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Projections onto translation-invariant subspaces of $ H\sp 1({\bf R})$


Authors: Dale E. Alspach and David C. Ullrich
Journal: Trans. Amer. Math. Soc. 313 (1989), 571-588
MSC: Primary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1989-0974510-2
MathSciNet review: 974510
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Abstract: Recently I. Klemes has characterized the complemented translation-invariant subspaces of $ {H^1}(\mathbb{T})$. In this paper we investigate the case of $ {H^1}(\mathbb{R})$. The main results are that the hull of a complemented translation-invariant subspace is $ \varepsilon $-separated for some $ \varepsilon > 0$, and that an $ \varepsilon $-separated subset of $ {\mathbb{R}^ + }$ which is in the ring generated by cosets of closed subgroups of $ \mathbb{R}$ (intersected with $ {\mathbb{R}^ + }$) and lacunary sequences is the hull of a complemented ideal.


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  • [A] D. E. Alspach, A characterization of the complemented translation-invariant subspaces of $ {H^1}(\mathbb{R})$, Trans. Amer. Math. Soc. (to appear). MR 967091 (89k:46040)
  • [AM] D. E. Alspach and A. Matheson, Projections onto translation-invariant subspaces of $ {L_1}(\mathbb{R})$, Trans. Amer. Math. Soc. 277 (1983), 815-823. MR 694390 (85e:46017)
  • [AMR] D. E. Alspach, A. Matheson and J. M. Rosenblatt, Separating sets with the Fourier-Stieltjes transform, J. Funct. Anal, (to appear).
  • [DS] N. Dunford and J. T. Schwartz, Linear operators: General theory, Pure and Appl. Math. 7, Interscience, New York, 1958.
  • [He] H. Helson, Harmonic analysis, Addison-Wesley, Reading, Mass., 1983. MR 729682 (85e:43001)
  • [K] I. Klemes, The idempotent multipliers of $ {H^1}(T)$, Canad. J. Math. 39 (1987), 1223-1234. MR 918595 (88k:42007)
  • [Ko] P. Koosis, Introduction to $ {H_p}$ spaces, London Math. Soc. Lecture Notes 40, Cambridge Univ. Press, Cambridge, 1980. MR 565451 (81c:30062)
  • [MPS] O. C. McGehee, L. Pigno, and B. Smith, Hardy's Inequality and the $ {L^1}$ norm of exponential sums, Ann. of Math. 113 (1981), 613-618. MR 621019 (83c:43002b)
  • [R1] H. P. Rosenthal, Projection onto translation-invariant subspaces of $ {L_p}(G)$, Mem. Amer. Math. Soc. No. 63 (1966). MR 0211198 (35:2080)
  • [R2] -, On the existence of approximate identities in ideals of group algebras, Ark. Mat 7 (1967), 185-191. MR 0239362 (39:719)
  • [Ru1] W. Rudin, Fourier analysis on groups, Wiley-Interscience, New York, 1962. MR 0152834 (27:2808)
  • [Ru2] -, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)

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DOI: https://doi.org/10.1090/S0002-9947-1989-0974510-2
Article copyright: © Copyright 1989 American Mathematical Society

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