Symplectic wavelet transformation.

Usually a wavelet transform is based on dilated-translated wavelets. We propose a symplectic-transformed-translated wavelet family psi()(r,s)(z-kappa) (r,s are the symplectic transform parameters, |s|(2)-|r|(2)=1, kappa is a translation parameter) generated from the mother wavelet psi and the corresponding wavelet transformation W(psi)f(r,s;kappa)=integral(infinity)(-infinity)(d(2)z/pi)f(z)psi()(r,s)(z-kappa). This new transform possesses well-behaved properties and is related to the optical Fresnel transform in quantum mechanical version.