Piecewise polynomial approximation methods in the theory of Nikol'ski\u\i--Besov spaces

I. P. Irodova

Contemporary Mathematics. Fundamental Directions, Functional analysis, Tome 50 (2013), pp. 3-159.

Bibliographie

[1] Berg I., Lëfstrem I., Interpolyatsionnye prostranstva, Mir, M., 1980 | MR

[2] Besov O. V., “O nekotorom semeistve funktsionalnykh prostranstv. Teoremy vlozheniya i prodolzheniya”, Dokl. AN SSSR, 126 (1959), 1163–1165 | MR | Zbl

[3] Besov O. V., “O nekotorykh usloviyakh prinadlezhnosti k $L_p$ proizvodnykh periodicheskikh funktsii”, v. 1, Nauch. dokl. vyssh. shk., 1959, 12–17

[4] Besov O. V., “Issledovaniya odnogo semeistva funktsionalnykh prostranstv v svyazi s teoremami vlozheniya i prodolzheniya”, Tr. MIAN, 60, 1961, 42–81 | MR | Zbl

[5] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR | Zbl

[6] Birman M. Sh., Solomyak M. Z., “Kusochno-polinomialnye priblizheniya funktsii klassov $W_p^\alpha$”, Mat. sb., 73(115):3 (1967), 331–355 | MR | Zbl

[7] Brudnyi Yu. A., “Kusochno-polinomialnaya approksimatsiya i lokalnye priblizheniya”, Dokl. AN SSSR, 201:1 (1971), 16–18 | MR

[8] Brudnyi Yu. A., “Prostranstva, opredelyaemye s pomoschyu lokalnykh priblizhenii”, Tr. Mosk. Mat. ob-va, 24, 1971, 69–132 | MR | Zbl

[9] Brudnyi Yu. A., “O perestanovke gladkoi funktsii”, Usp. mat. nauk, 27:2 (1972), 165–166 | MR | Zbl

[10] Brudnyi Yu. A., “Nelineinaya splain-approksimatsiya i funktsii ogranichennoi variatsii”, Dokl. AN SSSR, 215:3 (1974), 511–513 | MR

[11] Brudnyi Yu. A., “Approksimatsionnye prostranstva”, Geometriya lineinykh prostranstv i teoriya operatorov, YarGU, Yaroslavl, 1977, 3–30 | MR

[12] Brudnyi Yu. A., “Ratsionalnaya approksimatsiya i teorema vlozheniya”, Dokl. AN SSSR, 247:2 (1979), 269–272 | MR

[13] Brudnyi Yu. A., “Adaptivnaya approksimatsiya funktsii s osobennostyami”, Tr. Mosk. Mat. ob-va, 55, 1994, 149–242 | MR | Zbl

[14] Brudnyi Yu. A., Ganzburg M. I., “Ob odnoi ekstremalnoi zadache dlya mnogochlenov $n$ peremennykh”, Izv. AN SSSR. Ser. Mat., 37:2 (1973), 344–355 | MR | Zbl

[15] Brudnyi Yu. A., Gopengauz I. E., “Priblizhenie kusochno-polinomialnymi funktsiyami”, Izv. AN SSSR. Ser. mat., 27:4 (1963), 723–746 | MR | Zbl

[16] Brudnyi Yu. A., Irodova I. P., “Nelineinaya splain-approksimatsiya i $B$-prostranstva”, Tr. Mezhd. konf. po teorii priblizhenii (Kiev, 1983), Nauka, M., 1987, 71–75

[17] Brudnyi Yu. A., Irodova I. P., “Nelineinaya splain-approksimatsiya i $B$-prostranstva”, Algebra i analiz, 4:4 (1992), 45–79 | MR | Zbl

[18] Brudnyi Yu. A., Kruglyak N. Ya., “Ob odnom semeistve approksimatsionnykh prostranstv”, Issledovaniya po teorii funktsii mnogikh veschestvennykh peremennykh, YarGU, Yaroslavl, 1978, 15–43 | MR

