×

zbMATH — the first resource for mathematics

Approximation and entropy numbers of embeddings between approximation spaces. (English) Zbl 1396.41027
The authors present significant results concerning the degrees of compactness for the embeddings of the classical two-indexed approximation subspaces in the usual Euclidean spaces, following the approximation and the entropy numbers. As applications, one obtains the compact embeddings for some Besov spaces.

MSC:
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
46M35 Abstract interpolation of topological vector spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Almira, JM; Luther, U, Compactness and generalized approximation spaces, Numer. Funct. Anal. Optim., 23, 1-38, (2002) · Zbl 1076.41020
[2] Almira, JM; Luther, U, Generalized approximation spaces and applications, Math. Nachr., 263, 3-35, (2004) · Zbl 1118.41017
[3] Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge Univ. Press, Cambridge (1987) · Zbl 0617.26001
[4] Brudnyi, Ju.A., Krugljak, N.Ja.: A family of approximation spaces. In: Studies in the Theory of Functions of Several Real Variables, vol. 2, pp. 15-42. Yaroslav. Gos. Univ., Yaroslavl (1978) (in Russian)
[5] Butzer, P.L., Scherer, K.: Approximationsprozesse und Interpolationsmethoden. Mannheim, Zürich (1968) · Zbl 0177.08501
[6] Caetano, AM; Gogatishvili, A; Opic, B, Sharp embeddings of Besov spaces involving only logarithmic smoothness, J. Approx. Theory, 152, 188-214, (2008) · Zbl 1161.46017
[7] Caetano, AM; Leopold, H-G, On generalized Besov and Triebel-Lizorkin spaces of regular distributions, J. Funct. Anal., 264, 2676-2703, (2013) · Zbl 1291.46031
[8] Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge (1990) · Zbl 0705.47017
[9] Cobos, F; Domínguez, O, Embeddings of Besov spaces of logarithmic smoothness, Studia Math., 223, 193-204, (2014) · Zbl 1325.46039
[10] Cobos, F; Domínguez, O, Approximation spaces, limiting interpolation and Besov spaces, J. Approx. Theory, 189, 43-66, (2015) · Zbl 1326.46020
[11] Cobos, F; Domínguez, O, On Besov spaces of logarithmic smoothness and Lipschitz spaces, J. Math. Anal. Appl., 425, 71-84, (2015) · Zbl 1320.46028
[12] Cobos, F; Domínguez, O, On the relationship between two kinds of Besov spaces with smoothness near zero and some other applications of limiting interpolation, J. Fourier Anal. Appl., 22, 1174-1191, (2016) · Zbl 1365.46027
[13] Cobos, F; Domínguez, O; Triebel, H, Characterizations of logarithmic Besov spaces in terms of differences, Fourier-analytical decompositions, wavelets and semi-groups, J. Funct. Anal., 270, 4386-4425, (2016) · Zbl 1356.46026
[14] Cobos, F; Kühn, T, Approximation and entropy numbers in Besov spaces of generalized smoothness, J. Approx. Theory, 160, 56-70, (2009) · Zbl 1177.47029
[15] Cobos, F; Milman, M, On a limit class of approximation spaces, Numer. Funct. Anal. Optim., 11, 11-31, (1990) · Zbl 0729.41033
[16] Cobos, F; Resina, I, Representation theorems for some operator ideals, J. Lond. Math. Soc., 39, 324-334, (1989) · Zbl 0645.47037
[17] Cobos, F., Resina, I.: On some operator ideals defined by approximation numbers. In: Geometric Aspects of Banach Spaces, London Mathematical Society Lecture Note Series, vol. 140, pp. 133-139. Cambridge University Press, Cambridge (1989) · Zbl 0677.47029
[18] DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993) · Zbl 0797.41016
[19] DeVore, R.A., Popov, V.A.: Interpolation of approximation spaces. In: Constructive Theory of Functions (Varna, 1987), pp. 110-119. Publ. House Bulgar. Acad. Sci., Sofia (1988) · Zbl 1353.46018
[20] DeVore, RA; Riemenschneider, SD; Sharpley, RC, Weak interpolation in Banach spaces, J. Funct. Anal., 33, 58-94, (1979) · Zbl 0433.46062
