Whitney's theorem in the $L^p$-metric, $0$

È. A. Storozhenko   ; Yu. V. Kryakin

Sbornik. Mathematics, Tome 186 (1995) no. 3, pp. 435-445

Résumé

In the original proof of the theorem on local approximation of functions by algebraic polynomials, Whitney estimated the deviation of the function from an interpolating polynomial with equally spaced nodes by means of finite differences. In this article we show how Whitney's idea of choosing nodes depending on a parameter with subsequent averaging can be applied to functions in $L^p$. The methods indicated allow one to obtain an estimate for the speed of approximating functions given on au arc of the unit circle by trigonometric polynomials or splines.

Bibliographie

[1] Burkill H., “Cesaro–Perron absolute periodic functions”, Proc. London Math. Soc., 3 (1952), 150–174 | DOI | MR

[2] Whitney H., “On functions with bounded $n$-th differences”, J. Math. Pures. appl., 9:36 (1957), 67–95 | MR

[3] Whitney H., “On bounded functions with bounded $n$-th differences”, Proc. Amer. Math. Soc., 10:3 (1959), 480–481 | DOI | MR | Zbl

[4] Brudnyi Yu. A., “Ob odnoi teoreme lokalnykh nailuchshikh priblizhenii”, Uch. zap. Kazanskogo universiteta, 124:6 (1964), 43–49 | MR

[5] Storozhenko E. A., “Priblizhenie algebraicheskimi mnogochlenami funktsii klassa $L^p$, $0

1$”, Izv. AN SSSR. Ser. matem., 41 (1977), 652–662 | MR | Zbl

[6] Ivanov K., Takev V., “$O(n\,\ln n)$ bounds of constants of H. Whitney”, C. R. Acad. Bul. Sci., 38:9 (1985), 1129–1131 | MR | Zbl

[7] Binev P., “$O(n)$ bounds of Whitney constants”, C. R. Acad. Bul. Sci., 38:10 (1985), 1315–1317 | MR

[8] Sendov Bl., “The constants of H. Whitney are bounded”, C. R. Acad. Bul. Sci., 38:10 (1985), 1299–1302 | MR | Zbl

[9] Sendov Bl., Takev M., “The theorem of Whitney for integral norm”, C. R. Acad. Bul. Sci., 39:10 (1986), 35–38 | MR | Zbl

[10] Kryakin Yu. V., Kovalenko L. G., “O teoreme Uitni v klassakh $L^p$”, Izv. vuzov. Matematika, 1992, no. 1, 69–77 | MR | Zbl

[11] Wronich Z., “Moduli of smoothness associated with Chebyshov systems and approximation by $L$-splynes”, Konstruktivnaya teoriya funktsii'84, Sofiya, 1986, 906–916

[12] Brudnyi Yu. A., “Priblizhenie funktsii $n$-peremennykh kvazimnogochlenami”, Izv. AN SSSR. Ser. matem., 34:3 (1970), 564–583 | MR | Zbl

[13] Storozhenko E. A., Osvald P., “Teorema Dzheksona v prostranstvakh $L^p(\mathbb R^k)$, $0\nobreak p\nobreak 1$”, Sib. matem. zhurn., 19:4 (1984), 888–901

[14] Takev M., “Theorem of Whitney Type in $\mathbb R^n$”, Constructive theory of functions, Proc. of the inter. conf. of funct. (Varna, 1988, May 24–31), Sofia, 1988, 441–447 | MR | Zbl

[15] Osvald P., “Priblizhenie splainami v metrike $L^p$, $0

1$”, Math. Nachr., 94 (1980), 69–96 | DOI | MR | Zbl

[16] Ulyanov P. L., “O ryadakh po sisteme Khaara”, Matem. sb., 63(105):3 (1964), 356–391 | MR | Zbl

[17] Oswald P., “Ungleishungen vom Jackson–Typ fur die algebraishe beste Approximation in $L^p$”, J. Appr. Theory, 23:2 (1978), 65–75 | DOI | MR

[18] Tamrazov P. M., Gladkosti i polinomialnye priblizheniya, Naukova dumka, Kiev, 1975 | MR | Zbl

[19] Endryus G., Teoriya razbienii, Nauka, M., 1982 | MR

[20] Storozhenko E. A., “Ob odnoi zadache Khardi–Littlvuda”, Matem. sb., 119 (161):4 (1982), 564–583 | MR | Zbl

[21] Kryakin Yu. V., “O teoreme Khardi i Littlvuda”, Matem. zametki, 53:2 (1993), 78–83 | MR | Zbl

[22] Kryakin Yu. V., Priblizhenie funktsii na edinichnoi okruzhnosti v prostranstvakh $L^p$ i $H^p$, Dis. ...kand. fiz.-matem. nauk, Odessa, 1985