Email this article On orthogonal cubic Schoenberg splines
- Authors: Leontiev V.L.1
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Affiliations:
- Peter the Great St. Petersburg Polytechnic University
- Issue: Vol 27, No 4 (2025)
- Pages: 411-421
- Section: Mathematics
- Published: 26.11.2025
- URL: https://ogarev-online.ru/2079-6900/article/view/300950
- DOI: https://doi.org/10.15507/2079-6900.27.202504.411-421
- ID: 300950
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Abstract
The modification of the mother cubic Schoenberg spline is carried out using four cubic Schoenberg splines having finite supports, the sizes of which are smaller compared to the size of the finite support of the mother spline. As a result, eight grid sets of orthogonal cubic Schoenberg splines with real values are constructed. A theorem on the order of approximation of any function of the Sobolev space by linear combinations of constructed orthogonal cubic Schoenberg splines is proved. It is shown that the order of approximation by Schoenberg splines, also modified by Schoenberg splines, is significantly higher than the order of approximation by Schoenberg splines modified by step functions, and coincides with the order of approximation by classical cubic Schoenberg splines. The defect of the modified Schoenberg spline is equal to one, as that of the classical Schoenberg spline. A modified spline is a continuous function in which there are no breaks in the first and second derivatives at the points where the parts of the mother spline and the parts of the splines used for modification meet.
About the authors
Victor L. Leontiev
Peter the Great St. Petersburg Polytechnic University
Author for correspondence.
Email: leontiev_vl@spbstu.ru
ORCID iD: 0000-0002-8669-1919
SPIN-code: 6568-4866
Scopus Author ID: 57210749321
References
- Schoenberg I. J. Contributions to problem of approximation of equidistant data by analytic functions // Quart. Appl. Math. 4 (1946). P. 45-99, 112-141.
- Strang G., Fix G. Teoriya metoda konechnih elementov. Mir, М., 1977 (In Russ.), 349 s.
- Leontiev V. L. Ortogonalnie splyni i specialnie funkcii v metodah vichislitelnoy mehaniki i matematiki. POLITEH-PRESS, SpB, 2021 (In Russ.), 466 s.
- Leontiev V. L., Mihylov I. S. O postroenii potenciala vzaimodeystvia atomov, osnovannom na ortogonalnih finitnih funkciah // Nano- i mikrosistemnaia tehnika. 9:134 (2011) (In Russ.). S. 48-50.
- Shurenko A. V., Leontiev V. L. Finitnie funkcii v algoritmah kriptografii // Prikladnaia diskretnaia matematika. 36 (2017). (In Russ.). S. 73-83. DOI: https://doi.org/10.17223/20710410/36/6.
- Leontiev V. L. On orthogonalization of Schoenberg splines // Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva. 27:2 (2025). (In Russ.). S. 111-126. DOI: https://doi.org/10.15507/2079-6900.27.202502.111-126.
- Schoenberg I. J. Spline Functions and the problem of Graduation // Proceedings of the National Academy of Sciences of USA. 52:4 (1964, Oct). P. 947-950. doi: 10.1073/pnas.52.4.947
- Alekseev V. G., Suhodoev V. A. Schoenberg’s polynomial B-splines of odd degrees: A brief review of application // Computational Mathematics and Mathematical Physics, 52:10 (2012). P. 1331–1341. DOI: https://doi.org/10.1134/S096554251
- Alekseev V. G. Schoenberg B-splyni i ih primenenia v radiotehnike i v smegnih s ney disciplinah // Radiotehnika. 12:12 (2003) (In Russ.). S. 21-23.
- Volkov Yu. S., Subbotin Yu. N. 50 years to Schoenberg's problem on the convergence of spline interpolation // Trudy Instituta Matematiki i Mekhaniki UrO RAN. 20:1 (2014). P. 52–67. DOI: https://doi.org/10.1134/S0081543815020236
- Marchuk G. I., Agoshkov V. I. Vvedenie v proekcionno-setochnie metodi. Nauka, М., 1981 (In Russ.). 416 s.
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