Geodesic
Polynomials, orthogonal on grids from unit circle and number axis
Daghestan Electronic Mathematical Reports, Tome 1 (2014), pp. 1-55.

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

In current paper we investigate the asymptotic properties of polynomials, orthogonal on arbitrary (not necessarily uniform) grids from an unit circle or segment $[-1,1]$. When the grid of nodes $\Omega_N^T=\left\{e^{i\theta_0},e^{i\theta_1},\ldots,e^{i\theta_{N-1}}\right\}$ belongs to the unit circle $|w|=1$ we consider polynomials $\varphi_{0,N}(w),\varphi_{1,N}(w),\ldots,$ $\varphi_{N-1,N}(w)$, orthogonal in the following sense: $$ \frac1{2\pi}\int\limits_{-\pi}^\pi \varphi_{n,N}(e^{i\theta})\overline{\varphi_{m,N}(e^{i\theta})}\,d\sigma_N(\theta)= $$ $$ \frac1{2\pi}\sum\limits^{N-1}_{j=0} \varphi_{n,N}(e^{i\theta_j})\overline{\varphi_{m,N}(e^{i\theta_j})} \Delta\sigma_N(\theta_j)=\delta_{nm}, $$ where $\Delta\sigma_N(\theta_j)=\sigma_N(\theta_{j+1})-\sigma_N(\theta_j), j=0,\ldots,N-1$. In case, when $\Delta\sigma_N(\theta_j)=h(\theta_j)\Delta\theta_j$, the asymptotic formula for $\varphi_{n,N}(w)$ is established, which in turn, used for investigation of asymptotic properties of polynomials which are orthogonal on grids from $[-1,1]$.
Keywords: unit circle, number axis, polynomials orthogonal on grids, asymptotic formulas. 
@article{DEMR_2014_1_a0,
     author = {I. I. Sharapudinov},
     title = {Polynomials, orthogonal on grids from unit circle and number axis},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {1--55},
     publisher = {mathdoc},
     volume = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2014_1_a0/}
}
TY  - JOUR
AU  - I. I. Sharapudinov
TI  - Polynomials, orthogonal on grids from unit circle and number axis
JO  - Daghestan Electronic Mathematical Reports
PY  - 2014
SP  - 1
EP  - 55
VL  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DEMR_2014_1_a0/
LA  - ru
ID  - DEMR_2014_1_a0
ER  - 
%0 Journal Article
%A I. I. Sharapudinov
%T Polynomials, orthogonal on grids from unit circle and number axis
%J Daghestan Electronic Mathematical Reports
%D 2014
%P 1-55
%V 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DEMR_2014_1_a0/
%G ru
%F DEMR_2014_1_a0
I. I. Sharapudinov. Polynomials, orthogonal on grids from unit circle and number axis. Daghestan Electronic Mathematical Reports, Tome 1 (2014), pp. 1-55. http://geodesic.mathdoc.fr/item/DEMR_2014_1_a0/

[1] Chebyshev P.L., “Ob interpolirovanii velichin ravnootstoyaschikh (1875)”, Poln. sobr. soch., v. 3, Izd. AN SSSR, M., 1948, 66–87 | MR

[2] Charlier C.V.L., Arkiv for math. astron. o. fysik. 1905/06.2, No 20

[3] Krawtchouk M.F., “Sur une generalisation des polynomes d Hermite”, Comptes Rendus de 1 Acad. des., no. 189, 620–622, Paris | Zbl

[4] Meixner J., “Orthogonale Polynom systeme mit einer besonderen Gestalt der erzeugenden Function”, Journ. of the London Mathematical Sociate, 9 (1934), 6–13 | DOI | MR

[5] Hahn W., “Uber orthogonalpolynomsisteme, dieq–Differenzengleihungen genugen”, Math. Nachr., 2 (1949), 4–34 | DOI | MR | Zbl

[6] Weber M., Erdeleyi A., “On the finite difference analogue of Rodrigues formula”, Amer. Math. Month, 59 (1952), 163–168 | DOI | MR | Zbl

[7] Karlin S., McGregor J.L., “The Hahn polnomials, formulas and an application”, Scripta Math., 26 (1961), 33–46 | MR | Zbl

[8] Delsart F., Algebraicheskii podkhod k skhemam otnoshenii teorii kodirovaniya., Mir, M., 1976 | MR

[9] Dunkl C., “A Krawtchouk Polynomials Addition Theorem and Treath Products of Simmetric Grups”, Indiana Univ. Math. J., 25 (1976), 335–358 | DOI | MR | Zbl

[10] Askey R., Wilson J.A., “A set of orthogonal polynomials that generalyze the Racach coofficients or 6j-symbols”, SIAM J. Math. Anal., 10 (1979), 1008–1016 | DOI | MR | Zbl

[11] Stanton D., “Some q–Krawtchouk Polynomials on Shevalley Grups”, Amer. J. Math., 1980, no. 102, 625–662 | DOI | MR | Zbl

