds., BirkhoΓΌser, Basel, 1972, 57β72
C. de Boor, Good approximation with variable knots, II, in Conference on the numerical solution of differential equations, Dundee, 1973, Springer Lecture Notes, vol. 363, 1974, 12β20
- Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 309β319. MR 399708, DOI https://doi.org/10.1080/00036817408839073
- H. G. Burchard and D. F. Hale, Piecewise polynomial approximation on optimal meshes, J. Approximation Theory 14 (1975), no. 2, 128β147. MR 374761, DOI https://doi.org/10.1016/0021-9045%2875%2990084-2
- H. G. Burchard, On the degree of convergence of piecewise polynomial approximation on optimal meshes, Trans. Amer. Math. Soc. 234 (1977), no. 2, 531β559. MR 481758, DOI https://doi.org/10.1090/S0002-9947-1977-0481758-4
D. S. Dodson, Optimal order approximation by polynomial spline functions, Ph.D. thesis, Purdue Univ., Lafayette, Ind., 1972
- Kurt Jetter and Gerald Lange, Die Eindeutigkeit $L_{2}$-optimaler polynomialer Monosplines, Math. Z. 158 (1978), no. 1, 23β34 (German, with English summary). MR 467094, DOI https://doi.org/10.1007/BF01214562
- Donald E. McClure, Nonlinear segmented function approximation and analysis of line patterns, Quart. Appl. Math. 33 (1975/76), 1β37. MR 463769, DOI https://doi.org/10.1090/S0033-569X-1975-0463769-X
- Jerome Sacks and Donald Ylvisaker, Designs for regression problems with correlated errors. III, Ann. Math. Statist. 41 (1970), 2057β2074. MR 270530, DOI https://doi.org/10.1214/aoms/1177696705
- Jerome Sacks and Donald Ylvisaker, Statistical designs and integral approximation, Proc. Twelfth Biennial Sem. Canad. Math. Congr. on Time Series and Stochastic Processes; Convexity and Combinatorics (Vancouver, B.C., 1969) Canad. Math. Congr., Montreal, Que., 1970, pp. 115β136. MR 0277069
- I. J. Schoenberg, Monosplines and quadrature formulae, Theory and Applications of Spline Functions (Proceedings of Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1968) Academic Press, New York, 1969, pp. 157β207. MR 0241865
- Grace Wahba, On the regression design problem of Sacks and Ylvisaker, Ann. Math. Statist. 42 (1971), 1035β1053. MR 279955, DOI https://doi.org/10.1214/aoms/1177693331
- David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, N. J., 1941. MR 0005923
D. Barrow, C. Chui, P. Smith, and J. Ward, Unicity of best mean approximation by second order splines with variable knots, Math. Comp., to appear
C. de Boor, On calculating with B-splines, J. Approximation Theory 6, 50β62 (1972)
C. de Boor, The quasi-interpolant as a tool in elementary polynomial spline theory, in Approximation theory (G. G. Lorentz, ed.), Academic Press, New York, 1973, 269β276
C. de Boor, Good approximation by splines with variable knots, in Spline functions and approximation theory, A. Meir and A. Sharma, eds., BirkhoΓΌser, Basel, 1972, 57β72
C. de Boor, Good approximation with variable knots, II, in Conference on the numerical solution of differential equations, Dundee, 1973, Springer Lecture Notes, vol. 363, 1974, 12β20
H. G. Burchard, Splines (with optimal knots) are better, J. Applicable Analysis 3, 309β319 (1974)
H. G. Burchard and D. F. Hale, Piecewise polynomial approximation on optimal meshes, J. Approximation Theory 14, 128β147 (1975)
H. G. Burchard, On the degree of convergence of piecewise polynomial approximation on optimal meshes, Trans. Amer. Math. Soc. 234, 531β559 (1977)
D. S. Dodson, Optimal order approximation by polynomial spline functions, Ph.D. thesis, Purdue Univ., Lafayette, Ind., 1972
K. Jetter and G. Lange, Die Eindeutigkeit ${L_2}$-optimaler polynomialer Monospline, Math. Z. (to appear).
D. McClure, Nonlinear segmented function approximation and analysis of line patterns, Quart. Appl. Math. 33, 1β37 (1975)
J. Sacks and D. Ylvisaker, Designs for regression problems with correlated errors III, Ann. Math. Statist. 41, 2057β2074 (1970)
J. Sacks and D. Ylvisaker, Statistical designs and integral approximation, in Proceedings of the 12th Biennial Canadian Mathematical Society Seminar (Ronald Pyke, ed.), 1971, 115β136
I. J. Schoenberg, Monosplines and quadrature formulae, in Theory and applications of spline functions (T. N. E. Greville, ed.), Academic Press, New York, 1969, 157-207
G. Wahba, On the regression problem of Sacks and Ylvisaker, Ann. Math. Statist. 42, 1035β1053 (1971)
D. V. Widder, The Laplace transform, Princeton University Press, Princeton, 1946
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
41A15
Retrieve articles in all journals
with MSC:
41A15
Additional Information
Article copyright:
© Copyright 1978
American Mathematical Society