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We demonstrate theoretically a new method to accurately interpolate the complex reflection spectrum of fiber Bragg gratings with a finite length at any desired frequency resolution. The required sampling resolution is significantly smaller than can be expected by directly using the sampling theorem for obtaining a low-error characterization of the reflection spectrum. A further decrease in the required sampling resolution by a factor of two is obtained by sampling both the complex reflection and the complex transmission functions. The new reconstruction technique may enable to significantly reduce the time needed to characterize fiber Bragg gratings and to interrogate fiber Bragg sensors.