[12] Koornwinder T.H., “Clebsh-Gordan coefficients for SU(2) and Hahn polynomials”, Niew arch. wisk., 29 (1981), 140–155 | MR | Zbl

[13] Levenshtein V.I., “Granitsy dlya upakovok metricheskikh prostranstv i nekotorye ikh prilozheniya”, Problemy kibernetiki, 40 (1983), 43–110 | Zbl

[14] Sharapudinov I.I., “On the Hahn polynomials application for the optimal tabulation of functions”, Constructive theory of functions, Proc. of the Int. conf. on const. theory of functions (Varna, May 27 – june 2, 1984), 782–787 | MR | Zbl

[15] Nikiforov A.F., Suslov S.K., Uvarov V.B., Klassicheskie ortogonalnye mnogochleny diskretnoi peremennoi, Nauka, M., 1985 | MR

[16] Bakhvalov N.S., Kobelkov G.M., Noskov Yu.V., O prakticheskom vychislenii znachenii ortogonalnykh mnogochlenov nepreryvnogo i diskretnogo argumenta, Preprint No 158 Otdela vych. mat. AN SSSR, M., 1987 | MR

[17] Banai E., Ito T., Algebraicheskaya kombinatorika. Skhemy otnoshenii, Mir, M., 1987 | MR

[18] Van Assche W., “Asimptotics for orthogonal polynomials”, Lect. Notes Math., 1265, 1987, 1–201 | MR

[19] Sharapudinov I.I., “Asimptotic Formula Having no Remainder Term for the Orthogonal Hahn Polynomials of Discrete Variable”, Mathematica Bolkanica. New Series, 2(4) (1988), 314–318 | MR | Zbl

[20] Sharapudinov I.I., “Asimptoticheskie svoistva polinomov Kravchuka”, Matem. zametki, 44:2 (1988), 682–693 | MR | Zbl

[21] Sharapudinov I.I., “Asimptoticheskie svoistva ortogonalnykh mnogochlenov Khana diskretnoi peremennoi”, Matem. sbornik, 180:9 (1989), 1259–1277 | Zbl

[22] Minko A.A., Petunin Yu.I., “Skhodimost metoda naimenshikh kvadratov v ravnomernoi metrike”, Sib. Mat. Zh., 31:2 (1990), 111–122 | MR

[23] Sharapudinov I.I., “Asimptoticheskie svoistva i vesovye otsenki mnogochlenov Chebysheva–Khana”, Matem. sbornik, 183:3 (1991), 408–420

[24] Sharapudinov I.I., “Asimptoticheskie svoistva polinomov Kravchuka”, Matem. zametki, 44:2 (1988), 682–693 | MR | Zbl

[25] Feinsilver P. and Schott R., “Krawtchouk polynomials and finiteprobability theory”, Probability measures on groups, X, Plenum (Oberwolfach, 1990), New York, 1991, 129–135 | MR | Zbl

[26] Sharapudinov I.I., “Ob asimptotike mnogochlenov Chebysheva, ortogonalnykh na konechnoi sisteme tochek”, Vestnik MGU. Seriya 1, 1992, no. 1, 29–35 | MR

[27] Sharapudinov I.I., “O skhodimosti metoda naimenshikh kvadratov”, Matem. zametki, 53:3 (1993), 131–143 | MR | Zbl

[28] Krasikov I. and Litsyn S., “On integral zeros of Krawtchouk polynomials”, J. Combin. Theory Ser. A, 74:1 (1996), 71–99 | DOI | MR | Zbl

[29] Sharapudinov I.I., Mnogochleny, ortogonalnye na setkakh, Izd-vo Dag. gos. ped. un-ta, Makhachkala, 1997

[30] Dzhamalov A.Sh., “Ob asimptotike polinomov Meiksnera”, Matem. zametki, 62:4 (1997), 624–625 | DOI | MR | Zbl

[31] Ismail M.E.H., Simeonov P., “Strong asymptotics for Krawtchouk polynomials”, J. Comput. Appl. Math., 1998, no. 100(2), 121–144 | DOI | MR | Zbl

[32] Mukundan R., Ramakrishnan K.R., Moment functions in image analysis: theory and applications, World Scientific Publishing Co. Pte. Ltd., 1998 | MR | Zbl

[33] Deift P., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, NYU lectures, AMS, 2000, 261 pp. | MR

[34] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[35] Suetin P.K., Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979 | MR

[36] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, v. 2, Nauka, M., 1974 | MR

[37] Geronimus Ya.L., Mnogochleny, ortogonalnye na okruzhnosti i na otrezke, Fizmatgiz, M., 1958 | MR

[38] Simon B., “Orthogonal Polynomials on the unit circle. Part 1, Part 2”, Colloquium publications, 54 (2004), American Mathematical Society | MR

[39] Zigmund A., Trigonometricheskie ryady, v. 2, Mir, M., 1965 | MR