In a previous post, we discussed the orthogonality properties of Walsh functions and showed that they form a complete orthonormal system in . In this post we discuss the rate of decay of the Walsh coefficients when the function has bounded variation of fractional order and we investigate pointwise convergence of the Walsh series and pointwise convergence of the Walsh series to the function. We consider only Walsh functions in base although the results can be generalized to Walsh functions over groups. Information on Walsh functions over groups can be found in this post. For the necessary background information see the previous post on Walsh functions here. A table of contents for the posts on Walsh functions can be found here.
Let . For we define the fractional variation
where the supremum is taken over all and all partitions
If and the function is continuously differentiable, then the mean value theorem implies that for all there exists a such that Hence the Riemann integrability of implies that
In the following we prove a bound on for functions with bounded variation, that is, for such that Since for all we can also obtain a bound on the Walsh coefficients.
Let and with Then for any we have
For and let
Using Parseval’s identity for the Walsh function system we obtain
Let . The last summand can be estimated by
Using Cauchy-Schwarz’ inequality we have
where the last inequality follows as the Cauchy-Schwarz inequality is an equality for two vectors which are linearly dependent. Let be the value of for which the last term takes on its maximum, then
The above also implies that
where the last expression is just the modulus of continuity.
The Lemma above can also be generalized to arbitrary bases and dimensions This result can be found in the book J.D. and F. Pillichshammer, Digital Nets and Sequences. Cambridge University Press, Cambridge, 2010, in Chapter 13.
Pointwise convergence of Walsh series
For functions for which for some we obtain pointwise convergence of the Walsh series. For some background information on pointwise convergence see here.
Let be integrable and such that for some Then
and the Walsh series
Since we have
Hence implies the pointwise convergence of the Walsh series. It remains to show that
Since is integrable we have Further, using the Cauchy Schwarz inequality, we obtain
since and therefore
Pointwise convergence of the Walsh series to the function
Now we consider in which situations the Walsh series of a function converges to the function at every point. We have the following result.
Let be continuous. Assume that Then the Walsh series of converges to at every point that is, we have
Let Then for any and we have
Hence, for continuous we have
Let be fixed. Then the partial sums form a Cauchy sequence since and therefore the sums form a Cauchy sequence. Thus
and the result follows.
The last two theorems imply the following result.
Let be continuous and for some Then