Geodesic
On quasilinear interpolation by minimal splines
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVI, Tome 524 (2023), pp. 94-111 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The paper studies quasilinear interpolation by minimal splines that are constructed on nonuniform grids with multiple nodes. Asymptotic representations for normalized splines are obtained.The sharpness of biorthogonal approximation and the order of accuracy of quasilinear interpolation with respect to the grid step are established. Results of numerical experiments on approximating some test functions, which demonstrate the effect of choosing a generating vector function in constructing the corresponding minimal spline, are presented. 
@article{ZNSL_2023_524_a7,
     author = {L. P. Livshits and A. A. Makarov and S. V. Makarova},
     title = {On quasilinear interpolation by minimal splines},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {94--111},
     year = {2023},
     volume = {524},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a7/}
}
TY  - JOUR
AU  - L. P. Livshits
AU  - A. A. Makarov
AU  - S. V. Makarova
TI  - On quasilinear interpolation by minimal splines
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2023
SP  - 94
EP  - 111
VL  - 524
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a7/
LA  - ru
ID  - ZNSL_2023_524_a7
ER  - 
%0 Journal Article
%A L. P. Livshits
%A A. A. Makarov
%A S. V. Makarova
%T On quasilinear interpolation by minimal splines
%J Zapiski Nauchnykh Seminarov POMI
%D 2023
%P 94-111
%V 524
%U http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a7/
%G ru
%F ZNSL_2023_524_a7
L. P. Livshits; A. A. Makarov; S. V. Makarova. On quasilinear interpolation by minimal splines. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVI, Tome 524 (2023), pp. 94-111. http://geodesic.mathdoc.fr/item/ZNSL_2023_524_a7/

[1] Yu. S. Volkov, V. V. Bogdanov, “O pogreshnosti priblizheniya prosteishei lokalnoi approksimatsiei splainami”, Sib. matem. zh., 61:5 (2020), 1000–1008 | MR | Zbl

[2] Yu. K. Demyanovich, “O postroenii prostranstv lokalnykh funktsii na neravnomernoi setke”, Zap. nauchn. semin. LOMI, 124 (1983), 140–163 | Zbl

[3] Yu. K. Demyanovich, Lokalnaya approkcimatsiya na mnogoobrazii i minimalnye cplainy, SPb, 1994

[4] Yu. K. Demyanovich, Vspleski i minimalnye cplainy, SPb, 2003

[5] Yu. K. Demyanovich, Teoriya cplain-vspleskov, SPb, 2013

[6] Yu. K. Demyanovich, V. O. Dron, O. N. Ivantsova, “Ob approksimatsii $B_{\varphi }$-splainami”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 3 (2013), 67–72

[7] Yu. K. Demyanovich, A. A. Makarov, “Neobkhodimye i dostatochnye usloviya neotritsatelnosti koordinatnykh trigonometricheskikh splainov vtorogo poryadka”, Vestn. S.-Peterb. un-ta. Ser. 1, 4(62):1 (2017), 9–16 | MR

[8] A. Yu. Demyanovich, V. N. Malozemov, “Funktsii Dzhenkinsa i minimalnye splainy”, Vestn. S.-Peterb. un-ta. ser. 1, 2:8 (1998), 38–43 | Zbl

[9] Yu. K. Demyanovich, S. G. Mikhlin, “O setochnoi approksimatsii funktsii sobolevskikh prostranstv”, Zap. nauchn. semin. LOMI, 35 (1973), 6–11 | Zbl

[10] Yu. S. Zavyalov, B. I. Kvasov, V. L. Miroshnichenko, Metody splain-funktsii, M., 1980 | MR

[11] E. K. Kulikov, A. A. Makarov, “Ob approksimatsii giperbolicheskimi splainami”, Zap. nauchn. semin. POMI, 472 (2018), 179–194

[12] E. K. Kulikov, A. A. Makarov, “O kvadratichnykh minimalnykh splainakh s kratnymi uzlami”, Zap. nauchn. semin. POMI, 482 (2019), 220–230

[13] E. K. Kulikov, A. A. Makarov, “O modifitsirovannom metode splain-kollokatsii resheniya integralnogo uravneniya Fredgolma”, Diff. ur. prots. upr., 4 (2021), 211–223 | Zbl

