I want to know the relationship between the wavelet coefficients of a time series before and after linear interpolated.
Suppose we have a time series
$x(0),x(1),x(2),\cdots$.
When this time series is linear interpolated, the result becomes
$x(0),\alpha x(0)+(1-\alpha)x(1),x(1),\alpha x(1)+(1-\alpha)x(2),\cdots$.
where $\alpha$ is a real constant.
If the detailed and approximate coefficients of $x(n)$ are $d_{j,m}$ and $a_{j,k}$, respectively, can the wavelet coefficients for the interpolated time series be written as expressions of $d_{j,k}$,$a_{j,k}$?
Any suggestion or reference would be greatly appreciated.