Geodesic
Local Smoothing of Uniformly Smooth Maps
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 3, pp. 44-52.

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We solve the problem on the uniform approximation of uniformly continuous (smooth) maps by maps having the maximum possible local and uniform smoothness. In particular, we prove that each uniformly continuous map of the Hilbert space $l_2$ into itself can be approximated by locally infinitely differentiable maps having a Lipschitz derivative.
Keywords: approximation, smoothing, local smoothness, uniform smoothness, Lipschitz derivative. 
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I. G. Tsar'kov. Local Smoothing of Uniformly Smooth Maps. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 3, pp. 44-52. http://geodesic.mathdoc.fr/item/FAA_2006_40_3_a4/

[1] Dzh. Ills, “Osnovaniya globalnogo analiza”, UMN, 24:3 (147) (1969), 157–210 | MR

[2] J. Kurzweil, “On approximation in real Banach spaces”, Studia Math., 14:2 (1954), 214–231 | DOI | MR

[3] F. A. Valentine, “A Lipschitz conditions preserving extension for a vector function”, Amer. J. Math., 67:1 (1945), 83–93 | DOI | MR | Zbl

[4] A. S. Nemirovskii, Gladkaya i polinomialnaya approksimatsiya nepreryvnykh funktsii na gilbertovom prostranstve, Diss. kand. fiz.-mat. nauk, MGU, mekh-mat, 1973

[5] S. M. Semenov, “O simmetricheskikh funktsiyakh klassa $D^2_u(H)$”, Funkts. analiz i ego pril., 6:3 (1972), 85–86 | MR | Zbl

[6] S. M. Semenov, Simmetricheskie funktsii na prostranstvakh $L_p$, Diss. kand. fiz.-mat. nauk, MGU, mekh-mat, 1973

[7] A. S. Nemirovskii, S. M. Semenov, “O polinomialnoi approksimatsii na gilbertovom prostranstve”, Matem. sb., 92:2 (1973), 257–281 | MR | Zbl

[8] I. G. Tsarkov, “Sglazhivanie ravnomerno nepreryvnykh otobrazhenii v prostranstvakh $L_p$”, Matem. zametki, 54:3 (1993), 123–140 | MR | Zbl

[9] I. G. Tsarkov, “Sglazhivanie abstraktnykh funktsii”, Matem. sb., 185:11 (1994), 119–144 | MR | Zbl

[10] I. G. Tsarkov, “Lineinye metody v nekotorykh zadachakh sglazhivaniya”, Matem. zametki, 56:6 (1994), 64–87 | MR | Zbl

[11] I. G. Tsarkov, “Priblizhenie vektornoznachnykh funktsii mnogochlenami”, Funkts. analiz i ego pril., 29:3 (1995), 93–95 | DOI | MR | Zbl

[12] I. G. Tsarkov, “O prodolzhenii i sglazhivanii vektornoznachnykh funktsii”, Izv. RAN, 59:4 (1995), 187–220 | MR | Zbl

[13] I. G. Tsarkov, “Nekotorye voprosy prodolzheniya”, Matem. zametki, 58:6 (1995), 906–916 | MR | Zbl

[14] I. G. Tsarkov, “O gladkikh vyborkakh iz mnozhestv pochti chebyshevskikh tsentrov”, Vestnik MGU, ser. 1, matem., mekh., 1996, no. 2, 92–94 | MR | Zbl

[15] I. G. Tsarkov, “Sglazhivanie funktsii na gilbertovom share”, Trudy MIRAN, 219, 1997, 410–419 | MR | Zbl

[16] I. G. Tsarkov, “Sglazhivanie funktsii v konechnomernykh prostranstvakh”, Izv. RAN, 61:1 (1997), 199–214 | MR | Zbl

[17] I. G. Tsarkov, “Prodolzhenie gilbertovoznachnykh lipshitsevykh otobrazhenii”, Vestnik MGU, ser. 1, matem., mekh., 1999, no. 6, 9–16 | MR | Zbl

[18] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, vol. 1, Colloquium Publications, 48, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[19] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, I, II, Springer-Verlag, Berlin, 1996 | MR | Zbl

[20] I. Singer, Bases in Banach Spaces, I, Springer-Verlag, 1970 | MR | Zbl