Recent developments concerning entropy and approximation numbers

Edmunds, David E.

Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (1994), p. 33-76 / Harvested from

Détail

MR MR1322309 | Zbl 0831.47014

EUDML-ID : urn:eudml:doc:220329
Mots clés:

Bibliographie

Birman, M.S.; Solomyak, M.Z. Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, 10th Math. School Kiev 1974, pp. 5–189. | MR 0562305 | Zbl 0426.46020

Edmunds, D.E.; Gurka, P.; Opic, B. Double exponential integrability of convolution operators in generalised Lorentz–Zygmund spaces, to appear. | MR 1336431

Edmunds, D.E.; Ilyin, A.A. Approximation numbers of embeddings of Sobolev spaces, to appear. | MR 1376290 | Zbl 0841.46022

Edmunds, D.E.; Krbec, M. Two limiting cases of Sobolev imbeddings, to appear. | MR 1331250 | Zbl 0835.46027

Edmunds, D.E.; Triebel, H. Logarithmic Sobolev spaces and their applications to spectral theory, Proc. London Math. Soc., to appear. | MR 1337470 | Zbl 0835.46028

Edmunds, D.E.; Triebel, H. Function spaces, entropy numbers, differential operators, (forthcoming book). | MR 1410258 | Zbl 1143.46001

Franke, J.; Runst, Th. Regular elliptic boundary value problems in Besov–Triebel–Lizorkin spaces, to appear. | MR 1349041 | Zbl 0843.35026

Haroske, D. Approximation numbers in weighted function spaces, to appear.

Sickel, W.; Triebel, H. Hölder inequalities and sharp embeddings in function spaces of $B_{pq}^s$ and $F_{pq}^s$ type, to appear.

Solomyak, M.Z. Piecewise polynomial approximation of functions from Sobolev spaces, revisited, to appear. | MR 1423362 | Zbl 0885.46033

Solomyak, M.Z. Two-sided estimates on singular values for a class of integral operators on the semi-axis, to appear. | Zbl 0817.47024

Triebel, H. Relations between approximation numbers and entropy numbers, to appear in J. Approx. Theory. | MR 1284581 | Zbl 0811.47020

Triebel, H.; Winkelvoss, H. Intrinsic atomic characterizations of function spaces on domains, to appear. | Zbl 0843.46022

Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. Pure Appl. Math. 12 (1959), 623–727. (1959) | MR 0125307 | Zbl 0093.10401

Agmon, S. On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119–147. (1962) | MR 0147774 | Zbl 0109.32701

Birman, M.S.; Solomyak, M.Z. Piecewise polynomial approximations of functions of the classes $W_p^\alpha $ , Mat. Sb. 73 (1967), 331–355. (1967) | MR 0217487

Trudinger, N.S. On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. (1967) | MR 0216286 | Zbl 0163.36402

Birman, M.S.; Solomyak, M.Z. Spectral asymptotics of non-smooth elliptic operators, I, Trans. Moscow Math. Soc. 27 (1972), 1–52. (1972) | MR 1157648

Rosenbljum, G.V. The distribution of the discrete spectrum for singular differential operators, Soviet Math. Dokl. 13 (1972), 245–249. (1972)

Strichartz, R.S. A note on Trudinger’s extension of Sobolev’s inequalities, Indiana Univ. Math. J. 21 (1972), 841–8. (1972) | MR 0293389 | Zbl 0241.46028

Birman, M.S.; Solomyak, M.Z. Spectral asymptotics of non-smooth elliptic operators, II, Trans. Moscow Math. Soc. 28 (1973), 1–32. (1973) | MR 1157648

Lieb, E.W. Bounds on the eigenvalues of the Laplace and Schrödinger operators, Bull. Amer. Math. Soc. 82 (1976), 751–753. (1976) | MR 0407909 | Zbl 0329.35018

Rosenbljum, G.V. Distribution of the discrete spectrum of singular differential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1 (1976), 75–86. (Russian) (1976) | MR 0430557

Simon, B. Analysis with weak trace ideals and the number of bound states of Schrödinger operators, Trans. Amer. Math. Soc. 224 (1976), 367–380. (1976) | MR 0423128 | Zbl 0348.47017