[14] E. K. Kulikov, A. A. Makarov, “Ob odnom metode resheniya integralnogo uravneniya Fredgolma pervogo roda”, Zap. nauchn. semin. POMI, 514 (2022), 113–125

[15] A. A. Makarov, “O postroenii splainov maksimalnoi gladkosti”, Probl. matem. anal., 60 (2011), 25–38 | Zbl

[16] A. A. Makarov, “Biortogonalnye sistemy funktsionalov i matritsy dekompozitsii dlya minimalnykh splainov”, Ukr. mat. visn., 9:2 (2012), 219–236 | Zbl

[17] A. A. Makarov, “O dvoistvennykh funktsionalakh k minimalnym splainam”, Zap. nauchn. semin. POMI, 453 (2016), 198–218

[18] S. G. Mikhlin, “Variatsionno-setochnaya approksimatsiya”, Zap. nauchn. semin. LOMI, 48 (1974), 32–188 | MR | Zbl

[19] V. S. Ryabenkii, Ob ustoichivosti konechno-raznostnykh uravnenii, Dis. kand. fiz.-mat. nauk, M., 1952 | Zbl

[20] G. Streng, Dzh. Fiks, Teoriya metoda konechnykh elementov, M., 1977 | MR

[21] B. M. Shumilov, E. A. Esharov, N. K. Arkabaev, “Postroenie i optimizatsiya prognozov na osnove rekurrentnykh splainov pervoi stepeni”, Sib. zh. vychisl. matem., 13:2 (2010), 227–241 | Zbl

[22] I. V. Yuyukin, “Interpolyatsiya navigatsionnoi funktsii splainom lagranzheva tipa”, Vestn. Gos. un-ta morsk. rechn. flota, 12:1 (2020), 57–70

[23] C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 2001 | MR | Zbl

[24] M. Buhmann, J. Jäger, Quasi-Interpolation, Cambridge University Press, 2022 | MR

[25] G. Y. Deniskina, Y. I. Deniskin, Y. I. Bityukov, “About some computational algorithms for local approximation splines, based on the wavelet transformation and convolution”, Lect. Notes Electr. Eng., 729, 2021, 182–191 | DOI

[26] J. J. Goel, “Construction of basis functions for numerical utilization of Ritz's method”, Numer. Math., 12 (1968), 435–447 | DOI | MR | Zbl

[27] O. Kosogorov, A. Makarov, “On some piecewise quadratic spline functions”, Lect. Notes Comput. Sci., 10187, 2017, 448–455 | DOI | MR | Zbl

[28] E. Kulikov, A. Makarov, “On de Boor–Fix type functionals for minimal splines”, Topics in Classical and Modern Analysis, Applied and Numerical Harmonic Analysis, 2019, 211–225 | DOI | MR

[29] E. Kulikov, A. Makarov, “On biorthogonal approximation of solutions of some boundary value problems on Shishkin mesh”, AIP Conf. Proc., 2302 (2020), 110005 | DOI

[30] A. A. Makarov, “On example of circular arc approximation by quadratic minimal splines”, Poincare J. Anal. Appl., 2(II) (2018), 103–107 | DOI | MR | Zbl

[31] V. L. Leontiev, “Orthogonal splines in approximation of functions”, Math. Statistic, 8:2 (2020), 167–172 | DOI

[32] I. J. Schoenberg, “Contributions to the problem of approximation of equidistant data by analytic functions”, Quart. Appl. Math., 4 (1946), 45–99 ; 112–141 | DOI | MR | Zbl

[33] L. L. Schumaker, Spline Functions: Computational Methods, Society for Industrial and Applied Mathematics, 2015 | MR | Zbl

[34] G. Strang, G. Fix, “Fourier analysis of the finite element method in Ritz–Galerkin theory”, Stud. Appl. Math., 48:3 (1969), 265–273 | DOI | MR | Zbl

[35] A. I. Zadorin, “Interpolation method for a function with a singular component”, Lect. Notes Comp. Sci., 5434, 2009, 612–619 | DOI | Zbl