An exact Jackson--Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces

A. G. Babenko

Izvestiya. Mathematics , Tome 62 (1998) no. 6, pp. 1095-1119.

Résumé

Let $L^2_{\alpha,\beta}$ be the Hilbert space of real-valued functions on $[0,\pi]$ with scalar product $$ (F,G)=\int_{0}^{\pi}F(x)G(x)\biggl(\sin\dfrac{x}{2}\biggr)^{2\alpha+1} \biggl(\cos\dfrac{x}{2}\biggr)^{2\beta+1}\,dx,\qquad \alpha>-1,\quad \beta>-1, $$ and norm $\|F\|=(F,F)^{1/2}$. We prove in the case when $\alpha>\beta\geqslant-1/2$ the following exact Jackson–Stechkin inequality $$ E_{n-1} (F)\leqslant\omega_r\bigl(F,2x_{n}^{\alpha,\beta}\bigr),\quad F\in L^2_{\alpha,\beta}, $$ between the best of $F$ by cosine-polynomials of order $n-1$ and its generalized modulus of continuity of (real) order $r\geqslant 1$: $n\geqslant\max\bigl\{2,1+ \frac{\alpha-\beta}{2}\bigr\}$ if $\beta>-\frac12$ , $n\geqslant 1$ if $\beta=-\frac12$ , where $x_{n}^{\alpha,\beta}$ is the first positive zero of the Jacobi cosine-polynomial $P_{n}^{(\alpha,\beta)}(\cos x)$. We deduce from this inequality similar inequalities for mean-square approximations of functions of several variables given on projective spaces.

Bibliographie

[1] Babenko A. G., “Tochnoe neravenstvo Dzheksona–Stechkina v prostranstve $L^2$ funktsii na mnogomernoi sfere”, Matem. zametki, 60:3 (1996), 333–355 | MR | Zbl

[2] Babenko A. G., “Tochnoe neravenstvo Dzheksona v prostranstve $L_2$ s vesom Yakobi”, Materialy mezhdunarodnoi konferentsii i Chebyshevskikh chtenii, posvyaschennykh 175-letiyu so dnya rozhdeniya P. L. Chebysheva, T. 1, Izd-vo MGU, M., 1996, 40–43

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

[4] Levenshtein V. I., “Granitsy dlya upakovok metricheskikh prostranstv i nekotorye ikh prilozheniya”, Problemy kibernetiki, no. 40, Nauka, M., 1980, 44–110

[5] Löfstrëm J., Peetre J., “Approximation Theorems Connected with Generalized Translations”, Math. Ann., 181 (1969), 255–268 | DOI | MR

[6] Grünwald A. K., “Uber “begrenzte” Derivationen und deren Anwendung”, Z. angew. Math. und Phys., 12 (1867), 441–480

[7] Letnikov A. V., “Teoriya differentsirovaniya s proizvolnym ukazatelem”, Matem. sb., 3 (1868), 1–68

[8] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[9] Rustamov Kh. P., “O priblizhenii funktsii na sfere”, Izv. RAN. Ser. matem., 57:5 (1993), 127–148 | MR | Zbl

[10] Rustamov Kh. P., “Moduli gladkosti vysshikh poryadkov, svyazannye s razlozheniem Fure–Yakobi, i priblizhenie funktsii algebraicheskimi mnogochlenami”, Dokl. RAN, 344:5 (1995), 593–596 | MR | Zbl

[11] Stechkin S. B., “O poryadke nailuchshikh priblizhenii nepreryvnykh funktsii”, Izv. AN SSSR. Ser. matem., 15:3 (1951), 219–242 | MR | Zbl

[12] Potapov M. K., “O priblizhenii algebraicheskimi mnogochlenami v integralnoi metrike s vesom Yakobi”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekhan., 1983, no. 4, 43–52 | MR | Zbl

[13] Chernykh N. I., “O nailuchshem priblizhenii periodicheskikh funktsii trigonometricheskimi polinomami v $L_2$”, Matem. zametki, 2:5 (1967), 513–522 | MR

