Discrete wavelet transform

Given a scaling function ?0, mother wavelet ?0, and maximum scale level J, DWT is a mapping from x(t) to the approximation coefficients ax and detail coefficients dx:

x(t) ?{ax(J, k), dx(j, k), j ? [1, J], k ? Z}

ax(j, k) = ?x, ?j,k ?, dx(j, k) = ?x, ?j,k ?

?j,k = 2-j/2 ?0 (2-jt - k), ?j,k = 2-j/2 ?0 (2-jt - k), k ? Z