Abstract
N. Wiener characterized the cyclic vectors (with respect to translations) in $\ell^p(\mathbb{Z})$ and $L^p(\mathbb{R})$, $p=1,2$, in terms of the zero set of the Fourier transform. He conjectured that a similar characterization should be true for $1 < p < 2$. Our main result contradicts this conjecture.
-
[atkinson] K. E. Atkinson, An Introduction to Numerical Analysis, Second ed., New York: John Wiley & Sons, 1989.
@book {atkinson, MRKEY = {1007135},
AUTHOR = {Atkinson, Kendall E.},
TITLE = {An Introduction to Numerical Analysis},
EDITION = {Second},
PUBLISHER = {John Wiley \& Sons},
ADDRESS = {New York},
YEAR = {1989},
PAGES = {xvi+693},
ISBN = {0-471-62489-6},
MRCLASS = {65-01},
MRNUMBER = {1007135},
ZBLNUMBER = {0718.65001},
} -
[azuma]
K. Azuma, "Weighted sums of certain dependent random variables," Tôhoku Math. J., vol. 19, pp. 357-367, 1967.
@article {azuma, MRKEY = {0221571},
AUTHOR = {Azuma, Kazuoki},
TITLE = {Weighted sums of certain dependent random variables},
JOURNAL = {Tôhoku Math. J.},
FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
VOLUME = {19},
YEAR = {1967},
PAGES = {357--367},
ISSN = {0040-8735},
MRCLASS = {60.40},
MRNUMBER = {0221571},
MRREVIEWER = {Bernard Harris},
DOI = {10.2748/tmj/1178243286},
ZBLNUMBER = {0178.21103},
} -
[bary] N. K. Bary, A Treatise on Trigonometric Series, Oxford: Pergamon Press, 1964.
@book{bary,
author={Bary, N. K.},
TITLE={A Treatise on Trigonometric Series},
PUBLISHER={Pergamon Press},
ADDRESS={Oxford},
YEAR={1964},
MRNUMBER = {0171116},
ZBLNUMBER={0129.28002},
} -
[beurling:spectres] A. Beurling, "Sur les spectres des fonctions," in Analyse Harmonique, Paris: Centre National de la Recherche Scientifique, 1949, vol. 15, pp. 9-29.
@incollection {beurling:spectres, MRKEY = {0033367},
AUTHOR = {Beurling, Arne},
TITLE = {Sur les spectres des fonctions},
BOOKTITLE = {Analyse {H}armonique},
SERIES = {Colloques Internationaux du Centre National de la Recherche Scientifique},
VOLUME={15},
PAGES = {9--29},
PUBLISHER = {Centre National de la Recherche Scientifique},
ADDRESS = {Paris},
YEAR = {1949},
MRCLASS = {42.4X},
MRNUMBER = {0033367},
MRREVIEWER = {B. Sz.-Nagy},
ZBLNUMBER = {0040.21102},
} -
@article {beurling:closure, MRKEY = {0043248},
AUTHOR = {Beurling, Arne},
TITLE = {On a closure problem},
JOURNAL = {Ark. Mat.},
FJOURNAL = {Arkiv för Matematik},
VOLUME = {1},
YEAR = {1951},
PAGES = {301--303},
ISSN = {0004-2080},
MRCLASS = {42.4X},
MRNUMBER = {0043248},
MRREVIEWER = {L. Schwartz},
DOI = {10.1007/BF02591366},
ZBLNUMBER = {0042.35402},
} -
[edwards]
R. E. Edwards, "Spans of translates in $L^{p}\,(G)$," J. Austral. Math. Soc., vol. 5, pp. 216-233, 1965.
@article {edwards, MRKEY = {0192279},
AUTHOR = {Edwards, R. E.},
TITLE = {Spans of translates in {$L\sp{p}\,(G)$}},
JOURNAL = {J. Austral. Math. Soc.},
FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics},
VOLUME = {5},
YEAR = {1965},
PAGES = {216--233},
ISSN = {0263-6115},
MRCLASS = {42.56},
MRNUMBER = {0192279},
MRREVIEWER = {C. S. Herz},
ZBLNUMBER = {0156.37501},
DOI = {10.1017/S1446788700026781},
} -
[herz]
