On the computation of 𝐾-functionals

Irodova, I.

doi:10.1090/S1061-0022-2010-01107-5

On the computation of $K$-functionals
HTML articles powered by AMS MathViewer

by I. P. Irodova
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 21 (2010), 579-599

DOI: https://doi.org/10.1090/S1061-0022-2010-01107-5

Published electronically: May 20, 2010

PDF | Request permission

Abstract:

A new approach to the calculation of the sharp order of a $K$-functional is suggested. This approach employs the techniques of dyadic spaces.

References

I. P. Irodova, Dyadic Besov spaces, Algebra i Analiz 12 (2000), no. 3, 40–80 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 3, 379–405. MR 1778190
O. V. Besov, V. P. Il′in, and S. M. Nikol′skiĭ, Integral′nye predstavleniya funktsiĭ i teoremy vlozheniya, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0430771
Albert Cohen, Ronald DeVore, Pencho Petrushev, and Hong Xu, Nonlinear approximation and the space $\textrm {BV}(\textbf {R}^2)$, Amer. J. Math. 121 (1999), no. 3, 587–628. MR 1738406
N. Ya. Kruglyak and D. M. Nevskiĭ, Almost optimality of dyadic approximation algorithms in $L_p([0,1]^n)$, and real interpolation, Algebra i Analiz 14 (2002), no. 4, 63–90 (Russian); English transl., St. Petersburg Math. J. 14 (2003), no. 4, 583–602. MR 1935918
T. G. Bychkova, Optimal atomic decomposition in dyadic Hardy spaces $H_p$ and interpolation of spaces $H_p$, Proc. of All-Russian Sci. Conf., Yaroslavl′, 2003, pp. 50–69. (Russian)
Jaak Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Duke University, Mathematics Department, Durham, N.C., 1976. MR 0461123
I. P. Irodova, Properties of the scale of spaces $B^{\lambda \theta }_{p}$ for $0<p<1$, Dokl. Akad. Nauk SSSR 250 (1980), no. 2, 273–275 (Russian). MR 557768
S. M. Nikol′skiĭ, Priblizhenie funktsiĭ mnogikh peremennykh i teoremy vlozheniya, “Nauka”, Moscow, 1977 (Russian). Second edition, revised and supplemented. MR 506247
I. P. Irodova, Generalization of the Marchaud inequality, Studies in the theory of functions of several real variables, Yaroslav. Gos. Univ., Yaroslavl′1984, pp. 63–69, 152 (Russian). MR 830218
I. P. Irodova, Properties of functions defined by the rate of decrease of a piecewise-polynomial approximation, Studies in the theory of functions of several variables, Yaroslav. Gos. Univ., Yaroslavl′1980, pp. 92–117 (Russian). MR 712504
Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), no. 4, 737–785. MR 1175690, DOI 10.2307/2374796
R. A. DeVore, P. Petrushev, and X. M. Yu, Nonlinear wavelet approximation in the space $C(\textbf {R}^d)$, Progress in approximation theory (Tampa, FL, 1990) Springer Ser. Comput. Math., vol. 19, Springer, New York, 1992, pp. 261–283. MR 1240786, DOI 10.1007/978-1-4612-2966-7_{1}1
Rong Qing Jia, A Bernstein-type inequality associated with wavelet decomposition, Constr. Approx. 9 (1993), no. 2-3, 299–318. MR 1215774, DOI 10.1007/BF01198008
Ju. A. Brudnyĭ, Spaces that are definable by means of local approximations, Trudy Moskov. Mat. Obšč. 24 (1971), 69–132 (Russian). MR 0390747
È. A. Storoženko and P. Osval′d, Jackson’s theorem in the spaces $L^{p}(\textbf {R}^{k})$, $0<p<1$, Sibirsk. Mat. Ž. 19 (1978), no. 4, 888–901, 956 (Russian). MR 0493121
