In this paper a regularized spectral estimation formula and a regularized iterative algorithm
for band-limited extrapolation are presented. The ill-posedness of the problem is taken into account. First
a Fredholm equation is regularized. Then it is transformed to a differential equation in the
case where the time interval is R. A fast algorithm to solve the differential equation
by the finite differences is given and a regularized spectral estimation formula is obtained.
Then a regularized iterative extrapolation algorithm is introduced and compared with the
Papoulis and Gerchberg algorithm. A time-frequency regularized extrapolation algorithm
is presented in the two-dimensional case. The Gibbs phenomenon is analyzed.
Then the time-frequency regularized extrapolation algorithm is applied to image restoration
and compared with other algorithms.
