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

Let

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.

Lemma

Let and with Then for any we have

**Proof**

Let then

For and let

For let

Using Parseval’s identity for the Walsh function system we obtain

Let . The last summand can be estimated by

Therefore

We have

and therefore

Using Cauchy-Schwarz’ inequality we have

which implies

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

Remark

The above also implies thatwhere 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.

Theorem

Let be integrable and such that for some Thenand the Walsh series

converges pointwise.

**Proof**

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

Therefore

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.

Theorem

Let be continuous. Assume that Then the Walsh series of converges to at every point that is, we have

**Proof**

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.

Corollary

Let be continuous and for some Then