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psi_hat_exact<-function(filter,J=10){
## Computes the discrete Fourier transform of the wavelet associated to
## the given filter
##
## The wavelet is given by the function makescalingfunction of Fay et al
## (2009).
##
## The length of the Fourier transform is equal to the length of the grid
## where the wavelet is evaluated
##
## INPUT filter Quadratic mirror filter of the wavelet
## J Number of scales where the wavelet is evaluated
##
## OUTPUT fct Values of the discrete Fourier transform of the wavelet
## grid Frequencies where the Fourier transform is evaluated
##
## Achard & Gannaz (2014)
##________________________________________________________________________________
Jmax <- J-3
res <- scaling_function(filter,J)
phi <- res$phi
psi <- res$psi
L <- length(psi)
psi <- psi/sqrt(2^J)
Nfft <- 2^ceiling(log(L)/log(2))
Te <- (length(filter)-1)*2^Jmax ## support-size
psi <- c(psi,rep(0,(Nfft-L)))
psih <- fft(psi)/sqrt(pi*Te)
grid <- pi*Te*c(seq((1/Nfft),1/2,(1/Nfft)),seq((-1/2),(-1/Nfft),(1/Nfft)))
list(psih=psih,grid=grid)
}
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