In this contribution, we introduce the concept of Fourier ambiguity resolution. We show how it is rooted in the principle of integer equivariant (IE) estimation and in its periodic representation. As a result, we present a general Fourier representation of IE-estimators. As the IE-class is the largest class of estimators used in GNSS ambiguity resolution, the periodic representation opens up a broad spectrum of new applications, both in the field of parameter estimation and in that of statistical testing. The representation also applies to the integer class, with its popular estimators of integer-rounding, integer-bootstrapping, and integer least-squares, as well as to their integer-aperture variants. In this contribution, we consider the periodic representation of the best integer equivariant (BIE) estimator. It is shown how this minimum mean squared error IE-estimator can be represented in both the spatial and frequency domains and how preference for one of the two representations should be based on the GNSS carrier-phase ambiguity precision. We also present a hybrid form of the BIE-estimator and show how the spatial and frequency representations can be mixed so as to do justice to the practical situation when carrier-phase ambiguity vectors consist of ambiguities having a wide range of varying precision.