C. S. Herz, "A note on the span of translations in $L^p$," Proc. Amer. Math. Soc., vol. 8, pp. 724-727, 1957.
@article {herz, MRKEY = {0088604},
AUTHOR = {Herz, C. S.},
TITLE = {A note on the span of translations in {$L\sp p$}},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {8},
YEAR = {1957},
PAGES = {724--727},
ISSN = {0002-9939},
MRCLASS = {44.0X},
MRNUMBER = {0088604},
MRREVIEWER = {R. E. Fullerton},
DOI = {10.2307/2033288},
ZBLNUMBER = {0078.10601},
} -
[kahane:synthese] J. Kahane, "Sur la synthèse harmonique dans $l^{\infty }$," An. Acad. Brasil. Ci., vol. 32, pp. 179-189, 1960.
@article {kahane:synthese, MRKEY = {0123892},
AUTHOR = {Kahane, Jean-Pierre},
TITLE = {Sur la synthèse harmonique dans {$l\sp{\infty }$}},
JOURNAL = {An. Acad. Brasil. Ci.},
FJOURNAL = {Anais da Academia Brasileira de Ciências},
VOLUME = {32},
YEAR = {1960},
PAGES = {179--189},
ISSN = {0001-3765},
MRCLASS = {42.58},
MRNUMBER = {0123892},
MRREVIEWER = {C. S. Herz},
ZBLNUMBER = {0161.11301},
} -
[kahane:absolument] J. Kahane, Séries de Fourier Absolument Convergentes, New York: Springer-Verlag, 1970.
@book {kahane:absolument, MRKEY = {0275043},
AUTHOR = {Kahane, Jean-Pierre},
TITLE = {Séries de {F}ourier Absolument Convergentes},
SERIES = {Ergeb. Math. Grenzgeb.},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1970},
PAGES = {viii+169},
MRCLASS = {42.12},
MRNUMBER = {0275043},
MRREVIEWER = {G. I. Gaudry},
ZBLNUMBER = {0195.07602},
} -
[kahane-salem] J. Kahane and R. Salem, Ensembles Parfaits et Séries Trigonométriques, Second ed., Paris: Hermann, 1994.
@book {kahane-salem, MRKEY = {1303593},
AUTHOR = {Kahane, Jean-Pierre and Salem, Rapha{ë}l},
TITLE = {Ensembles Parfaits et Séries Trigonométriques},
EDITION = {Second},
PUBLISHER = {Hermann},
ADDRESS = {Paris},
YEAR = {1994},
PAGES = {245},
ISBN = {2-7056-6193-X},
MRCLASS = {42-02 (43-02)},
MRNUMBER = {1303593},
MRREVIEWER = {Colin C. Graham},
ZBLNUMBER = {0856.42001},
} -
[katznelson] Y. Katznelson, An Introduction to Harmonic Analysis, 2nd ed., New York: Dover Publications, 1976.
@book {katznelson, MRKEY = {0422992},
AUTHOR = {Katznelson, Yitzhak},
TITLE = {An Introduction to Harmonic Analysis},
EDITION = {2nd},
PUBLISHER = {Dover Publications},
ADDRESS = {New York},
YEAR = {1976},
PAGES = {xiv+264},
MRCLASS = {43-02 (42-02)},
MRNUMBER = {0422992},
ZBLNUMBER = {0352.43001},
} -
[kaufman] R. Kaufman, "$M$-sets and distributions," in Pseudofunctions and Helson Sets, Paris: Soc. Math. France, 1973, vol. 5, pp. 225-230.
@incollection {kaufman, MRKEY = {0404975},
AUTHOR = {Kaufman, Robert},
TITLE = {{$M$}-sets and distributions},
BOOKTITLE = {Pseudofunctions and {H}elson Sets},
PAGES = {225--230},
SERIES={Astérisque},
VOLUME={5},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1973},
MRCLASS = {42A48 (43A45)},
MRNUMBER = {0404975},
MRREVIEWER = {Colin C. Graham},
ZBLNUMBER = {0281.43006},
} -
[kinukawa]
