Abstract
Anselone and Laurent [1], Schoenberg [12], and de Boor [2] have given interesting applications of splines to smoothing of histograms by matching the integral means of splines between successive knots with the same means of a given function. Subbotin [14] has also considered integral means of cardinal splines (see also [7]). These investigations have been continued in a recent paper by Shevaldin [13]. It is the purpose of our paper to study the corresponding problem for classes of periodic functions which are constructed from translates of a generating function with respect to a uniform mesh. We apply our method to periodic splines of arbitrary degree and trigonometric polynomials.
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References
P.M. Anselone and P.-J. Laurent: A general method for the construction of interpolating or smoothing splines, Numer. Math. 12(1968), 66–82.
C. de Boor: Appendix to: I.J. Schoenberg, Splines and histograms, in “Spline Functions and Approximation Theory” (A. Meir and A. Sharma, eds.), ISNM 21, 329-358, Birkhäuser Verlag, Basel, 1973.
L. Collatz und W. Quade: Zur Interpolationstheorie der reellen periodischen Funktionen, Sitzungsberichte der Preußischen Akademie der Wissenschaften 30(1938),383–429.
F.-J. Delvos: Periodic interpolation on uniform meshes, Journal of Approximation Theory (to appear).
F.-J. Delvos: Interpolation of odd periodic functions on uniform meshes, in “Delay Equations, Approximation and Application” (G. Meinardus, G. Nürnberger, eds.), ISNM 74, 105-121, Birkhäuser Verlag, Basel, 1985.
F.-J. Delvos: Bernoulli functions and periodic B-splines, Computing (to appear).
F.-J. Delvos: Limits of abstract splines, in “Inverse and Ill-posed Problems” (H. W. Engl and C. W. Groetsch, eds.), Academic Press, New York, 1987 (to appear).
F. Locher: Interpolation on uniform meshes by translates of one function and related attentuation factors, Mathematics of Computation 37(1981), 403–416.
G. Meinardus: Periodische Spline-Funktionen, in “Spline-Functions, Karlsruhe 1975” (K. Böhmer, G. Meinardus, W. Schempp, eds.), Lecture Notes in Mathematics 501(1976), 177-199.
G. Merz: Interpolation mit periodischen Spline-Funktionen I, Journal of Approximation Theory 30(1980), 11–19.
H. ter Morsche: On the existence and uniqueness of interpolating splines of arbitrary degree, in “Spline-Funktio= nen” (K. Böhmer, G. Meinardus, W. Schempp, eds.), Bibliographisches Institut, Mannheim, 1974.
I. J. Schoenberg: Splines and histograms, in “Spline Functions and Approximation Theory” (A. Meir and A. Sharma, eds.), ISNM 21, Birkhäuser Verlag, Basel, 1973.
V. T. Shevaldin: Some problems of extremal interpolation in the mean for linear differential operators, Trudy Mat. Inst. Steklov 164(1983), 233–273; English translation in Proc. Steklov Inst. Math. 164(1985).
Yu. Subbotin: Extremal problems of functional interpolation and splines for interpolation in the mean, Trudy Mat. Inst. Steklov 138(1975), 118–173, English translation in Proc. Steklov Inst. Math. 138(1975).
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Delvos, FJ. (1987). Periodic Area Matching Interpolation. In: Collatz, L., Meinardus, G., Nürnberger, G. (eds) Numerical Methods of Approximation Theory/Numerische Methoden der Approximationstheorie. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 81. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6656-9_5
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DOI: https://doi.org/10.1007/978-3-0348-6656-9_5
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