On divergence of sinc-approximations everywhere on~$(0,\pi)$
Algebra i analiz, Tome 22 (2010) no. 4, pp. 232-256.

Voir la notice de l'article provenant de la source Math-Net.Ru

Some properties of sinc-approximations of continuous functions on a segment are studied.
Keywords: sinc-approximation, interpolation, Whittaker cardinal functions, convergence.
@article{AA_2010_22_4_a7,
     author = {A. Yu. Trynin},
     title = {On divergence of sinc-approximations everywhere on~$(0,\pi)$},
     journal = {Algebra i analiz},
     pages = {232--256},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_4_a7/}
}
TY  - JOUR
AU  - A. Yu. Trynin
TI  - On divergence of sinc-approximations everywhere on~$(0,\pi)$
JO  - Algebra i analiz
PY  - 2010
SP  - 232
EP  - 256
VL  - 22
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2010_22_4_a7/
LA  - ru
ID  - AA_2010_22_4_a7
ER  - 
%0 Journal Article
%A A. Yu. Trynin
%T On divergence of sinc-approximations everywhere on~$(0,\pi)$
%J Algebra i analiz
%D 2010
%P 232-256
%V 22
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2010_22_4_a7/
%G ru
%F AA_2010_22_4_a7
A. Yu. Trynin. On divergence of sinc-approximations everywhere on~$(0,\pi)$. Algebra i analiz, Tome 22 (2010) no. 4, pp. 232-256. http://geodesic.mathdoc.fr/item/AA_2010_22_4_a7/

[1] Zhuk A. S., Zhuk V. V., “Nekotorye ortogonalnosti v teorii priblizheniya”, Zap. nauch. semin. POMI, 314, 2004, 83–123 | MR | Zbl

[2] Schmeisser G., Stenger F., “Sinc approximation with a Gaussian multiplier”, Sampl. Theory Signal Image Process, 6:2 (2007), 199–221 | MR | Zbl

[3] Ignjatović A., “Local approximations based on orthogonal differential operators”, J. Fourier Anal. Appl., 13:3 (2007), 309–330 | DOI | MR | Zbl

[4] Gelb A., “Reconstruction of piecewise smooth functions from non-uniform grid point data”, J. Sci. Comput., 30:3 (2007), 409–440 | DOI | MR | Zbl

[5] Annaby M. H., Tharwat M. M., “Sinc-based computations of eigenvalues of Dirac systems”, BIT, 47 (2007), 699–713 | DOI | MR | Zbl

[6] Stenger F., Numerical methods based on sinc and analytic functions, Springer Ser. in Comput. Math., 20, Springer-Verlag, New York, 1993 | MR | Zbl

[7] Higgins J. R., “Five short stories about the cardinal series”, Bull. Amer. Math. Soc. (N.S.), 12:1 (1985), 45–89 | DOI | MR | Zbl

[8] Novikov I. Ya., Stechkin S. B., “Osnovy teorii vspleskov”, Uspekhi mat. nauk, 53:6 (1998), 53–128 | MR | Zbl

[9] Novikov I. Ya., Stechkin S. B., “Osnovnye konstruktsii vspleskov”, Fundam. i prikl. mat., 3:4 (1997), 999–1028 | MR | Zbl

[10] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, AFTs, M., 1999 | MR | Zbl

[11] Dobeshi I., Desyat lektsii po veivletam, Regulyarnaya i khaoticheskaya dinamika, Izhevsk, 2001, 464 pp.

[12] Butzer P. L., Hinsen G., “Reconstruction of bounded signals from pseudo-periodic, irregularly spaced samples”, Signal Process, 17 (1989), 1–17 | DOI | MR

[13] Higgins J. R., “Sampling theorems and the contour integral method”, Appl. Anal., 41 (1991), 155–169 | DOI | MR | Zbl

[14] Hinsen G., “Irregular sampling of bandlimited $L^p$-functions”, J. Approx. Theory, 72 (1993), 346–364 | DOI | MR | Zbl

[15] Kramer H. P., “A generalized sampling theorem”, J. Math. Phys., 38 (1959), 68–72 | MR | Zbl

[16] Zayed A. I., Hinsen G., Butzer P. L., “On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm–Liouville problems”, SIAM J. Appl. Math., 50:3 (1990), 893–909 | DOI | MR | Zbl

[17] McArthur K. M., Bowers K. L., Lund J., “The sinc method in multiple space dimensions: model problems”, Numer. Math., 56 (1990), 789–816 | DOI | MR | Zbl

[18] Ebata M., Eguchi M., Koizumi Sh., Kumahara K., “On sampling formulas on symmetric spaces”, J. Fourier Anal. Appl., 12:1 (2006), 1–15 | DOI | MR | Zbl