[19] Bychkova T. G., “Optimalnoe razlozhenie po atomam v diadicheskikh prostranstvakh Khardi $H_p$ i interpolyatsiya prostranstv $H_p$”, Materialy Vserossiiskoi nauchnoi konferentsii, Yaroslavl, 2003, 50–69

[20] Garnet Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1980 | MR

[21] Golovkin K. K., “Ob odnom obobschenii interpolyatsionnoi teoremy Martsinkevicha”, Tr. MIAN, 102, 1967, 5–28 | MR | Zbl

[22] Golubov B. I., “O modifitsirovannom silnom dvoichnom integrale i proizvodnoi”, Mat. sb., 193:4 (2002), 37–60 | DOI | MR | Zbl

[23] Golubov B. I., Elementy dvoichnogo analiza, MGUP, M., 2005 | MR

[24] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[25] Irodova I. P., “O svoistvakh shkaly prostranstv $B_p^{\lambda\theta}$ pri $0

1$”, Dokl. AN SSSR, 250:2 (1980), 273–275 | MR | Zbl

[26] Irodova I. P., “Svoistva funktsii, zadannykh skorostyu ubyvaniya kusochno-polinomialnoi approksimatsii”, Issledovaniya po teorii funktsii mnogikh veschestvennykh peremennykh, YarGU, Yaroslavl, 1980, 92–117 | MR | Zbl

[27] Irodova I. P., “O svoistvakh shkaly prostranstv $B_p^{\lambda\theta}$ pri $0

1$”, Issledovaniya po teorii funktsii mnogikh veschestvennykh peremennykh, YarGU, Yaroslavl, 1982, 57–79

[28] Irodova I. P., Sovmestnoe priblizhenie funktsii i ee proizvodnykh v $L_p\left([0,1]^n\right)$ s pomoschyu nelineinykh splainov, Dep. v VINITI No 1135–82, 1982

[29] Irodova I. P., “Obobschenie neravenstva Marsho”, Issledovaniya po teorii funktsii mnogikh veschestvennykh peremennykh, YarGU, Yaroslavl, 1984, 64–70

[30] Irodova I. P., “Nekotorye svoistva diadicheskikh prostranstv Besova”, Funkts. pr-va. Diff. operatory. Problemy mat. obrazovaniya, Trudy mezhd. konf. RUDN, v. 1, RUDN, M., 1998, 78–81

[31] Irodova I. P., “O neravenstvakh Dzheksona i Bernshteina dlya diadicheskikh prostranstv Besova”, Izv. Tulskogo gos. un-ta. Ser. Matematika. Mekhanika. Informatika, 4:1 (1998), 83–86 | MR

[32] Irodova I. P., “Diadicheskie prostranstva Besova”, Algebra i analiz, 12:3 (2000), 40–80 | MR | Zbl

[33] Irodova I. P., “Neravenstvo Dzheksona dlya diadicheskikh nediagonalnykh prostranstv Besova”, Optimization of finite element approximation and splines and wavelets, Proc. of the 2-nd Int. Conf. OFEA' 2001 (June 25–29, 2001, St.-Petersburg), 2001, 138–139

[34] Irodova I. P., “Algoritm postroeniya gladkogo splaina”, Matematika v sovremennom mire, Materialy 2-i Rossiiskoi nauchno-prakticheskoi konferentsii, Kaluga, 2004, 100–105

[35] Irodova I. P., “Ob opisanii modulya nepreryvnosti v terminakh kusochno-polinomialnoi approksimatsii”, Izv. Tulskogo gos. un-ta. Ser. Mat. Mekh. Inform., 11:1 (2005), 148–155

[36] Irodova I. P., “O diadicheskikh prostranstvakh Nikolskogo–Besova”, Funktsionalnye prostranstva, teoriya priblizhenii, nelineinyi analiz, Tezisy dokladov mezhdunarodnoi konferentsii, posvyaschennoi 100-letiyu akademika S. M. Nikolskogo, Moskva, 2005, 115

[37] Irodova I. P., “Kusochno-polinomialnye priblizheniya i modul nepreryvnosti”, Matematika v Yaroslavskom universitete, Sb. obzornykh statei k 30-letiyu mat. fak., YarGU, Yaroslavl, 2006, 227–244

