Abstract
We obtain upper estimates of the entropy numbers of a compact Hardy integral operator in weighted spaces of Besov–Triebel–Lizorkin type with small smoothness parameters.
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K. F. Andersen and H. P. Heinig, Weighted norm inequalities for certain integral operators, SIAM J. Math., 14 (1983), 834–844.
G. Battle, A block spin construction of ondelettes. Part I: Lemarié functions, Comm. Math. Phys., 110 (1987), 601–615.
G. Battle, A block spin construction of ondelettes. Part II: QFT connection, Comm. Math. Phys., 114 (1988), 93–102.
G. Bennett, Some elementary enequalities. III, Quart. J. Math. Oxford Ser. (2), 42 (1991), 149–174.
B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge University Press (Cambridge, UK, 1990).
C. K. Chui, An Introduction to Wavelets, Academic Press (New York, 1992).
D. E. Edmunds and J. Lang, Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case, J. Funct. Anal., 206 (2004), 149–166.
D. E. Edmunds and J. Lang, Approximation numbers and Kolmogorov widths of Hardy-type operators in a non-homogeneous case, Math. Nachr., 279 (2006), 727–742.
D. E. Edmunds and J. Lang, Operators of Hardy type, J. Comput. Appl. Math., 208 (2007), 20–28.
D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge University Press (1996).
M. L. Goldman, H. P. Heinig and V. D. Stepanov, On the principle of duality in Lorentz spaces, Canad. J. Math., 48 (1996), 959–979.
A. Haar, Zur theorie der orthogonalen Funktionsysteme, Math. Ann., 69 (1910), 331–371.
D. Haroske, Embeddings of some weighted function spaces on Rn entropy and approximation numbers, Ann. Univ. Craiova Ser. Mat. Informat., 24 (1997), 1–44.
D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators. I, Math. Nachr., 167 (1994), 131–156.
D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators. II, Math. Nachr., 168 (1994), 109–137.
D. Haroske and H. Triebel, Wavelet bases and entropy numbers in weighted function spaces, Math. Nachr., 278 (2005), 108–132.
T. Kühn, H.-G. Leopold, W. Sickel and L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces, Jenaer Schriften zur Mathematik und Informatik, Math/Inf/13/03 (2003).
T. Kühn, H.-G. Leopold, W. Sickel and L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces, I, Constr. Approx., 23 (2006), 61–77.
T. Kühn, H.-G. Leopold, W. Sickel and L. Skrzypczak, Entropy numbers of embeddings of weighted Besov spaces, II, Proc. Edinb. Math. Soc., 49 (2006), 331–359.
J. Lang, Improved estimates for the approximation numbers of Hardy-type operator, J. Approx. Theory, 121 (2003), 61–70.
J. Lang, Estimates for n-widths of the Hardy-type operators, J. Approx. Theory, 140 (2006), 141–146.
J. Lang, A. Nekvinda and O. Mendez, Asymptotic behavior of the approximation numbers of the Hardy-type operator from L p into L q (case 1 < p ≤ q ≤ 2 or 2 ≤ p ≤ q < ∞), J. Inequal. Pure Appl. Math., 5 (2004), Article 18.
P. G. Lemarie, Une nouvelle base d’ondelettes de L 2(ℝn), J. Math. Pures Appl., 67 (1988), 227–236.
M. A. Lifshits and W. Linde, Approximation and entropy numbers of Volterra operators with application to Brownian motion, Mem. Amer. Math. Soc., 157 (2002), 1–87.
E. N. Lomakina and V. D. Stepanov, Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann–Liouville operator, Siberian Adv. Math., 17 (2007), 1–36.
A. Malecka, Haar functions in weighted Besov and Triebel–Lizorkin spaces, J. Approx. Theory, 200 (2015), 1–27.
Ya. Novikov and S. B. Stechkin, Basic constructions of wavelets, Fundam. Prikl. Mat., 3 (1997), 999–1028.
Ya. Novikov and S. B. Stechkin, Basic wavelet theory, Russian Math. Surveys, 53 (1998), 1159–1231.
A. Pietsch, Eigenvalues and s-numbers, Akad. Verlagsgesellschaft Geest & Portig (Leipzig, 1987).
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Part II, Springer-Verlag (Berlin–Heidelberg, 1978).
H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley (Chichester, 1987).
I. J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math., 4 (1946), 45–99, 112–141.
T. Schott, Funtion spaces with exponential weights, I, Math. Nachr., 189 (1998), 221–242.
W. Sickel and H. Triebel, Hölder inequalities and sharp embeddings in function spaces of \(B_{pq}^s\) and \(F_{pq}^s\) type, Z. Anal. Anwend., 14 (1995), 105–140.
L. Skrzypczak, On approximation numbers of Sobolev embeddings of weighted function spaces, J. Approx. Theory, 136 (2005), 91–107.
H. Triebel, Theory of Function Spaces, Birkhäuser Verlag (Basel, 1983).
H. Triebel, Theory of Function Spaces, II, Birkhäuser Verlag (Basel, 1992).
H. Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, European Math. Soc. Publishing House (Zurich, 2010).
A. A. Vasil’eva, Entropy numbers of embedding operators for weighted Sobolev spaces, Mat. Zametki, 98 (2015), 937–940 (in Russsian); translation in Math. Notes, 98 (2015), 982–985.
A. Wojciechowska, Local means and wavelets in function spaces with local Muckenhoupt weights, in: Function Spaces IX, Banach Center Publications 92 (2011), pp. 399–412.
A. Wojciechowska, Multidimensional wavelet bases in Besov and Lizorkin–Triebel spaces, PhD. thesis, Adam Mickiewicz University (Poznań, 2012).
P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press (Cambridge, UK, 1997).
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The research of the second author was supported by the Russian Science Foundation (project RSF-DST: 16-41-02004).
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Nasyrova, M.G., Ushakova, E.P. Wavelet Bases and Entropy Numbers of Hardy Operator. Anal Math 44, 543–576 (2018). https://doi.org/10.1007/s10476-017-0603-9
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DOI: https://doi.org/10.1007/s10476-017-0603-9