[19] Boumenir A., “Computing eigenvalues of Lommel-type equations by the sampling method”, J. Comput. Anal. Appl., 2:4 (2000), 323–332 | MR | Zbl

[20] Mohsen A., El-Gamel M., “A sinc-collocation method for the linear Fredholm integro-differential equations”, Z. Angew. Math. Phys., 58:3 (2007), 380–390 | DOI | MR | Zbl

[21] Hackbusch W., Khoromskij B. N., “Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions”, Computing, 76 (2006), 177–202 | DOI | MR | Zbl

[22] El-Gamel M., Cannon J. R., “On the solution of a second order singularly-perturbed boundary value problem by the sinc-Galerkin method”, Z. Angew. Math. Phys., 56 (2005), 45–58 | DOI | MR | Zbl

[23] Walter G. G., Shen X., “Wavelets based on prolate spheroidal wave functions”, J. Fourier Anal. Appl., 10:1 (2004), 1–26 | DOI | MR | Zbl

[24] Sun Q., “Frames in spaces with finite rate of innovatio”, Adv. Comput. Math., 28 (2008), 301–329 | DOI | MR | Zbl

[25] Li H. A., Fang G. S., “Sampling theorem of Hermite type and aliasing error on the Sobolev class of functions”, Front. Math. China, 1:2 (2006), 252–271 | DOI | MR

[26] Butzer P. L., Stens R. L., “A modification of the Whittaker–Kotelnikov–Shannon sampling series”, Aequationes Math., 28 (1985), 305–311 | DOI | MR | Zbl

[27] Butzer P. L., Higgins J. R., Stens R. L., “Classical and approximate sampling theorems: studies in the $L^p(\mathbb R)$ and the uniform norm”, J. Approx. Theory, 137 (2005), 250–263 | DOI | MR | Zbl

[28] Trynin A. Yu., “Ob approksimatsii analiticheskikh funktsii operatorami Lagranzha–Shturma–Liuvillya”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Tez. dokl. 10-i Saratov. zim. shk. (27 yanv. – 2 fevr. 2000 g.), Saratov. un-t, Saratov, 2000, 140–141

[29] Trynin A. Yu., “Ob otsenke approksimatsii analiticheskikh funktsii interpolyatsionnym operatorom po sinkam”, Matematika. Mekhanika, Saratov. un-t, Saratov, 2005, 124–127

[30] Berrut Jean-Paul, “A formula for the error of finite sinc-interpolation over a finite interval”, Numer. Algorithms, 45 (2007), 369–374 | DOI | MR | Zbl

[31] Trynin A. Yu., “Otsenki funktsii Lebega i formula Nevai dlya sinc-priblizhenii nepreryvnykh funktsii na otrezke”, Sib. mat. zh., 48:5 (2007), 1155–1166 | MR | Zbl

[32] Trynin A. Yu., “Kriterii potochechnoi i ravnomernoi skhodimosti sink-priblizhenii nepreryvnykh funktsii na otrezke”, Mat. sb., 198:10 (2007), 141–158 | MR | Zbl

[33] Sklyarov V. P., “On the best uniform sinc-approximation on a finite interval”, East J. Approx., 14:2 (2008), 183–192 | MR

[34] Trynin A. Yu., “Kriterii ravnomernoi skhodimosti sinc-priblizhenii na otrezke”, Izv. vuzov. Mat., 2008, no. 6, 66–78 | MR | Zbl

[35] Trynin A. Yu., Sklyarov V. P., “Error of sinc approximation of analytic functions on an interval”, Sampl. Theory Signal Image Process, 7:3 (2008), 263–270 | MR | Zbl

[36] Grünwald G., “Über Divergenzerscheinungen der Lagrangeschen Interpolationspolynome stetiger Funktionen”, Ann. of Math. (2), 37 (1936), 908–918 | DOI | MR | Zbl

[37] Marcinkiewicz J., “Sur la divergence des polynômes d'interpolation”, Acta Litterarum as Scientiarum, 8 (1937), 131–135 | Zbl

[38] Privalov A. A., “O raskhodimosti interpolyatsionnykh protsessov Lagranzha po uzlam Yakobi na mnozhestve polozhitelnoi mery”, Sib. mat. zh., 17:4 (1976), 837–859 | MR | Zbl

[39] Carleson L., “On convergence and growth of partial sumas of Fourier series”, Acta Math., 116:1–2 (1966), 135–157 | DOI | MR | Zbl

[40] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[41] Nikolskii S. M., Kurs matematicheskogo analiza, v. 2, Nauka, M., 1983