[38] Irodova I. P., “O vychislenii $K$-funktsionala pary $B$-prostranstv”, Izv. Tulskogo gos. un-ta. Ser. Mat. Mekh. Inform., 12:1 (2006), 109–123

[39] Irodova I. P., “Diadicheskie proizvodnye i ikh svoistva”, Izv. Tulskogo gos. un-ta. Ser. Mat. Mekh. Inform., 14:1 (2008), 29–36

[40] Irodova I. P., “O diadicheskikh prostranstvakh Nikolskogo–Besova i ikh svyazi s klassicheskimi prostranstvami”, Mat. zametki, 83:5 (2008), 683–695 | DOI | MR | Zbl

[41] Irodova I. P., “O neravenstve tipa neravenstva Bernshteina”, Model. i analiz inform. sist., 15:4 (2008), 31–41 | MR

[42] Irodova I. P., “O vychislenii $K$-funktsionalov”, Algebra i analiz, 21:4 (2009), 95–125 | MR | Zbl

[43] Irodova I. P., “O neravenstve tipa neravenstva Dzheksona v diadicheskom prostranstve VMO”, Model. i analiz inform. sist., 16:3 (2009), 29–46

[44] Irodova I. P., Nevskii M. V., “Diadicheskie prostranstva Besova i drugie voprosy teorii priblizheniya”, Matematika v Yaroslavskom universitete, Sb. obzornykh statei, YarGU, Yaroslavl, 2001, 115–131 | MR | Zbl

[45] Kruglyak N. Ya., Nevskii D. M., “Pochti optimalnost algoritmov diadicheskoi approksimatsii $L^p([0, 1]^n)$ i veschestvennaya interpolyatsiya”, Algebra i analiz, 14:4 (2002), 63–90 | MR | Zbl

[46] Kudryavtsev L. D., Nikolskii S. M., “Prostranstva differentsiruemykh funktsii mnogikh peremennykh i teoremy vlozheniya”, Itogi nauki i tekhn. Sovrem. probl. mat. Fundam. napravl., 26, 1988, 5–157 | MR | Zbl

[47] Maiorov V. E., “O nailuchshem priblizhenii klassov $W^r_1(I^s)$ v prostranstve $L_\infty(I^s)$”, Mat. zametki, 19:5 (1976), 699–706 | MR | Zbl

[48] Nevskii M. V., O skorosti kusochno-polinomialnoi approksimatsii v klassakh Orlicha, Dep. v VINITI No 3225-V86, 05.05.1986

[49] Netrusov Yu. V., “Nelineinaya approksimatsiya funktsii iz prostranstva Besova–Lorentsa v ravnomernoi metrike”, Zap. nauch. sem. LOMI, 204, 1993, 61–81 | MR | Zbl

[50] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[51] Oskolkov K. I., “Poligonalnaya approksimatsiya funktsii dvukh peremennykh”, Mat. sb., 107(149):4 (1978), 601–612 | MR | Zbl

[52] Peleshenko B. I., “Strukturnye teoremy dlya odnogo klassa funktsionalnykh prostranstv”, Zap. nauch. sem. LOMI, 39, 1974, 191–193 | MR | Zbl

[53] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1972

[54] Storozhenko E. A., Osvald P., “Teoremy Dzheksona v prostranstvakh $L_p,$ $0

1$”, Sib. mat. zh., 19:4 (1978), 630–639

[55] Subbotin Yu. N., “Priblizhenie splainami i gladkie bazisy v $E(0,2\pi)$”, Mat. zametki, 12:1 (1972), 43–51 | MR | Zbl

[56] Subbotin Yu. N., Chernykh N. I., “Poryadok nailuchshikh splain-priblizhenii nekotorykh klassov funktsii”, Mat. zametki, 7:1 (1970), 31–42 | MR | Zbl

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

[58] Tribel Kh., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[59] Tribel Kh., Teoriya funktsionalnykh prostranstv, Mir, M., 1986 | MR

[60] Ulyanov P. L., “Teoremy vlozheniya i sootnosheniya mezhdu nailuchshimi priblizheniyami (modulyami nepreryvnosti) v raznykh metrikakh”, Mat. sb., 81(123):1 (1970), 104–131 | MR | Zbl

