ON THE ORDER OF MAGNITUDE OF WALSH-FOURIER TRANSFORM

For a Lebesgue integrable complex-valued function f defined on R+ := [0, infinity) let (f) over cap be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that (f) over cap (y) -> 0 as y -> infinity. 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(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 is an element of N.