[14] Chernykh N. I., “O neravenstve Dzheksona v $L_2$”, Tr. MIAN SSSR, 88, Nauka, M., 1967, 71–74 | MR

[15] Arestov V. V., Chernykh N. I., “On the $L_2$-approximation of periodic functions by trigonometric polynomials”, Approximation and functions spaces, Proc. Conf. (Gdan'sk, 1979), North-Holland, Amsterdam, 1981, 25–43 | MR

[16] Yudin V. A., “Mnogomernaya teorema Dzheksona v $L_2$”, Matem. zametki, 29:2 (1981), 309–315 | MR | Zbl

[17] Popov V. Yu., “O tochnykh konstantakh v neravenstvakh Dzheksona dlya nailuchshikh sfericheskikh srednekvadratichnykh priblizhenii”, Izv. vuzov. Matematika, 1981, no. 12, 67–78 | Zbl

[18] Popov V. Yu., “Mnogomernye priblizheniya v $L_2(T_m)$”, Teoriya funktsii i priblizhenii. Trudy 3-i Saratovskoi zimnei Shkoly, Mezhvuzovskii nauchnyi sbornik. Ch. 3 (27 yanvarya – 7 fevralya 1986 g.), Saratovskii universitet, Saratov, 1988, 22–25

[19] Popov V. Yu., “Priblizhenie na sfere v $L_2$”, DAN SSSR, 301:4 (1988), 793–797

[20] Babenko A. G., “O tochnoi konstante v neravenstve Dzheksona v $L^2$”, Matem. zametki, 39:5 (1986), 651–664 | MR | Zbl

[21] Babenko A. G., “O neravenstve Dzheksona v prostranstve $L^2$”, Approksimatsiya v konkretnykh i abstraktnykh banakhovykh prostranstvakh, UNTs AN SSSR, Sverdlovsk, 1987, 4–14 | MR

[22] Shalaev V. V., “O poperechnikakh v $L_2$ klassov differentsiruemykh funktsii, opredelyaemykh modulyami nepreryvnosti vysshikh poryadkov”, Ukr. matem. zhurn., 43:1 (1991), 125–129 | MR | Zbl

[23] Shakenova M. Zh., “O tochnom neravenstve mezhdu nailuchshimi priblizheniyami i modulem gladkosti polozhitelnogo poryadka v $L_p[0,2\pi ]^2$”, Primenenie metodov teorii funktsii i funktsionalnogo analiza k zadacham matematicheskoi fiziki, Tez. dokl., Almaty, 1993, 178–179

[24] Arestov V. V., Popov V. Yu., “Neravenstvo Dzheksona na sfere $L_2$”, Izv. vuzov. Matematika, 1995, no. 8(399), 13–20 | MR | Zbl

[25] Ivanov V. I., Smirnov O. I., “O teoreme Dzheksona v prostranstve $l_2({\mathbb Z}_2^n)$”, Matem. zametki, 60:3 (1996), 390–405 | MR | Zbl

[26] Gorbachev D. V., “Tochnye konstanty Dzheksona na gruppe $\mathrm {SU}(2)$”, Algebra i analiz, Tez. dokl. shkoly-konferentsii, posvyaschennoi 100-letiyu so dnya rozhdeniya B. M. Gagaeva (16–22 iyunya 1997 g., g. Kazan), Izd-vo Kazanskogo matem. ob-va, Kazan, 1997, 61–63

[27] Smirnov O. I., “O konstantakh Dzheksona v prostranstve $l_2({\mathbb Z}_2^n)$”, Algebra i analiz, Tez. dokl. shkoly-konferentsii, posvyaschennoi 100-letiyu so dnya rozhdeniya B. M. Gagaeva (16–22 iyunya 1997 g., g. Kazan), Izd-vo Kazanskogo matem. ob-va, Kazan, 1997, 199–200

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

[29] Kabatyanskii G. A., Levenshtein V. I., “O granitsakh dlya upakovok na sfere i v prostranstve”, Problemy peredachi informatsii, 14:1 (1978), 3–25 | MR | Zbl

