Representation of a crinkled arc
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- by Richard A. Vitale PDF
- Proc. Amer. Math. Soc. 52 (1975), 303-304 Request permission
Abstract:
Johnson [A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375-376] has shown that under suitable normalizations all crinkled arcs are unitarily equivalent. Using this result, we find a general series expansion for a crinkled arc: \[ f(t) = \sqrt {2} \sum _{n=1}^\infty x_n \tfrac {\sin (n - 1/2)\pi t}{(n - 1/2)\pi } , \] where $\{ {x_n}\}$ is an orthonormal set.References
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R. Ash, Information theory, Interscience Tracts in Pure and Appl. Math., no. 19, Interscience, New York, 1962. MR 37 #5049.
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- Gordon G. Johnson, A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375–376. MR 259574, DOI 10.1090/S0002-9939-1970-0259574-7
- G. G. Johnson, Hilbert space problem four, Amer. Math. Monthly 78 (1971), 525–527. MR 285896, DOI 10.2307/2317762
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 52 (1975), 303-304
- MSC: Primary 46C10
- DOI: https://doi.org/10.1090/S0002-9939-1975-0388056-1
- MathSciNet review: 0388056