A note on Walsh-Fourier series

Young, Wo Sang

doi:10.1090/S0002-9939-1976-0410247-2

A note on Walsh-Fourier series
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Proc. Amer. Math. Soc. 59 (1976), 305-310 Request permission

Abstract:

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

References

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A remarkable series of orthogonal functions

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