M. Kinukawa, "A note on the closure of translations in $L^{p}$," Tôhoku Math. J., vol. 18, pp. 225-231, 1966.
@article {kinukawa, MRKEY = {0200669},
AUTHOR = {Kinukawa, Masakiti},
TITLE = {A note on the closure of translations in {$L\sp{p}$}},
JOURNAL = {Tôhoku Math. J.},
FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
VOLUME = {18},
YEAR = {1966},
PAGES = {225--231},
ISSN = {0040-8735},
MRCLASS = {42.18},
MRNUMBER = {0200669},
MRREVIEWER = {C. S. Herz},
DOI = {10.2748/tmj/1178243449},
ZBLNUMBER = {0156.35901},
} -
[korner] T. Körner, "A pseudofunction on a Helson set. I and II," in Pseudofunctions and Helson sets, Paris: Soc. Math. France, 1973, vol. 5, pp. 3-224 and 231.
@incollection {korner, MRKEY = {0404974},
AUTHOR = {K{ö}rner, Thomas},
TITLE = {A pseudofunction on a {H}elson set. {I} and {II}},
BOOKTITLE = {Pseudofunctions and {H}elson sets},
PAGES = {3--224 and 231--239},
SERIES={Astérisque},
VOLUME={5},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1973},
MRCLASS = {42A48 (43A45)},
MRNUMBER = {0404974},
MRREVIEWER = {Colin C. Graham},
ZBLNUMBER = {0281.43004},
} -
[lev-olevskii:piatetski]
N. Lev and A. Olevskii, "Piatetski-Shapiro phenomenon in the uniqueness problem," C. R. Math. Acad. Sci. Paris, vol. 340, iss. 11, pp. 793-798, 2005.
@article {lev-olevskii:piatetski, MRKEY = {2139889},
AUTHOR = {Lev, Nir and Olevskii, Alexander},
TITLE = {Piatetski-{S}hapiro phenomenon in the uniqueness problem},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
VOLUME = {340},
YEAR = {2005},
NUMBER = {11},
PAGES = {793--798},
ISSN = {1631-073X},
MRCLASS = {42A38},
MRNUMBER = {2139889},
DOI = {10.1016/j.crma.2005.04.031},
ZBLNUMBER = {1063.43006},
} -
[lev-olevskii:generators]
N. Lev and A. Olevskii, "No characterization of generators in $l^p (1<p<2)$ by zero set of Fourier transform," C. R. Math. Acad. Sci. Paris, vol. 346, iss. 11-12, pp. 645-648, 2008.
@article {lev-olevskii:generators, MRKEY = {2423270},
AUTHOR = {Lev, Nir and Olevskii, Alexander},
TITLE = {No characterization of generators in {$l\sp p (1 JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
VOLUME = {346},
YEAR = {2008},
NUMBER = {11-12},
PAGES = {645--648},
ISSN = {1631-073X},
MRCLASS = {42A55 (42A05 43A15)},
MRNUMBER = {2423270},
MRREVIEWER = {Colin C. Graham},
DOI = {10.1016/j.crma.2008.04.017},
ZBLNUMBER = {1155.46015},
} -
[malliavin] P. Malliavin, "Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts," Inst. Hautes Études Sci. Publ. Math., vol. 1959, pp. 85-92, 1959.
@article {malliavin, MRKEY = {0107129},
AUTHOR = {Malliavin, Paul},
TITLE = {Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {1959},
YEAR = {1959},
PAGES = {85--92},
ISSN = {0073-8301},
MRCLASS = {42.00},
MRNUMBER = {0107129},
MRREVIEWER = {C. S. Herz},
ZBLNUMBER = {0101.09403},
} -
[newman]
D. J. Newman, "The closure of translates in $l^{p}$," Amer. J. Math., vol. 86, pp. 651-667, 1964.
@article {newman, MRKEY = {0164193},
AUTHOR = {Newman, D. J.},
TITLE = {The closure of translates in {$l\sp{p}$}},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {86},
YEAR = {1964},
PAGES = {651--667},
ISSN = {0002-9327},
MRCLASS = {42.58},
MRNUMBER = {0164193},
MRREVIEWER = {C. S. Herz},
DOI = {10.2307/2373029},
ZBLNUMBER = {0147.11301},
} -
[piatetski] I. I. Pyateckiui-vSapiro, "Supplement to the work “On the problem of uniqueness of expansion of a function in a trigonometric series.”," Moskov. Gos. Univ. Uč. Zap. Mat., vol. 165(7), pp. 79-97, 1954.
