Sharp Jackson--Stechkin inequality in $L^2$ for multidimensional spheres

A. G. Babenko

Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 333-355.

Résumé

In this paper we prove the Jackson–Stechkin inequality $$ E_{n-1}(f)\omega_r(f,2\tau_{n,\lambda}), \qquad n\ge1, \quad m\ge5, \quad r\ge1, $$ $f\in L^2(\mathbb S^{m-1})$, $f\not\equiv\textrm{const}$, which is sharp for each $n=2,3,\dots$; here $E_{n-1}(f)$ is the best approximation of a function $f$ by spherical polynomials of degree $\le n-1$, $\omega_r(f,\tau)$ is the $r$th modulus of continuity of $f$ based on the translations $$ s_tf(x)=\frac 1{|\mathbb S^{m-2}|}\int_{\mathbb S^{m-2}}f(x\cos t+\xi\sin t)\,d\xi, \qquad t\in\mathbb R, \quad x\in\mathbb S^{m-1}, $$ $\mathbb S^{m-2}=\mathbb S^{m-2}_x=\bigl\{\xi\in \mathbb S^{m-1}:x\cdot\xi=0\bigr\}$, $|\mathbb S^{m-2}|$ is the measure of the unit Euclidean sphere $\mathbb S^{m-2}$, $\lambda=(m-2)/2$ and $\tau_{n,\lambda}$ is the first positive zero of the Gegenbauer cosine polynomial $C^\lambda_n(\cos t)$ .

Bibliographie

[1] Jackson D., “On approximation by trigonometric sums and polinomials”, Trans. Amer. Math. Soc., 13 (1912), 491–515 | DOI | MR

[2] Akhiezer N. I., Lektsii po teorii approksimatsii, Gostekhizdat, M., 1947 | Zbl

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

[4] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[5] Korneichuk N. P., “Tochnaya konstanta v teoreme D. Dzheksona o nailuchshem ravnomernom priblizhenii nepreryvnykh periodicheskikh funktsii”, Dokl. AN SSSR, 145:3 (1962), 514–515 | MR

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

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

[8] Korneichuk N. P., “O tochnoi konstante v neravenstve Dzheksona dlya nepreryvnykh periodicheskikh funktsii”, Matem. zametki, 32:5 (1982), 669–674 | MR | Zbl

[9] 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

[10] Chernykh N. I., “Neravenstvo Dzheksona v $L_p(0,2\pi)$ $(1\le p2$) s tochnoi konstantoi”, Tr. MIRAN, 198 (1992), 232–241, Nauka, M. | Zbl

[11] Zhuk V. V., Approksimatsiya periodicheskikh funktsii, Izd-vo LGU, L., 1982 | Zbl

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

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

[14] 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, Tezisy dokl., Almaty, 1993, 178–179

[15] Gronwall T. H., “On the degree of convergence of Laplace's series”, Trans. Amer. Math. Soc., 15 (1914), 1–30 | DOI | MR

[16] Nikolskii S. M., Lizorkin P. I., “Priblizhenie sfericheskimi polinomami”, Tr. MIAN, 166, Nauka, M., 1984, 186–200 | MR | Zbl

[17] Fëdorov V. M., “Priblizhenie funktsii na sfere”, Vestn. MGU. Ser. 1. Matem., mekh., 1990, no. 1, 15–23 | Zbl

[18] Lizorkin P. I., “O priblizhenii funktsii na sfere $\sigma$. O prostranstvakh $B^\alpha_{p,q}(\sigma)$”, Dokl. RAN, 331:5 (1993), 555–558 | Zbl

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

[20] Vebster A., Segë G., Differentsialnye uravneniya v chastnykh proizvodnykh matematicheskoi fiziki, ONTI GTTI, M.–L., 1934

[21] Kushnirenko G. G., “O priblizhenii funktsii, zadannykh na edinichnoi sfere, konechnymi sfericheskimi summami”, Nauchnye dokl. vysshei shkoly. Fiziko-matem. nauki, 3:4 (1958), 47–53 | MR

[22] Kushnirenko G. G., “Nekotorye voprosy priblizheniya nepreryvnykh funktsii na edinichnoi sfere konechnymi sfericheskimi summami”, Tr. Kharkovskogo politekhnicheskogo in-ta. Ser. inzhenerno-fizich., 25:3 (1959), 3–22

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

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

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

[26] Shalaev V. V., “Tochnye otsenki priblizheniya nepreryvnykh na sfere funktsii lineinymi operatorami tipa svertki”, Ukr. matem. zh., 43:4 (1991), 565–567 | MR | Zbl

[27] Arestov V. V., Popov V. Yu., “Neravenstvo Dzheksona na sfere $L_2$”, Mezhdunarodnaya konf. “Teoriya priblizheniya i zadachi vychislitelnoi matematiki”, Tezisy dokl., DDU, Dnepropetrovsk, 1993, 8

[28] Popov V. Yu., “Mnogomernye priblizheniya v $L_2(T_m)$”, Teoriya funktsii i priblizhenii, Trudy 3-i Saratovskoi zimnei shkoly. Mezhvuzovskii nauchn. sb., Ch. 3 (27 yanvarya – 7 fevralya 1986 g.), SGU, Saratov, 1988, 22–25

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

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

[31] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[32] Berens H., Butzer P. L., Pawelke S., “Limitierungsverfahren von Reihen mehrdimensionaver Kugelfunktionen und deren Saturationsverhalten”, Publ. Res. Inst. Math. Sci. (Kyoto). Ser. A, 4 (1968), 202–268 | MR

[33] Ivanov V. A., “K voprosu o svoistvakh modulei nepreryvnosti dlya funktsii na sfere”, Differents. uravneniya, 23:3 (1987), 481–487 | MR | Zbl

[34] Bochner S., “Positive zonal functions on spheres”, Proc. Nat. Acad. Sci. USA, 40 (1954), 1141–1147 | DOI | MR | Zbl

[35] Pawelke S., “Ein Satz vom Jacksonschen Typ für algebraische Polynome”, Acta Sci. Math., 33 (1972), 323–336 | MR | Zbl

[36] Potapov M. K., “O priblizhenii algebraicheskimi mnogochlenami v integralnoi metrike s vesom Yakobi”, Vestn. MGU. Ser. 1. Matem., mekh., 1983, no. 4, 43–52 | MR | Zbl

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

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

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

[40] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[41] Levenshtein V. I., “Granitsy dlya upakovok metricheskikh prostranstv i nekotorye ikh prilozheniya”, Probl. kibern., 40 (1983), 43–110 | MR

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