Birman, M.S.; Solomyak, M.Z. The asymptotics of the spectrum of pseudodifferential operators with anisotropically homogeneous symbols, I, II, Vestnik Leningrad Univ. Mat. 13 (1977), 13–21. (Russian) (1977)

Cwikel, M. Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. 106 (1977), 93–100. (1977) | MR 0473576 | Zbl 0362.47006

Triebel, H. Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, 1978. (1978) | MR 0503903 | Zbl 0387.46033

Simon, B. Trace ideals and their applications, Cambridge University Press, Cambridge, 1979. (1979) | MR 0541149 | Zbl 0423.47001

Bennett, C.; Rudnick, K. On Lorentz-Zygmund spaces, Dissertationes Math. 175 (1980), 1–72. (1980) | MR 0576995 | Zbl 0456.46028

Carl, B.; Triebel, H. Inequalities between eigenvalues, entropy numbers and related quantities in Banach spaces, Math. Ann. 251 (1980), 129–133. (1980) | MR 0585959

Pietsch, A. Operator ideals, North-Holland, Amsterdam, 1980. (1980) | MR 0582655 | Zbl 0455.47032

Carl, B. Entropy numbers, $s$-numbers and eigenvalue problems, J. Funct. Anal. 41 (1981), 290–306. (1981) | MR 0619953 | Zbl 0466.41008

Carl, B. Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems, Proc. Roy. Soc. Edinburgh Sect. A 90 (1981), 63–70. (1981) | MR 0636064 | Zbl 0508.47041

Tashkian, G.M. The classical formula of the asymptotics of the spectrum of elliptic equations, degenerate at the boundary of the domain, Mat. Zametki 30 (1981), no. 6, 871–880. (Russian) (1981) | MR 0641661

Glushin, E.D. Norms of random matrices and diameter of finite-dimensional sets, Mat. Sb. 120 (1983), 180–189. (Russian) (1983) | MR 0687610

Triebel, H. Theory of function spaces, Birkhäuser, Basel, 1983. (1983) | MR 0781540 | Zbl 0546.46028

Frazier, M.; Jawerth, B. Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777–799. (1985) | MR 0808825 | Zbl 0551.46018

Edmunds, D.E.; Tylli, H.-O. On the entropy numbers of an operator and its adjoint, Math. Nachr. 126 (1986), 231–239. (1986) | MR 0846577 | Zbl 0625.47015

König, H. Eigenvalue distributions of compact operators, Birkhäuser, Basel, 1986. (1986) | MR 0889455

Linde, R. $s$-numbers of diagonal operators and Besov embeddings, Proc. 13th Winter School, Suppl. Rend. Circ. Mat. Palermo 2 (1986), 83–110. (1986) | MR 0894275

Schechter, M. Spectra of partial differential operators (2nd edition), North Holland, Amsterdam, 1986. (1986) | MR 0869254

Edmunds, D.E.; Evans, W.D. Spectral theory and differential operators, Oxford University Press, Oxford, 1987. (1987) | MR 0929030 | Zbl 0628.47017

Pietsch, A. Eigenvalues and $s$-numbers, Cambridge University Press, Cambridge, 1987. (1987) | MR 0890520 | Zbl 0615.47019

Adams, D.R. A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385–398. (1988) | MR 0960950 | Zbl 0672.31008

Bennett, C.; Sharpley, R. Interpolation of operators, Academic Press, New York, 1988. (1988) | MR 0928802 | Zbl 0647.46057

Cobos, F.; Edmunds, D.E. Clarkson’s inequalities, Besov spaces and Triebel–Sobolev spaces, Z. Anal. Anwendungen 7 (1988), 229–232. (1988) | MR 0951121 | Zbl 0663.46026

Edmunds, D.E.; Evans, W.D.; Harris, D.J. Approximation numbers of certain Volterra integral operators, J. London Math. Soc. 37 (1988), no. 2, 471–489. (1988) | MR 0939123 | Zbl 0658.47049

Bourgain, J.; Pajor, A.; Szarek, S.J.; Tomczak, N. -Jaegerman On the duality problem for entropy numbers of operators, Geometric aspects of functional analysis (1987–88), Lecture Notes in Mathematics, Vol. 1376, Springer, Berlin, 1989, pp. 50–63. (1989) | MR 1008716

