Abstract
It is shown that the double sequence { λ m n } \{ {\lambda _{mn}}\} with λ m n = 1 {\lambda _{mn}} = 1 if n ⩽ m n \leqslant m and 0 0 otherwise is an L p {L^p} multiplier for the Walsh system in two dimensions only if p = 2 p = 2 . This result is then used to show that the one-dimensional trigonometric system and the Walsh system are nonequivalent bases of the Banach space L p [ 0 , 1 ] {L^p}[0,\;1] , and hence have different L p {L^p} multipliers, 1 > p > ∞ , p ≠ 2 1 > p > \infty ,\;p \ne 2 .
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