Geodesic
On Asymptotics of the~Jump of~Highest Derivative for a~Polynomial Spline
Matematičeskie trudy, Tome 5 (2002) no. 1, pp. 66-73.

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In this article, we obtain $2[n/2]+2$ terms ($[\boldsymbol{\cdot}]$ stands for the integer part) of the asymptotic expansion of the error $$ \bigl(S^{(n)}({}\,\overline{\kern-.3mm x}_i+0)-S^{(n)}({}\,\overline{\kern-.3mm x}_i-0)\bigr)\big/h-f^{(n+1)}({}\,\overline{\kern-.3mm x}_i), $$ where $S(x)$ is a periodic spline of degree $n\ge 0$ and deficiency 1 that interpolates a periodic sufficiently smooth function $f(x)$ at the nodes $x_i$ ($i=0,\pm1,\dots$) of a uniform mesh of width $h$. The nodes of the spline are the points ${}\,\overline{\kern-.3mm x}_i=x_i+h\bigl(1+(-1)^n\bigr)/4$. The expansion coefficients are represented explicitly in terms of the values of the Bernoulli polynomials at 0 for $n$ odd and 1/2 for $n$ even. 
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B. S. Kindalev. On Asymptotics of the~Jump of~Highest Derivative for a~Polynomial Spline. Matematičeskie trudy, Tome 5 (2002) no. 1, pp. 66-73. http://geodesic.mathdoc.fr/item/MT_2002_5_1_a3/

[1] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980

[2] Kvasov B. I., Interpolyatsiya kvadraticheskimi splainami, Preprint No 3, ITPM SO AN SSSR, Novosibirsk, 1981 | MR

[3] Kindalev B. S., “Asimptoticheskie formuly dlya splaina pyatoi stepeni i ikh primenenie”, Metody splain-funktsii, Sb. nauchn. tr. IM SO AN SSSR, Vychislitelnye sistemy, 87, Izd-vo In-ta matematiki, Novosibirsk, 1981, 18–24

[4] Kindalev B. S., “Asimptoticheskie formuly dlya splainov nechetnoi stepeni i approksimatsiya proizvodnykh vysokogo poryadka”, Metody splain-funktsii, Sb. nauchn. tr. IM SO AN SSSR, Vychislitelnye sistemy, 93, Izd-vo In-ta matematiki, Novosibirsk, 1982, 39–52 | MR

[5] Kindalev B. S., “O tochnosti priblizheniya periodicheskimi interpolyatsionnymi splainami nechetnoi stepeni”, Metody splain-funktsii v chislennom analize, Sb. nauchn. tr. IM SO AN SSSR, Vychislitelnye sistemy, 98, Izd-vo In-ta matematiki, Novosibirsk, 1983, 67–82

[6] Kindalev B. S., “Asimptotika pogreshnosti i superskhodimost periodicheskikh interpolyatsionnykh splainov chetnoi stepeni”, Splainy v vychislitelnoi matematike, Sb. nauchn. tr. IM SO AN SSSR, Vychislitelnye sistemy, 115, Izd-vo In-ta matematiki, Novosibirsk, 1986, 3–25

[7] Kindalev B. S., “Reshenie periodicheskoi kraevoi zadachi dlya differentsialnogo uravneniya $2m$-go poryadka s ispolzovaniem splaina stepeni $2m-1$”, Interpolyatsiya i approksimatsiya splainami, Sb. nauchn. tr. IM SO RAN, Vychislitelnye sistemy, 147, Izd-vo In-ta matematiki, Novosibirsk, 1992, 68–83

[8] Spravochnik po spetsialnym funktsiyam, ed. M. Abramovits, I. Stigan, Nauka, M., 1979

[9] Albasiny E. L., Hoskins W. D., “Explicit error bounds for periodic splines of odd order on a uniform mesh”, J. Inst. Math. Appl., 12:3 (1973), 303–318 | DOI | MR | Zbl

[10] Hoskins W. D., Meek D. S., “Linear dependence relations for polynomial splines at midknots”, BIT, 15 (1975), 272–276 | DOI | MR | Zbl

[11] Kindalev B. S., “Asymtotics of error for interpolating splines of even degree”, Constructive Theory of Functions'84, Sofia, 1984, 445–450

[12] Lucas T. R., “Error bounds for interpolating cubic splines under various end conditions”, SIAM J. Numer. Anal., 11:3 (1974), 569–584 | DOI | MR | Zbl

[13] Lucas T. R., “Asymptotic expansions for interpolating periodic splines”, SIAM J. Numer. Anal., 19:5 (1982), 1051–1066 | DOI | MR | Zbl