[21] Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces and Embeddings. Springer, Berlin (2004) · Zbl 1099.46002
[22] Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996) · Zbl 0865.46020
[23] Fehér, F; Grässler, G, On an extremal scale of approximation spaces, J. Comput. Anal. Appl., 3, 95-108, (2001) · Zbl 1033.46021
[24] Fernández-Martínez, P; Signes, T, Limiting ultrasymmetric sequence spaces, Math. Ineq. Appl., 19, 597-624, (2016) · Zbl 1353.46018
[25] Grafakos, L.: Classical Fourier Analysis. Springer, New York (2008) · Zbl 1220.42001
[26] Haroske, DD; Skrzypczak, L, Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, I. Rev. Mat. Complut., 21, 135-177, (2008) · Zbl 1202.46039
[27] Haroske, DD; Skrzypczak, L, Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights. II. general weights, Ann. Acad. Scient. Fennicae Math., 36, 111-138, (2011) · Zbl 1222.46027
[28] Haroske, DD; Triebel, H, Wavelet bases and entropy numbers in weighted function spaces, Math. Nachr., 278, 108-132, (2005) · Zbl 1078.46022
[29] König, H.: Eigenvalue Distribution of Compact Operators. Birkhäuser, Basel (1986) · Zbl 0618.47013
[30] Köthe, G.: Topological Vector Spaces I. Springer, Berlin (1969) · Zbl 0179.17001
[31] Kühn, T, Entropy numbers in weighted function spaces. the case of intermediate weights, Proc. Steklov Inst. Math., 255, 159-168, (2006) · Zbl 1351.42019
[32] Kühn, T, Entropy numbers in sequence spaces with an application to weighted function spaces, J. Approx. Theory, 153, 40-52, (2008) · Zbl 1145.47017
[33] Kühn, T; Leopold, H-G; Sickel, W; Skrzypczak, L, Entropy numbers of embeddings of weighted Besov spaces, Constr. Approx., 23, 61-77, (2006) · Zbl 1100.41015
[34] Kühn, T; Leopold, H-G; Sickel, W; Skrzypczak, L, Entropy numbers of embeddings of weighted Besov spaces II, Proc. Edinburgh Math. Soc., 49, 331-359, (2006) · Zbl 1103.41027
[35] Kühn, T; Leopold, H-G; Sickel, W; Skrzypczak, L, Entropy numbers of embeddings of weighted Besov spaces III. weights of logarithmic type, Math. Z., 255, 1-15, (2007) · Zbl 1144.41006
[36] Leopold, H.-G.: Embeddings and entropy numbers in Besov spaces of generalized smoothness. In: Hudzik, H., Skrzypczak, L. (eds.) Function Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 213, pp. 323-336. Marcel Dekker, New York (2000) · Zbl 0966.46018
[37] Nikolskii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975)
[38] Peetre, J; Sparr, G, Interpolation of normed abelian groups, Ann. Mat. Pure Appl., 92, 217-262, (1972) · Zbl 0237.46039
[39] Petrushev, P.P., Popov, V.A.: Rational Approximation of Real Functions. Encyclopedia of Mathematics and its Applications, vol. 28. Cambridge University Press, Cambridge (1987) · Zbl 0644.41010
[40] Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980) · Zbl 0434.47030
[41] Pietsch, A, Approximation spaces, J. Approx. Theory, 32, 115-134, (1981) · Zbl 0489.47008
[42] Pietsch, A, Tensor products of sequences, functions, and operators, Arch. Math., 38, 335-344, (1982) · Zbl 0464.47006
[43] Pustylnik, E, Ultrasymmetric sequence spaces in approximation theory, Collect. Math., 57, 257-277, (2006) · Zbl 1104.41020
[44] Pustylnik, E, A new class of approximation spaces, Rend. Circolo Mat. Palermo, 76, 517-532, (2005) · Zbl 1136.41311
[45] Schmeisser, H.-J., Runovski, K.: in preparation · Zbl 1136.41311
[46] Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987) · Zbl 0661.46024
[47] Temlyakov, V.N.: Approximation of Periodic Functions. Nova Science, New York (1994) · Zbl 0899.41001
[48] Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978) · Zbl 0387.46032
[49] Weisz, F, \(ℓ _1\)-summability of \(d\)-dimensional Fourier transforms, Constr. Approx., 34, 421-452, (2011) · Zbl 1234.42004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.