Sets of uniqueness for spherically convergent multiple trigonometric series
Authors:
J. Marshall Ash and Gang Wang
Journal:
Trans. Amer. Math. Soc. 354 (2002), 47694788
MSC (2000):
Primary 05C38, 15A15; Secondary 05A15, 15A18
Published electronically:
July 25, 2002
MathSciNet review:
1926836
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A subset of the dimensional torus is called a set of uniqueness, or set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are sets and also that sets are sets for every . In particular, , where is the Cantor set, is an set and hence a set. We will say that is a set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The abovementioned results hold also for sets. In addition, every set has measure , and a countable union of closed sets is a set.
 [AKR]
J. M. Ash, E. Rieders, and R. P. Kaufman, The CantorLebesgue property, Israel J. Math. 84 (1993), 179191. MR 94m:42007
 [AW]
J. M. Ash and G. Wang, A survey of uniqueness questions in multiple trigonometric series, A Conference in Harmonic Analysis and Nonlinear Differential Equations in Honor of Victor L. Shapiro, Contemporary Mathematics, 208 (1997), 3571. MR 99f:42019
 [AW1]
J. M. Ash and G. Wang, Some spherical uniqueness theorems for multiple trigonometric series, Annals of Math., 151 (2000), 133. MR 2001a:42009
 [AW2]
J. M. Ash and G. Wang, Sets of uniqueness of the power of the continuum, Proc. of the Georgian Academy of Sciences, 122 (2000), 1519. MR 2001k:42010
 [Be]
L. D. Berkovitz, Circular summation and localization of double trigonometric series, Trans. Amer. Math. Soc. 70 (1951), 323344. MR 12:697b
 [B]
J. Bourgain, Spherical summation and uniqueness of multiple trigonometric series, Internat. Math. Res. Notices (1996), 93107. MR 97b:42022
 [Co]
B. Connes, Sur les coefficients des séries trigonométriques convergentes sphériquement, C. R. Acad. Sci. Paris Sér. A 283 (1976), 159161. MR 54:10975
 [C]
R. Cooke, A CantorLebesgue theorem in two dimensions, Proc. Amer. Math. Soc. 30 (1971), 547550. MR 43:7847
 [KL]
A. S. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, 128, Cambridge University Press, CambridgeNew York, 1987. MR 90a:42008
 [KN]
L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974. MR 54:7415
 [PS]
I. I. PyatetskiShapiro, On the problem of uniqueness of expansion of a function in a trigonometric series (in Russian), Ucenye Zapiski Moskovskogo Gosudarstvennogo Universiteta, 155, Matematika 5 (1952), 5472.
 [PS1]
I. I. PyatetskiShapiro, Supplement to the work `On the problem, etc.' (in Russian), Ucenye Zapiski Moskovskogo Gosudarstvennogo Universiteta, 165, Matematika 7(1954), 7897.