@article {piatetski, MRKEY = {0067226},
AUTHOR = {Pyatecki{\u\i}-{Š}apiro, I. I.},
TITLE = {Supplement to the work ``{O}n the problem of uniqueness of expansion of a function in a trigonometric series.''},
JOURNAL = {Moskov. Gos. Univ. Uč. Zap. Mat.},
NOTE={English translation in {\it Selected Works of Ilya Piatetski-Shapiro},
{\it Amer. Math. Soc. Collected Works} {\bf 15},
2000},
VOLUME = {165(7)},
YEAR = {1954},
PAGES = {79--97},
MRCLASS = {42.4X},
MRNUMBER = {0067226},
MRREVIEWER = {A. Zygmund},
} -
[pollard]
H. Pollard, "The closure of translations in $L^p$," Proc. Amer. Math. Soc., vol. 2, pp. 100-104, 1951.
@article {pollard, MRKEY = {0041968},
AUTHOR = {Pollard, Harry},
TITLE = {The closure of translations in {$L\sp p$}},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {2},
YEAR = {1951},
PAGES = {100--104},
ISSN = {0002-9939},
MRCLASS = {42.4X},
MRNUMBER = {0041968},
MRREVIEWER = {L. H. Loomis},
DOI = {10.2307/2032628},
ZBLNUMBER = {0043.32903},
} -
[rosenblatt-shuman]
J. M. Rosenblatt and K. L. Shuman, "Cyclic functions in $L^p(\Bbb R)$, $1\leq p<\infty$," J. Fourier Anal. Appl., vol. 9, iss. 3, pp. 289-300, 2003.
@article {rosenblatt-shuman, MRKEY = {1988753},
AUTHOR = {Rosenblatt, Joseph M. and Shuman, Karen L.},
TITLE = {Cyclic functions in {$L\sp p(\Bbb R)$},
{$1\leq p<\infty$}},
JOURNAL = {J. Fourier Anal. Appl.},
FJOURNAL = {The Journal of Fourier Analysis and Applications},
VOLUME = {9},
YEAR = {2003},
NUMBER = {3},
PAGES = {289--300},
ISSN = {1069-5869},
MRCLASS = {42A38 (42A65 46E30 46F05)},
MRNUMBER = {1988753},
MRREVIEWER = {Stamatis Koumandos},
DOI = {10.1007/s00041-003-0015-z},
ZBLNUMBER = {1044.42005},
} -
[segal:span] I. E. Segal, "The span of the translations of a function in a Lebesgue space," Proc. Nat. Acad. Sci. U. S. A., vol. 30, pp. 165-169, 1944.
@article {segal:span, MRKEY = {0010642},
AUTHOR = {Segal, I. E.},
TITLE = {The span of the translations of a function in a {L}ebesgue space},
JOURNAL = {Proc. Nat. Acad. Sci. U. S. A.},
FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
VOLUME = {30},
YEAR = {1944},
PAGES = {165--169},
ISSN = {0027-8424},
MRCLASS = {42.4X},
MRNUMBER = {0010642},
MRREVIEWER = {S. Bochner},
ZBLNUMBER = {0063.06859},
} -
[segal:group]
I. E. Segal, "The group algebra of a locally compact group," Trans. Amer. Math. Soc., vol. 61, pp. 69-105, 1947.
@article {segal:group, MRKEY = {0019617},
AUTHOR = {Segal, I. E.},
TITLE = {The group algebra of a locally compact group},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {61},
YEAR = {1947},
PAGES = {69--105},
ISSN = {0002-9947},
MRCLASS = {20.0X},
MRNUMBER = {0019617},
MRREVIEWER = {A. Weil},
DOI = {10.2307/1990290},
ZBLNUMBER = {0032.02901},
} -
[wiener:tauberian]
N. Wiener, "Tauberian theorems," Ann. of Math., vol. 33, iss. 1, pp. 1-100, 1932.
@article {wiener:tauberian, MRKEY = {1503035},
AUTHOR = {Wiener, Norbert},
TITLE = {Tauberian theorems},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {33},
YEAR = {1932},
NUMBER = {1},
PAGES = {1--100},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {Contributed Item},
MRNUMBER = {1503035},
DOI = {10.2307/1968102},
ZBLNUMBER = {0004.05905},
}