What is the "fourth" Haar basis function of the 4×4 Haar basis?

Question

I know the three Haar basis functions

//mother wavelet:
[1 1 -1 -1]
//dilation:
[1 -1 0 0] 
//translation:
[0 0 1 -1]

However, the Haar basis consists of four basis elements: The mother wavelet, the translation, the dilation and another, fourth element, which I don't understand. According to this, this fourth element is

[1 1 1 1]

But how would this one look like, and how is it obtained (dilation/translation, whatever combination)? This confuses me.

Btw: is the Haar basis defined as

[ 1  1  1  1
  1  1 -1 -1
  1 -1  0  0
  0  0  1 -1]

or as the transpose of this?

Answer 1

If I understand the notation right, it looks like the [1 1 1 1] element is just the DC coefficient, i.e. the mean of the input signal.

By symmetry, the Haar wavelet integrates to 0 over its domain. Thus, if you didn't include the DC coefficient, adding a constant bias to the input signal would not change its Haar transform at all.