# Solutions to biorthogonality relations over more complicated fields than $\mathbb R$ and $\mathbb C$.

The biorthogonality relation for discrete wavelets can be formulated as follows:

$$\sum_{n\in \mathbb Z} a_n \tilde a_{n+2m} = 2\cdot \delta_{m,0}$$

for two sequences of numbers $$\{a_{-N},\cdots,a_N\},\{\tilde a_{-N},\cdots,\tilde a_N\}$$

Usually we consider solutions for which $$a_k,\tilde a_k \in \mathbb R$$.

Now to the question.. given that the equation system above in all essence is a linear equation system relating $$a$$ and $$\tilde a$$ to each other, can we expand to a higher or more advanced field of numbers? I am for example aware that there exist complex-valued discrete wavelets, (although I do not know how these are usually constructed).

If so, how would we do this in practice?