eesti teaduste
akadeemia kirjastus
SINCE 1952
Proceeding cover
of the estonian academy of sciences
ISSN 1736-7530 (Electronic)
ISSN 1736-6046 (Print)
Impact Factor (2022): 0.9
Periodic polynomial spline histopolation; pp. 246–251

Peeter Oja, Gul Wali Shah

Periodic polynomial spline histopolation with arbitrary placement of histogram knots is studied. Spline knots are considered coinciding with histogram knots. The main problem is the existence and uniqueness of the histopolant for any degree of spline and for any number of partition points. The results for arbitrary grid give as particular cases known assertions for the uniform grid but different techniques are used.


1. Ahlberg, J. H., Nilson, E. N., and Walsh, J. L. The Theory of Splines and Their Applications. Academic Press, New York–London, 1967.

2. Delvos, F-J. Periodic area matching interpolation. In Numerical Methods of Approximation Theory, Vol. 8. Oberwolfach, 1986; Internationale Schriftenreihe zur Numerischen Mathematik, Vol 81. Birkh¨auser, Basel, 1987, 54–66.

3. Delvos, F-J. Periodic interpolation on uniform meshes. J. Approx. Theory, 1989, 51, 71–80.

4. Dubeau, F. On band circulant matrices in the periodic spline interpolation theory. Linear Algebra and its Applications, 1985, 72, 177–182.

5. Dubeau, F. and Savoie, J. On circulant matrices for certain periodic spline and histospline projections. Bull. Austral. Math. Soc., 1987, 36, 49–59.

6. Dubeau, F. and Savoie, J. De l’interpolation á l’aide d’une fonction spline definie sur une partition quelconque. Ann. Sci. Math. Québec, 1992, 16, 25–33.

7. Kirsiaed, E., Oja, P., and Shah, G. W. Cubic spline histopolation. Math. Model. Anal., 2017, 22, 514–527.

8. Kobza, J. and Ženčák, P. Some algorithms for quartic smoothing splines. Acta Univ. Palacki. Olomuc, Mathematica, 1997, 36, 79–94.

9. Meinardus, G. and Merz, G. Zur periodischen Spline-Interpolation. In Spline-Funktionen. Bibliographisches Institut, Mannheim, 1974, 177–195.

10. Ter Morsche, H. On the existence and convergence of interpolating periodic spline functions of arbitrary degree. In Spline-Funktionen. Bibliographisches Institut Mannheim, 1974, 197–214.

11. Plonka, G. Periodic spline interpolation with shifted nodes. J. Approx. Theory, 1994, 76, 1–20.

12. Rana, S. S. Quadratic spline interpolation. J. Approx. Theory, 1989, 57, 300–305.

13. Schumaker, L. Spline Functions: Basic Theory. Wiley, NY, 1981.

14. Szyszka, U. Periodic spline interpolation on uniform meshes. Math. Nachr., 1991, 153, 109–121.

15. Zeilfelder, F. Hermite interpolation by periodic splines with equidistant knots. Commun. Appl. Anal., 1998, 2, 183–195.


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