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Generalized Amalgams, With Applications to Fourier Transform

Published online by Cambridge University Press:  20 November 2018

Hans G. Feichtinger
Hans G. Feichtinger*
Affiliation:
Department of Mathematics University of Maryland College Park, MD, 20742 USA
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A recent survey article by J. Fournier and J. Stewart (Bull.AMS 13 (1985), 1-21) explains how amalgams of Lp with lq (as function spaces over any locally compact abelian group G) can be used as an effective tool for the treatment of various problems in harmonic analysis. The present article may be seen as a complement to this survey, indicating further advantages that arise if one works with generalized amalgams (introduced in 1980 under the name of Wiener-type spaces by the author [10]). The main difference between amalgams and these more general spaces is the fact that they allow a more precise description of the local behavior of functions (or distributions) by rather arbitrary norms and that the conditions on the global behavior (of the quantity obtained using that chosen local norm) is described in a way that includes both growth and integrability conditions (not only lq-summability).

Keywords

46E3043A1543A25generalized amalgams(generalized) Fourier transformcomplex interpolationWiener type spacesHausdorff-Young inequalityp-localizableultra-distributions
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 42 , Issue 3 , 01 June 1990 , pp. 395 - 409
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Bergh, J., Löfström, J., Interpolation spaces (An Introduction), Grundl. math. Wiss. 223, Berlin- Heidelberg-New York, Springer, 1976.Google Scholar
2. Bloom, W.R., Strict local inclusion results between spaces of Fourier transforms, Pac. J. Math. 99 (1982), 265–270.Google Scholar
3. Björck, G., Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351–407.Google Scholar
4. Coifman, R.R. and Meyer, Y., Au-delà des opérateurs pseudo-différentiels, Astérisque, Soc. Math. France, n° 57.Google Scholar
5. Feichtinger, H.G., A characterization of Wiener's algebra on locally compact groups, Archiv f. Math. 29(1977), 136–140.Google Scholar
6. Feichtinger, H.G., Gewichtsfunktionen auf lokalkompakten Gruppen, Sitzber.d. österr. Adad. Wiss. 1S8 (1979), 451–471.Google Scholar
7. Feichtinger, H.G., A characterization of minimal homogeneous Banach spaces, Proc. Amer. Math. Soc. SI(1981), 55–61.Google Scholar
8. Feichtinger, H.G., (Jnespace de distributions tempérées sur les groupes localement compactes abelians, Compt. Rend. Acad. Sci. Paris, Ser. A, 290/17 (1980), 791–794.Google Scholar
9. Feichtinger, H.G., On a new Segal algebra, Monh. f. Math. 92 (1981), 269–289.Google Scholar
10. Feichtinger, H.G., Banach convolution algebras of Wiener s type, Proc. Conf. “Functions, Series, Operators”, Budapest, August 1980, Colloquia Math. Soc. J. Bolyai, North Holland Publ. Co., Amsterdam-Oxford-New York (1983), 509–524.Google Scholar
11. Feichtinger, H.G., Banach spaces of distributions of Wiener's type and interpolation, Proc. Conf. Oberwolfach, August 1980. Functional Analysis and Approximation. Ed. P. Butzer, B. Sz. Nagy and E. Görlich. Int. Ser. Num. Math. Vol. 69, Birkhäuser-Verlag. Basel-Boston-Stuttgart (1981), 153–165.Google Scholar
12. Feichtinger, H.G., Compactness in translation invariant Banach spaces of distributions and compact multipliers,, J. Math. Anal. Appl. 102 (1984), 289–327.Google Scholar
13. Feichtinger, H.G., A new family of functional spaces on Euclidean n-space. Proc. Conf. “Theory of Approximationof Functions”, 30.5-6.6.1983 Kiew.Google Scholar
14. Feichtinger, H.G., Modulation spaces on locally compact abelian groups. Techn. Report, Univ. Vienna, 1983 (5().p).Google Scholar
15. Feichtinger, H.G., Minimal Banach spaces and atomic representations, Publ. Math. Debrecen 33 (1986), announcement at Conf. in Debrecen Nov. 1984, Teor. Priblizh., UDC 517.98, p. 493-497, full article in 34 (1987), 231–240.Google Scholar
16. Feichtinger, H.G., Banach spaces of distributions defined by decomposition methods IL Math. Nachr. 132 (1987), 207–237.Google Scholar
17. Feichtinger, H.G., Atomic characterizations of modulation spaces through Gabor-type representations, Proc. Conf. “Constructive Function Theory”, Edmonton, July 1986, Rocky Mount. J. Math. 19 (1989), 113–126.Google Scholar
18. Feichtinger, H.G., Translation bounded quasimeasures and some of their application in harmonic analysis (80 p., in preparation).Google Scholar
19. Feichtinger, H.G. and Gröbner, P., Banach spaces of distributions defined by decompositionmethods, I. Math. Nachr. 123 (85), 97–120.Google Scholar
20. Feichtinger, H.G. and Gröchenig, K.H., A unified approach to atomic characterizations throughintegrable group representations. Proc. Conf. Lund, June 1986, Lect. Notes in Math. 1302 (1988), 52–73.Google Scholar
21. Fournier, J.J., Local complements to the Hausdorff'-Young theorem, Mich. Math. J. 20 (1973), 263–276.Google Scholar
22. Fournier, J.J. and Stewart, J., Amalgams, Bull. Amer. Math. Soc. 13 (1985), 1–21.Google Scholar
23. Larsen, R., An Introduction to the Theory of Multipliers, Grundl. math. Wiss. Bd. 175, New York-Heidelberg-Berlin, Springer, 1971.Google Scholar
24. Peetre, J., New Thoughts on Besov Spaces, Duke Univ. Press, Durham, 1976.Google Scholar
25. Reiter, H., Classical Harmonic Analysis and Locally Compact Groups, Oxford Univ. Press, 1968.Google Scholar
26. Stein, E.M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.Google Scholar
27. Tchamitchian, Ph., Géneralisation des algèbres de Beurling, Ann. Inst. Fourier, Grenoble, 3414 (1984), 151–168.Google Scholar
28. Triebel, H., Interpolation Theory, Function Spaces, Differential Operators, Berlin, VEB, Dt.Verlag d.Wiss, and North Holland, 1978.Google Scholar
29. Triebel, H., Fourier Analysis and Function Spaces, Teubner Texte zur Mathematik, Teubner Verlag, Leipzig, 1978.Google Scholar
30. Triebel, H., Theory of Function Spaces, Portig & Geest, Leipzig and Birkhauser, Basel-Boston-New York, 1983.Google Scholar
31. Triebel, H., Modulation spaces on the Euclidean n-space, Zeitschr.f.Anal. und Anwendungen 2 (1983), 443-457.Google Scholar