[30] Odlyzko A. M., Sloane N. J. A., “New bounds on the number of unit spheres that can touch a unit sphere in $n$ dimensions”, J. of Combinatorial Theory. Series A, 26 (1979), 210–214 | DOI | MR | Zbl

[31] Konvei Dzh., Sloen N., Upakovki sharov, reshetki i gruppy, T. 1, 2, Mir, M., 1990

[32] Yudin V. A., “Nizhnie otsenki dlya sfericheskikh dizainov”, Izv. AN SSSR. Ser. matem., 61:3 (1987), 213–223 | MR

[33] Stechkin S. B., “O nulyakh dzeta-funktsii Rimana”, Matem. zametki, 8:4 (1970), 419–429 | Zbl

[34] Arestov V. V., “Ob ekstremalnykh svoistvakh neotritsatelnykh trigonometricheskikh polinomov”, Tr. In-ta matem. i mekh. UrO RAN, 1 (1992), 50–70 | MR | Zbl

[35] Gasper G., “Positivity and the convolution structure for Jacobi series”, Ann. of Math., 93 (1971), 112–118 | DOI | MR | Zbl

[36] Gasper G., “Banach algebras for Jacobi series and positivity of a kernel”, Ann. of Math., 95 (1972), 261–280 | DOI | MR | Zbl

[37] Koornwinder T., “The addition formula for Jacobi polynomials and spherical harmonics”, SIAM J. Appl. Math., 25:2 (1973), 236–246 | DOI | MR | Zbl

[38] Koornwinder T., “Jacobi polynomials. II: An analytic proof of the product formula”, SIAM J. Math. Anal., 5:1 (1974), 125–137 | DOI | MR | Zbl

[39] Gasper G., Trebels W., “Multiplier criteria of Marcinkiewicz type for Jacobi expansions”, Trans. Amer. Math. Soc., 231:1 (1977), 117–132 | DOI | MR | Zbl

[40] Shapiro R. L., “Spetsialnye funktsii, svyazannye s predstavleniem gruppy $SU(n)$ klassa I otnositelno $SU(n-1)$ $(n\ge 3)$”, Izv. vuzov. Matematika, 1968, no. 4(71), 97–107 | MR | Zbl

[41] Le Gendre A. M., Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures, T. 2, Paris, 1817

[42] Vatson G. N., Teoriya besselevykh funktsii, Ch. 1, IL, M., 1949

[43] Ragozin D. L., “Constructive polynomial approximation on spheres and projective spaces”, Transactions of the Amer. Math. Soc., 162 (1971), 157–170 | DOI | MR

[44] Nikolskii S. M., Lizorkin P. I., “Approksimatsiya funktsii na sfere”, Izv. AN SSSR. Ser. matem., 51:3 (1987), 635–651 | MR | Zbl

[45] Fedorov V. M., “Priblizhenie funktsii na sfere”, Vestn. Mosk. un-ta. Ser. 1. Matematika. Mekhanika, 1990, no. 1, 15–23 | MR | Zbl

[46] Platonov S. S., “O teoremakh dzheksonovskogo tipa na kompaktnom simmetricheskom prostranstve ranga 1”, DAN, 353:4 (1997), 445–448 | MR | Zbl

[47] Ivanov V. I., Smirnov O. I., “Konstanty Dzheksona v prostranstvakh $L_2$ na metricheskikh kompaktakh”, Izv. Tulskogo gosud. un-ta. Ser. Matematika. Mekhanika. Informatika, 2:1 (1996), 93–118 | MR

[48] Ivanov V. I., “Konstanty Dzheksona v prostranstvakh $L_p$ na metricheskikh kompaktakh”, Algebra i analiz, Tez. dokl. shkoly-konferentsii, posvyaschennoi 100-letiyu so dnya rozhdeniya B. M. Gagaeva (16–22 iyunya 1997 g., g. Kazan), Izd-vo Kazanskogo matem. ob-va, Kazan, 1997, 98–101

[49] Khelgason S., Gruppy i geometricheskii analiz, Mir, M., 1987 | MR