A qualitative uncertainty principle for unimodular groups of type

Author:
Jeffrey A. Hogan

Journal:
Trans. Amer. Math. Soc. **340** (1993), 587-594

MSC:
Primary 43A30; Secondary 43A25

MathSciNet review:
1102222

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Abstract: It has long been known that if and the supports of and its Fourier transform are bounded then almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of and have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair . Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.

**[1]**W. O. Amrein and A. M. Berthier,*On support properties of 𝐿^{𝑝}-functions and their Fourier transforms*, J. Functional Analysis**24**(1977), no. 3, 258–267. MR**0461025****[2]**Michael Benedicks,*On Fourier transforms of functions supported on sets of finite Lebesgue measure*, J. Math. Anal. Appl.**106**(1985), no. 1, 180–183. MR**780328**, 10.1016/0022-247X(85)90140-4**[3]**Michael Cowling, John F. Price, and Alladi Sitaram,*A qualitative uncertainty principle for semisimple Lie groups*, J. Austral. Math. Soc. Ser. A**45**(1988), no. 1, 127–132. MR**940530****[4]**Jacques Dixmier,*𝐶*-algebras*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett; North-Holland Mathematical Library, Vol. 15. MR**0458185****[5]**Jeffrey A. Hogan,*A qualitative uncertainty principle for locally compact abelian groups*, Miniconferences on harmonic analysis and operator algebras (Canberra, 1987), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 16, Austral. Nat. Univ., Canberra, 1988, pp. 133–142. MR**953989****[6]**Ronald L. Lipsman,*Representation theory of almost connected groups*, Pacific J. Math.**42**(1972), 453–467. MR**0327975****[7]**Tamás Matolcsi and József Szűcs,*Intersection des mesures spectrales conjuguées*, C. R. Acad. Sci. Paris Sér. A-B**277**(1973), A841–A843 (French). MR**0326460****[8]**Calvin C. Moore,*Groups with finite dimensional irreducible representations*, Trans. Amer. Math. Soc.**166**(1972), 401–410. MR**0302817**, 10.1090/S0002-9947-1972-0302817-8**[9]**John F. Price and Alladi Sitaram,*Functions and their Fourier transforms with supports of finite measure for certain locally compact groups*, J. Funct. Anal.**79**(1988), no. 1, 166–182. MR**950089**, 10.1016/0022-1236(88)90035-3**[10]**Alain Robert,*Introduction to the representation theory of compact and locally compact groups*, London Mathematical Society Lecture Note Series, vol. 80, Cambridge University Press, Cambridge-New York, 1983. MR**690955****[11]**Walter Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR**0152834****[12]**G. Warner,*Harmonic analysis on semisimple Lie groups*. I and II, Springer, Berlin, 1972.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1102222-4

Keywords:
Fourier transform,
unimodular groups of type I

Article copyright:
© Copyright 1993
American Mathematical Society