Some extremal problems for algebraic polynomials in loaded spaces

B. P. Osilenker

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2010), pp. 53-65.

Résumé

Let $$ \Pi _N^{(r)}(x)=\sum_{k=N-r+1}^Na_k^0x^k+\sum_{j=0}^{N-r}a_jx^j \quad(a_N^{(0)}>0) $$ be an algebraic polynomial with fixed coefficients $a_k^{(0)}$. For the $l$th derivative of the mentioned polynomial we solve the following extremal problems: in a loaded Jacobi space with the inner product $$ \langle f,g\rangle=\frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}\int_{-1}^1fg(1-x)^\alpha(1+x)^\beta\,dx+Lf(1)g(1)+Mf(-1)g(-1), $$ $(L,M\ge0)$, find $\inf\langle D^l[\Pi_N^{(r)}(x)],D^l[\Pi_N^{(r)}(x)]\rangle$ ($D=\frac d{dx}$, $0\le l\le N-r$) and calculate extremal polynomials.

Détail

Keywords: extremal problem, loaded spaces, loaded orthogonal polynomials, algebraic polynomials, classical Jacobi polynomials.

Bibliographie

[1] Atkinson F., Nepreryvnye i diskretnye granichnye zadachi, Mir, M., 1968, 750 pp. | MR | Zbl

[2] Kurant R., Gilbert D., Metody matematicheskoi fiziki, T. 1, Gostekhizdat, M., 1951, 476 pp. | MR

[3] Krall A. M., Hilbert space, boundary value problems and orthogonal polynomials, Operator theory: Advances and applications, 133, Birkhäuser Verlag, Basel, 2002, 183 | MR

[4] Cmirnov V. I., Kurs vysshei matematiki, T. IV, GITTL, M., 1957, 812 pp.

[5] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1966, 724 pp. | MR | Zbl

[6] Bazov I. A., Zadorozhnyi A. I., “Ryady Fure po sisteme sobstvennykh funktsii zadachi o kolebaniyakh nagruzhennogo sterzhnya”, Mezhdunarodnyi simpozium “Ryady Fure i ikh prilozheniya”, Tezisy dokladov, Rostov-na-Donu, 2005, 9–10

[7] Marchelan F., Osilenker B. P., “Otsenki dlya polinomov, ortogonalnykh po otnosheniyu k skalyarnomu proizvedeniyu Lezhandra–Soboleva”, Matem. zametki, 62:6 (1997), 871–880 | MR | Zbl

[8] Khairov A. R., “O trekh novykh sistemakh ortogonalnykh mnogochlenov”, Mezhdunarodnaya konferentsiya “Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz”, Tezisy dokladov, Moskva, 2005, 238

[9] Bavinck H., “A note on the Koekoeks' differential equiation for generalized Jacobi polynomials”, J. Comp. Appl. Math., 115 (2000), 87–92 | DOI | MR | Zbl

[10] Fulton C. T., Pruess S., “Numerical methods for a singular eigenvalue problem in the boundary conditions”, J. Math. Anal. Appl., 71 (1979), 431–462 | DOI | MR | Zbl

[11] Koekoek R., “Differential equations for symmetric generalized ultraspherical polynomials”, Trans. Amer. Math. Soc., 345 (1994), 47–72 | DOI | MR | Zbl

[12] Koekoek J., Koekoek R., “Differential equations for generalized Jacobi polynomials”, J. Comp. Appl. Math., 126 (2000), 1–31 | DOI | MR | Zbl

[13] Koornwider T., “Orthogonal polynomials with weight function $(1-x)^\alpha(1+x)^\beta+M\delta(x+1)+N\delta(x-1)$”, Canad. Math. Bull., 27 (1984), 205–214 | MR

[14] Krall A. M., “Orthogonal polynomials satisfying fourth order differential equations”, Proc. Roy. Soc. Edinburgh A, 87 (1981), 271–288 | MR | Zbl

[15] Littlejohn L. L., “The Krall polynomials: a new class of orthogonal polynomials”, Quaest. Math., 5 (1982), 255–265 | MR | Zbl

[16] Osilenker B. P., “Generalized trace formula and asymptotics of the averaged Turan determinant for polynomials orthogonal with a discrete Sobolev inner product”, J. Approx. Theory, 141 (2006), 70–97 | DOI | MR | Zbl

[17] Osilenker B. P., “Ob odnoi ekstremalnoi zadache dlya algebraicheskikh polinomov v simmetrichnom diskretnom prostranstve Gegenbauera–Soboleva”, Matem. zametki, 82:3 (2007), 411–425 | MR | Zbl

[18] Tikhomirov V. M., “Teoriya priblizhenii. II”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 14, Moskva, 1987, 103–260 | MR | Zbl

[19] Sege G., Ortogonalnye mnogochleny, Fizmatlit, M., 1962, 500 pp.

[20] Suetin P. K., Klassicheskie ortogonalnye mnogochleny, Nauka, M., 1979, 416 pp. | MR | Zbl