The helical transform as a connection between ergodic theory and harmonic analysis
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- by Idris Assani and Karl Petersen
- Trans. Amer. Math. Soc. 331 (1992), 131-142
- DOI: https://doi.org/10.1090/S0002-9947-1992-1075378-9
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Abstract:
Direct proofs are given for the formal equivalence of the ${L^2}$ boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong $(2, 2)$ for the double maximal (ergodic) helical transform is extended to actions of ${\mathbb {R}^d}$ and ${\mathbb {Z}^d}$. It is also noted that the spectral measure of a measure-preserving flow has a continuity property at $\infty$, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak $(1, 1)$.References
- James Campbell and Karl Petersen, The spectral measure and Hilbert transform of a measure-preserving transformation, Trans. Amer. Math. Soc. 313 (1989), no. 1, 121–129. MR 958884, DOI 10.1090/S0002-9947-1989-0958884-4
- A.-P. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 349–353. MR 227354, DOI 10.1073/pnas.59.2.349
- Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. MR 199631, DOI 10.1007/BF02392815
- H. Davenport and H. Halberstam, The values of a trigonometrical polynomial at well spaced points, Mathematika 13 (1966), 91–96. MR 197427, DOI 10.1112/S0025579300004277
- Charles Fefferman, On the convergence of multiple Fourier series, Bull. Amer. Math. Soc. 77 (1971), 744–745. MR 435724, DOI 10.1090/S0002-9904-1971-12793-3 V. F. Gaposhkin, The Local Ergodic Theorem for groups of unitary operators and second order stationary processes, Math. USSR-Sb. 39 (1981), 227-242.
- Richard A. Hunt, On the convergence of Fourier series, Orthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 1967) Southern Illinois Univ. Press, Carbondale, Ill., 1968, pp. 235–255. MR 0238019
- Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
- Michael B. Marcus, $\xi$-radial processes and random Fourier series, Mem. Amer. Math. Soc. 68 (1987), no. 368, viii+181. MR 897272, DOI 10.1090/memo/0368
- Karl Petersen, Almost everywhere convergence of some nonhomogeneous averages, Almost everywhere convergence (Columbus, OH, 1988) Academic Press, Boston, MA, 1989, pp. 349–367. MR 1035255
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
- Norbert Wiener, The ergodic theorem, Duke Math. J. 5 (1939), no. 1, 1–18. MR 1546100, DOI 10.1215/S0012-7094-39-00501-6
Bibliographic Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 331 (1992), 131-142
- MSC: Primary 28D05; Secondary 42A20, 42A50
- DOI: https://doi.org/10.1090/S0002-9947-1992-1075378-9
- MathSciNet review: 1075378