On the order of magnitude of Walsh-Fourier transform

summary:For a Lebesgue integrable complex-valued function $f$ defined on $\mathbb R^+:=[0,\infty )$ let $\hat f$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat f(y)\to 0$ as $y\to \infty $. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1(\mathbb R^+)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on $\mathbb R^+$. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on $(\mathbb R^+)^N$, $N\in \mathbb N$