On the order of magnitude of Walsh-Fourier transform

Abstract

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 $\hat f(y)\to0$ 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 $(\mathbb R^+)^N$, $N\in\mathbb N$.

Keywords

• function of bounded variation over $\mathbb r^+$
• function of bounded variation over $(\mathbb r^+)^2$
• function of bounded variation over $(\mathbb r^+)^n$
• order of magnitude
• riemann-lebesgue lemma
• walsh-fourier transform