 [Sh]
V. L. Shapiro, Uniqueness of multiple trigonometric series, Ann. Math. 66 (1957), 467480. MR 19:854d
 [Sh1]
V. L. Shapiro, Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 4893. MR 28:1448
 [Sh2]
V. L. Shapiro, Removable sets for pointwise solutions of the generalized CauchyRiemann equations, Ann. Math. 92 (1970), 82101. MR 55:10819
 [Sh3]
V. L. Shapiro, Sets of uniqueness on the 2 torus, Trans. Amer. Math. Soc. 165 (1972), 127147. MR 46:7798
 [Z]
A. Zygmund, A CantorLebesgue theorem for double trigonometric series, Studia Math. 43 (1972), 173178. MR 47:711
 [Z1]
A. Zygmund, Trigonometric series, Vols. I, II, Cambridge University Press, CambridgeNew YorkMelbourne, 1977. MR 58:29731
 [AKR]
 J. M. Ash, E. Rieders, and R. P. Kaufman, The CantorLebesgue property, Israel J. Math. 84 (1993), 179191. MR 94m:42007
 [AW]
 J. M. Ash and G. Wang, A survey of uniqueness questions in multiple trigonometric series, A Conference in Harmonic Analysis and Nonlinear Differential Equations in Honor of Victor L. Shapiro, Contemporary Mathematics, 208 (1997), 3571. MR 99f:42019
 [AW1]
 J. M. Ash and G. Wang, Some spherical uniqueness theorems for multiple trigonometric series, Annals of Math., 151 (2000), 133. MR 2001a:42009
 [AW2]
 J. M. Ash and G. Wang, Sets of uniqueness of the power of the continuum, Proc. of the Georgian Academy of Sciences, 122 (2000), 1519. MR 2001k:42010
 [Be]
 L. D. Berkovitz, Circular summation and localization of double trigonometric series, Trans. Amer. Math. Soc. 70 (1951), 323344. MR 12:697b
 [B]
 J. Bourgain, Spherical summation and uniqueness of multiple trigonometric series, Internat. Math. Res. Notices (1996), 93107. MR 97b:42022
 [Co]
 B. Connes, Sur les coefficients des séries trigonométriques convergentes sphériquement, C. R. Acad. Sci. Paris Sér. A 283 (1976), 159161. MR 54:10975
 [C]
 R. Cooke, A CantorLebesgue theorem in two dimensions, Proc. Amer. Math. Soc. 30 (1971), 547550. MR 43:7847
 [KL]
 A. S. Kechris and A. Louveau, Descriptive set theory and the structure of sets of uniqueness, London Mathematical Society Lecture Note Series, 128, Cambridge University Press, CambridgeNew York, 1987. MR 90a:42008
 [KN]
 L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974. MR 54:7415
 [PS]
 I. I. PyatetskiShapiro, On the problem of uniqueness of expansion of a function in a trigonometric series (in Russian), Ucenye Zapiski Moskovskogo Gosudarstvennogo Universiteta, 155, Matematika 5 (1952), 5472.
 [PS1]
 I. I. PyatetskiShapiro, Supplement to the work `On the problem, etc.' (in Russian), Ucenye Zapiski Moskovskogo Gosudarstvennogo Universiteta, 165, Matematika 7(1954), 7897.
 [Sh]
 V. L. Shapiro, Uniqueness of multiple trigonometric series, Ann. Math. 66 (1957), 467480. MR 19:854d
 [Sh1]
 V. L. Shapiro, Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 4893. MR 28:1448
 [Sh2]
 V. L. Shapiro, Removable sets for pointwise solutions of the generalized CauchyRiemann equations, Ann. Math. 92 (1970), 82101. MR 55:10819
 [Sh3]
 V. L. Shapiro, Sets of uniqueness on the 2 torus, Trans. Amer. Math. Soc. 165 (1972), 127147. MR 46:7798
 [Z]
 A. Zygmund, A CantorLebesgue theorem for double trigonometric series, Studia Math. 43 (1972), 173178. MR 47:711
 [Z1]
 A. Zygmund, Trigonometric series, Vols. I, II, Cambridge University Press, CambridgeNew YorkMelbourne, 1977. MR 58:29731
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Additional Information
J. Marshall Ash
Affiliation:
Mathematics Department, DePaul University, Chicago, Illinois 60614
Email:
mash@math.depaul.edu
Gang Wang
Affiliation:
Mathematics Department, DePaul University, Chicago, Illinois 60614
Email:
gwang@math.depaul.edu
DOI:
http://dx.doi.org/10.1090/S0002994702030866
PII:
S 00029947(02)030866
Keywords:
Abel summation,
Baire category,
Fourier series,
generalized Laplacian,
Green's function,
$H^{J}$ sets,
multiple trigonometric series,
set of uniqueness,
spherical convergence,
subharmonic function,
uniqueness
Received by editor(s):
July 15, 1999
Received by editor(s) in revised form:
April 24, 2000
Published electronically:
July 25, 2002
Additional Notes:
This paper is in final form and no version of it will be submitted for publication elsewhere
Article copyright:
© Copyright 2002
American Mathematical Society