Cobos, F.; Fernandez, D.L. On interpolation of compact operators, Ark. Mat. 27 (1989), 211–217. (1989) | MR 1022277 | Zbl 0691.46047

Edmunds, D.E.; Triebel, H. Entropy numbers and approximation numbers in function spaces, Proc. London Math. Soc. 58 (1989), no. 3, 137–152. (1989) | MR 0969551 | Zbl 0629.46034

Carl, B.; Stephani, I. Entropy, compactness and the approximation of operators, Cambridge University Press, Cambridge, 1990. (1990) | MR 1098497 | Zbl 0705.47017

Cobos, F.; Edmunds, D.E.; Potter, A.J.B. Real interpolation and compact linear operators, J. Funct. Anal. 88 (1990), 351–365. (1990) | MR 1038446 | Zbl 0704.46049

Cwikel, M. Real and complex interpolation and extrapolation of compact operators, Technion, Haifa, 1990, preprint. (1990) | MR 1150590

Birman, M.S.; Karadzhov, G.E.; Solomyak, M.Z. Boundedness conditions and spectrum estimates for the operators $b(X)a(D)$ and their analogs, Adv. in Soviet Math. 7 (1991), 85–106. (1991) | MR 1306510 | Zbl 0768.47025

Birman, M.S.; Solomyak, M.Z. Estimates for the number of negative eigenvalues of the Schrödinger operator and its generalizations, Adv. in Soviet Math. 7 (1991), 1–55. (1991) | MR 1306507

Edmunds, D.E.; Triebel, H. Entropy numbers and approximation numbers in function spaces, II, Proc. London Math. Soc. 64 (1992), no. 3, 153–169. (1992) | MR 1132858 | Zbl 0778.46022

Triebel, H. Theory of function spaces II, Birkhäuser, Basel, 1992. (1992) | MR 1163193 | Zbl 0763.46025

Edmunds, D.E.; Tylli, H.-O. Entropy numbers of tensor products of operators, Ark. Mat. 31 (1993), 247–274. (1993) | MR 1263554 | Zbl 1047.47502

Fusco, N.; Lions, P.L.; Sbordone, C. Some remarks on Sobolev imbeddings in borderline cases, Preprint No. 25, Università degli Studi di Napoli ”Federico II”, 1993, pp. 7. (1993)

Triebel, H. Approximation numbers and entropy numbers of embeddings of fractional Besov–Sobolev spaces in Orlicz spaces, Proc. London Math. Soc. 66 (1993), 589–618. (1993) | MR 1207550 | Zbl 0792.46024

Edmunds, D.E.; Gurka, P.; Pick, L. Compactness of Hardy-type integral operators in weighted Banach function spaces, Studia Math. 109 (1994), 73–90. (1994) | MR 1267713 | Zbl 0821.46036

Edmunds, D.E.; Ilyin, A.A. On some multiplicative inequalities and approximation numbers, Quart. J. Math. Oxford 45 (1994), 159–179. (1994) | MR 1280691 | Zbl 0818.46027

Edmunds, D.E.; Stepanov, V.D. The measure of non-compactness and approximation numbers of certain Volterra integral operators, Math. Ann. 298 (1994), 41–66. (1994) | MR 1252816 | Zbl 0788.45013

Edmunds, D.E.; Triebel, H. Eigenvalue distributions of some degenerate elliptic operators: an approach via entropy numbers, Math. Ann. 299 (1994), 311–340. (1994) | MR 1275771 | Zbl 0804.35097

Haroske, D.; Triebel, H. Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators I, Math. Nachr. 167 (1994), 131–156. (1994) | MR 1285311 | Zbl 0829.46019

Haroske, D.; Triebel, H. Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators II, Math. Nachr. 168 (1994), 109–137. (1994) | MR 1282635 | Zbl 0829.46020

Solomyak, M.Z. Piecewise polynomial approximation of functions from $H^\ell ((0,1)^d)$, $2\ell =d,$ and applications to the spectral theory of the Schrödinger operator, Israel J. Math. 86 (1994), 253–275. (1994) | MR 1276138

Triebel, H. A localization property of $B_{pq}^s$ and $F_{pq}^s$ spaces, Studia Math. 109 (1994), 183–195. (1994) | MR 1269775