A class of complete orthogonal sequences of step functions
Authors:
J. L. Sox and W. J. Harrington
Journal:
Trans. Amer. Math. Soc. 157 (1971), 129-135
MSC:
Primary 42.15
DOI:
https://doi.org/10.1090/S0002-9947-1971-0275046-3
MathSciNet review:
0275046
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Abstract | References | Similar Articles | Additional Information
Abstract: A class of orthogonal sets of step functions is defined and each member is shown to be complete in 
. Pointwise convergence theorems are obtained for the Fourier expansions relative to these sets. The classical Haar orthogonal set is shown to be a set of this class and the class itself is seen to be a subclass of the ``generalized Haar systems'' defined recently by Price.
- [1] Philip Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), no. 1, 522–529. MR 1512499, https://doi.org/10.1007/BF01448860
- [2] Alfred Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 (1910), no. 3, 331–371 (German). MR 1511592, https://doi.org/10.1007/BF01456326
- [3] J. J. Price, An algebraic characterization of certain orthonormal systems, Proc. Amer. Math. Soc. 19 (1968), 268–273. MR 0225085, https://doi.org/10.1090/S0002-9939-1968-0225085-9
- [4] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1971-0275046-3
Keywords:
Haar functions,
Fourier analysis,
orthonormal set of functions,
complete in 
,
pointwise convergence,
generalized Haar system
Article copyright:
© Copyright 1971
American Mathematical Society
