Geodesic
On the Gibbs phenomenon for rational spline functions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 238-251 Cet article a éte moissonné depuis la source Math-Net.Ru

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In the case of functions $f(x)$ continuous on a given closed interval $[a,b]$ except for jump discontinuity points, the Gibbs phenomenon is studied for rational spline functions $R_{N,1}(x)=R_{N,1}(x,f,\Delta, g)$ defined for a knot grid $\Delta: a=x_0$ and a family of poles $g_i\not \in [x_{i-1},x_{i+1}]$ $(i=1,2,\dots,N-1)$ by the equalities $R_{N,1}(x)= [R_i(x)(x-x_{i-1})+R_{i-1}(x)(x_i-x)]/(x_i-x_{i-1})$ for $x\in[x_{i-1}, x_i]$ $(i=1,2,\dots,N)$. Here the rational functions $R_i(x)=\alpha_i+\beta_i(x-x_i)+\gamma_i/(x-g_i)$ $(i=1,2,\dots,N-1)$ are uniquely defined by the conditions $R_i(x_j)=f(x_j)$ $(j=i-1,i,i+1)$; we assume that $R_0(x)\equiv R_1(x)$, $R_N(x)\equiv R_{N-1}(x)$. Conditions on the knot grid $\Delta$ are found under which the Gibbs phenomenon occurs or does not occur in a neighborhood of a discontinuity point.
Mots-clés : interpolation spline
Keywords: rational spline, Gibbs phenomenon. 
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A.-R. K. Ramazanov; A.-K. K. Ramazanov; V. G. Magomedova. On the Gibbs phenomenon for rational spline functions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 2, pp. 238-251. http://geodesic.mathdoc.fr/item/TIMM_2020_26_2_a18/

[1] Matematicheskaya entsiklopediya, v. 1, Sovetskaya Entsiklopediya, M., 1977

[2] Bari N.K., Trigonometricheskie ryady, GIFML, M., 1961

[3] Jerri A.J., The Gibbs phenomenon in Fourier analysis, splines and wavelet approximations, Math. Appl., 446, Springer, Boston, 1998, 340 pp. | DOI | MR

[4] Golubov B.I., “On Gibbs phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series”, Analysis Mathematica, 4:4 (1978), 269–287 | DOI | MR | Zbl

[5] Olevska Yu.B., Olevskyi V.I., Shapka I.V., Naumenko T.S., “Application of two-dimensional Pade-type approximants for reducing the Gibbs phenomenon”, AIP Conf. Proc., 2164 (2019), 060014 | DOI

[6] Mohammad M., “On the Gibbs effect based on the quasi-affine dual tight framelets system generated using the mixed oblique extension principle”, Mathematics, 7:10 (2019), 952, 14 pp. | DOI

[7] Lin S., Xu Y., Chen Y., Chang C., Chen Y.E., Chen J., “Gibbs-phenomenon-reduced digital PWM for power amplifiers using pulse modulation”, IEEE Access, 7 (2019), 178788–178797 | DOI

[8] Subbotin Yu.N., “Variatsii na temu splainov”, Fundament. i prikl. matematika, 3:4 (1997), 1043–1058 | MR | Zbl

[9] Zavyalov Yu.S., Kvasov B. I., Miroshnichenko V. L., Metody splain–funktsii, Nauka, M., 1980, 352 pp. | MR

[10] Andreev A. S., “On interpolation by cubic splines of a function possessing discontinuities”, C. R. Acad. Bulg. Sci., 27 (1974), 881–884 | MR | Zbl

[11] Richards F. B., “A Gibbs phenomenon for spline functions”, J. Approximation Theory, 66 (1991), 344–351 | DOI | MR | Zbl

[12] Zhimin Zhang, Clyde F. Martin., “Convergence and Gibbs phenomenon in cubic spline interpolation of discontinuous functions”, J. Computational and Applied Mathematics, 87 (1997), 359–371 | DOI | MR | Zbl

[13] Kvasov B. I., Kobkov V. V., “Nekotorye svoistva kubicheskikh ermitovykh splainov s dopolnitelnymi uzlami”, Dokl. AN SSSR, 217:5 (1974), 1007–1010 | Zbl

[14] Ramazanov A.-R. K., Magomedova V. G., “Bezuslovno skhodyaschiesya interpolyatsionnye ratsionalnye splainy”, Mat. zametki, 103:4 (2018), 592–603 | DOI | Zbl

[15] Ramazanov A.-R. K., Magomedova V. G., “Splainy po trekhtochechnym ratsionalnym interpolyantam s avtonomnymi polyusami”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 16–28 | DOI