M. V. Nevskiĭ, On the rate of the piecewise-polynomial approximation for functions from Orlicz classes, Yaroslav. Gos. Univ., Yaroslavl′, 1986. (Manuscript deposed VINITI 06.05.86, No. 3225-B). (Russian)
I. P. Irodova, On certain properties of dyadic Besov spaces, Functional Spaces. Differential Operators. Problems of the Mathematical Education (Proc. Internat. Conf.). Vol. 1, Ross. Univ. Druzhby Narodov, Moscow, 1998, pp. 78–81. (Russian)
I. P. Irodova, Dyadic Nikol′skiĭ-Besov spaces and their relationship to classical spaces, Mat. Zametki 83 (2008), no. 5, 683–695 (Russian, with Russian summary); English transl., Math. Notes 83 (2008), no. 5-6, 624–634. MR 2451357, DOI 10.1134/S0001434608050052
Ronald A. DeVore and Vasil A. Popov, Interpolation of Besov spaces, Trans. Amer. Math. Soc. 305 (1988), no. 1, 397–414. MR 920166, DOI 10.1090/S0002-9947-1988-0920166-3
C. de Boor and G. J. Fix, Spline approximation by quasiinterpolants, J. Approximation Theory 8 (1973), 19–45. MR 340893, DOI 10.1016/0021-9045(73)90029-4
Ju. A. Brudnyĭ, Piecewise polynomial approximation and local approximations, Dokl. Akad. Nauk SSSR 201 (1971), 16–18 (Russian). MR 0290001
I. P. Irodova, Algorithm for a construction of the smooth spline, Mathematics in the Modern World (2nd Russian Scientific and Practical Conf.), Kaluga, 2004, pp. 100–105. (Russian)
—, On computation of the $K$-functional of a pair of $B$-spaces, Izv. Tul′skogo Gos. Univ. Ser. Mat. Mekh. Inf. 12 (2006), vyp. 1, 109–123. (Russian)
—, Dyadic derivatives and their properties, Izv. Tul′skogo Gos. Univ. Ser. Estestv. Nauk 1 (2008), 29–36. (Russian)
J. Peetre and G. Sparr, Interpolation of normed abelian groups, Ann. Mat. Pura Appl. (4) 92 (1972), 217–262. MR 322529, DOI 10.1007/BF02417949
Ju. A. Brudnyĭ and N. Ja. Krugljak, A family of approximation spaces, Studies in the theory of functions of several real variables, No. 2 (Russian), Yaroslav. Gos. Univ., Yaroslavl′1978, pp. 15–42 (Russian). MR 559913
Jaak Peetre and Erik Svensson, On the generalized Hardy’s inequality of McGehee, Pigno and Smith and the problem of interpolation between BMO and a Besov space, Math. Scand. 54 (1984), no. 2, 221–241. MR 757464, DOI 10.7146/math.scand.a-12055
F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
Ju. A. Brudnyĭ, Rational approximation and imbedding theorems, Dokl. Akad. Nauk SSSR 247 (1979), no. 2, 269–272 (Russian). MR 545347
Pencho P. Petrushev, Direct and converse theorems for spline and rational approximation and Besov spaces, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 363–377. MR 942281, DOI 10.1007/BFb0078887
Yu. A. Brudnyĭ and I. P. Irodova, Nonlinear spline approximation of functions and $B$-spaces, Proc. Internat. Conf. on the Theory of Approximation of Functions (Kiev, 1983), Nauka, Moscow, 1987, pp. 1–75. (Russian)
—, Approximation of functions of several variables by nonlinear splines, Theory of Approximation of Functions (All-Union Workshop): Thesis, Akad. Nauk Ukrain. SSR. Inst. Mat., Kiev, 1989, pp. 27. (Russian)
Yu. A. Brudnyĭ and I. P. Irodova, Nonlinear spline approximation of functions of several variables and B-spaces, Algebra i Analiz 4 (1992), no. 4, 45–79 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 4, 667–694. MR 1190782