In the Lebesgue spaces L p ([0, 1]d), 1 ≤ p ≤ ∞, for d ≥ 2, we define a multiple basis system of functions Hd = (h n ) ∞ n = 1 . This system has the main properties of the well-known one-dimensional Haar basis H. In particular, it is shown that the system Hd is a Schauder basis in the spaces L p ([0, 1]d), 1 ≤ p < ∞.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 9, pp. 1253–1264, September, 2015.
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Romanyuk, V.S. Multiple Haar Basis and its Properties. Ukr Math J 67, 1411–1424 (2016). https://doi.org/10.1007/s11253-016-1162-0
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DOI: https://doi.org/10.1007/s11253-016-1162-0