[61] Chernykh N. I., “Priblizhenie splainami s zadannoi plotnostyu raspredeleniya uzlov”, Tr. MIAN, 138, 1975, 174–197 | MR | Zbl

[62] Adams D., Hedberg L., Function spaces and potential theory, Springer, Berlin, 1996 | MR | Zbl

[63] Blasco O., Pott S., “Dyadic BMO on the bidisk”, Rev. Math. Iberoam., 21:2 (2005), 483–510 | DOI | MR | Zbl

[64] Buchard H. G., “Splines (with optimal knot) are better”, J. Appl. Anal., 3 (1974), 309–319 | MR

[65] Butler G. J., Richards F. B., “An $L^p$ saturation theorem for splines”, Canad. J. Math., 24:5 (1972), 957–966 | DOI | MR | Zbl

[66] Ciesielski Z., “Constructive function theory and spline systems”, Studia Math., 52 (1973), 277–302 | MR

[67] Cohen A., Devore R. A., Petrushev P. P., Xu H., “Nonlinear approximation and the space $\mathrm{BV}(\mathbb R^2)$”, Amer. J. Math., 121 (1999), 587–628 | DOI | MR | Zbl

[68] Curry H. B., Schönberg I. J., “On spline distributions and their limits the Polya distribution functions”, Bull. Amer. Math. Soc., 53:11 (1947), 1114

[69] Daly J. E., Fridli S., “Walsh multipliers for dyadic Hardy spaces”, Appl. Anal., 82:7 (2003), 689–700 | DOI | MR | Zbl

[70] De Boor C., “The quasi-interpolant as a tool in elementary polynomial spline theory”, Appoxim. theory, Proc. Int. Sympos. (Univ. Texas, Austin, Tex., 1973), Academic Press, N.Y., 1973, 266–276 | MR

[71] De Boor C., Practical guide to splines, Springer, N.Y., 1978 | Zbl

[72] De Boor C., Fix G. J., “Spline approximation by quasiinterpolants”, J. Approx. Theory, 8:1 (1973), 19–45 | DOI | MR | Zbl

[73] Devore R. A., Jawerth B., Popov V. A., “Compression of wavelet decompositions”, Amer. J. Math., 114 (1992), 737–785 | DOI | MR | Zbl

[74] Devore R. A., Petruschev P. P., Yu X. M., “Nonlinear wavelet approximation in the space $C(R^d)$”, Progress in Approximation Theory (Tampa, FL, 1990), Springer, N.Y., 1992, 261–283 | DOI | MR | Zbl

[75] Devore R. A., Popov V. A., “Free multivariate splines”, Constr. Approx., 3 (1987), 239–248 | DOI | MR | Zbl

[76] Devore R. A., Popov V. A., “Interpolation of Besov Spaces”, Trans. Amer. Math. Soc., 305 (1988), 397–414 | DOI | MR | Zbl

[77] Devore R. A., Richards F. B., “Saturation and inverse theorems for spline approximation”, Spline Functions and Approximation Theory, ISNM, 21, Basel, Birkhauser, 1972, 73–82 | MR

[78] Devore R. A., Scherer K., “A constructive theory for approximation by splines with an arbitrary sequence of knot sets”, Approxim. theory, Proc. Internat. Colloq. (Inst. Angew. Math., Univ. Bonn, Bonn, 1976), Springer, Berlin, 1976, 167–183 | DOI | MR

[79] Dorronsoro J. R., “Characterization of potential spaces”, Proc. Amer. Math. Soc., 95:1 (1985), 21–31 | DOI | MR | Zbl

[80] Garnett J. B., Jones P. W., “BMO from dyadic BMO”, Pacific J. Math., 99:2 (1982), 351–371 | DOI | MR | Zbl

[81] Geiss S., Müller R., Pillwein V., “A remark on extrapolation of rearrangement operators on dyadic $H_s^p$, $01$”, Studia Math., 171:2 (2005), 197–205 | DOI | MR | Zbl

