Abstract
Let F(x) be a function from ℝ to ℂ and let
be the spline function of degree 2m−1, with knots at the integers, that interpolates F(x) at all the integers. We know that S m (x), as defined by (1), exists if F(x) grows at most like a power | x |γ as x → ± ∞ (see [7]). Here we investigate conditions which will insure that S m (x) will converge to F(x) as m → ∞.
Sponsored by U.S. Army under contract No. DA—31–124—ARO—D—462.
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References
R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954.
P. R. Lipow and I. J. Schoenberg, Cardinal interpolation and spline functions III. Cardinal Hermite interpolation, J. Linear Algebra Appl. 6 (1973), 273–304. Also MRC T. S. Rep. #1113
W. Quade and L. Collatz, Zur Interpolationstheorie der reelen periodischen Funktionen, Sitzungsber. der Preuss. Akad. Wiss., Phys.-Math. Kl. XXX (1938), 383–429.
I. J. Schoenberg, Contributions to the problem of approximatiion of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946), 45–99, 112-141.
I. J. Schoenberg, On spline interpolation at all integer points of the real axis, Mathematica (Cluj) 10 (1968), 151–170.
I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory 2 (1969), 167–206.
I. J. Schoenberg, Cardinal interpolation and spline functions II. Interpolation of data of power growth, J. Approximation Theory 6 (1972), 404–420. Also MRC T.S. Rep. #1104.
I. J. Schoenberg, Cardinal interpolation and spline functions IV. The exponential Euler splines, in: Linear Operators and Approximation (Proc. of Oberwolfach Conference, August 14-22, 1971), ISNM 20 (1972), 382-404. Also MRC T.S. Rep. #1153
I.J. Schoenberg, Cardinal interpolation and spline functions VI. Semi-cardinal interpolation and quadrature formulae, MRC T.S.R. #1180, J. Analyse Math. 27 (1974), 159–204.
I. J. Schoenberg and A. Sharma, Cardinal interpolation and spline functions V. The B’Splines for cardinal Hermite interpolation, Linear Algebra Appl. 7 (1973), 1–42. Also MRC T.S. Rep. #1150
J. M. Whittaker, On the cardinal function of interpolation theory, Proc. Edinburgh Math. Soc. 1 (1929), 41–46.
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© 1988 Springer Science+Business Media New York
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Schoenberg, I.J. (1988). Cardinal Interpolation and Spline Functions VII. the Behavior of Cardinal Spline Interpolants as their Degree Tends to Infinity. In: de Boor, C. (eds) I. J. Schoenberg Selected Papers. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-0433-1_4
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DOI: https://doi.org/10.1007/978-1-4899-0433-1_4
Publisher Name: Birkhäuser, Boston, MA
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