On the order of magnitude of Walsh-Fourier transform

Abstract

For a Lebesgue integrable complex-valued function f defined on R := [0,∞) let f̂ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that f̂(y)→ 0 as y → ∞. 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(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 R. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on (R) , N ∈ N.