[82] Herz C. S., “Lipschitz spaces and Bernstein theorem of absolutely convergent Fourier transforms”, J. Math. Mech., 18:4 (1968), 283–323 | MR | Zbl

[83] John F., Nirenberg L., “On functions of bounded mean oscillation”, Comm. Pure Appl. Math., 14 (1961), 415–426 | DOI | MR | Zbl

[84] Marchoud A., “Sur les derivées et sur les différences des fonctions de variables réebles”, J. Math. Pures Appl., 6 (1927), 337–425

[85] Mei T., “BMO is the intersection of two translates of dyadic BMO”, C. R. Math. Acad. Sci. Paris, 336:12 (2003), 1003–1006 | DOI | MR | Zbl

[86] Oswald P., “On the degree for nonlinear spline approximation un Besov–Sobolev spaces”, J. Approx. Theory, 61 (1990), 131–157 | DOI | MR | Zbl

[87] Peetre J., “Espaces d'interpolation et théoréme de Soboleff”, Ann. Inst. Fourier, 16 (1966), 279–317 | DOI | MR | Zbl

[88] Peetre J., Thoughts on Besov spaces, Lund, Lund Univ., 1966

[89] Peetre J., “Remarques sur les espaces de Besov. Le cas $0

1$”, C. R. Math. Acad. Sci. Paris, 277 (1973), 947–949 | MR | Zbl

[90] Peetre J., New thoughts on Besov spaces, Durham, N.C., Duke Univ. Math., 1976 | MR | Zbl

[91] Peetre J., Sparr G., “Interpolation of normed Abelian groups”, Ann. Mat. Pura Appl. (4), 92:4 (1972), 217–262 | DOI | MR | Zbl

[92] Peetre J., Svensson E., “On the Generalized Hardy's inequality of Mcgehee, Pigno and Smith and the problem of interpolation between BMO and Besov Space”, Math. Scand., 54 (1984), 221–241 | MR | Zbl

[93] Petruschev P. P., “Direct and converse theorems for spline and rational approximation and Besov spaces”, Function Spaces and Applications (Lund), Lecture Notes in Math., 1302, Springer, Berlin, 1988, 363–377 | DOI | MR

[94] Pirher J., Ward L., “BMO from dyadic BMO on the bidisk”, J. Lond. Math. Soc. (2), 77:2 (2008), 524–544 | MR

[95] Popa N., “Isomorphism theorems for dyadic Hardy spaces generated by a rearrangement invariant space $X$ on unit interval”, Rev. Roumaine Math. Pures Appl., 35:3 (1990), 261–281 | MR | Zbl

[96] Popov V. A., “On the connection between rational and spline approximation”, C. R. Acad. Bulgare Sci., 27 (1974), 623–626 | MR | Zbl

[97] Popov V. A., Freud G., “Lower error bounds in the theory of spline functions”, Studia Sci. Math. Hungar., 6 (1971), 387–391 | MR

[98] Rong-Qing Jia, “A Bernstein-type inequality associated with wavelet decomposition”, Constr. Approx., 9 (1993), 299–318 | DOI | MR | Zbl

[99] Sagher Y., Shvartsman P., “On the John–Strömberg–Torchinsky characterization of BMO”, Fourier Anal. Appl., 4:4–5 (1998), 521–548 | DOI | MR | Zbl

[100] Scherer K., “Characterization of generalized Lipschitz classes by best approximation with splines”, SIAM J. Numer. Anal., 11:2 (1974), 283–304 | DOI | MR | Zbl

[101] Schipp F., “On equivalence of rearrangement of the Haar system in dyadic Hardy and BMO spaces”, Anal. Math., 16:16 (1990), 135–141 | DOI | MR | Zbl

[102] Stein E. M., “The characterization of functions arising as potentials”, Bull. Amer. Math. Soc., 67:1 (1961), 102–104 | DOI | MR

[103] Strichartz R. S., “Multipliers on fractional Sobolev spaces”, J. Math. Mech., 16 (1967), 1031–1042 | MR

[104] Yager D., “Saturation bei splines”, Approx. Quadratur. Numer. Math., 16 (1970), 129–140 | MR