Biological membrane assemblies with two membranes per unit cell and with well defined fluid layers are treated. In membrane diffraction the minus fluid model is used. A method for the determination of the zero-order amplitude using knowledge of the membrane-pair width is described. Sampling-theorem expressions for the Fourier transform of the individual membranes are derived. These expressions simplify the structural analysis and the interpretation of modi- fied and disordered membrane systems. The classical sampling-theorem expression for the Fourier transform has been used on a number of occasions in membrane diffraction and, in particular, it has been used in the X-ray analysis of one- dimensional swollen membrane systems. In the swel- ling method two or more sets of X-ray data with different repeat periods are obtained and these data sets lie on the same continuous intensity transform. The correct set of phases for the various data sets yields the same continuous Fourier transform via the sampling theorem. This method of interpolation re- quires the X-ray data sets to be on the same relative scale and it requires knowledge of the zero-order amplitude. The theory of Fourier analysis and the sampling expansion together with an exhaustive bibliography for all aspects of the sampling expansion is contained in a monograph by Jerri (1986). Let t(x) represent the electron density in a direction at right angles to the membrane surface and let T(X) represent its Fourier transform. Denote t(x) ~, T(X), where t(x) and T(X) are a Fourier transform pair and where x, X are real- and reciprocal-space coordinates. In the diffraction field, the derivation of the sampling-theorem ex- pressions follows the same well defined path. In the reciprocal-space domain, the electron density function t(x) is first expressed as a Fourier series. A second Fourier transformation leads to the sampling-theorem expression for T(X). We consider the case of mem- brane structure research. Biological membranes have the property that they readily form planar or con- centric multilayered structures. It is also recognized that the membrane multilayered systems generally contain well defined fluid layers. It is therefore conve- nient to study the minus fluid model (Worthington, 1969). Several sampling-theorem expressions for the minus fluid model have been presented (Worthington, King & Mclntosh, 1973). These expressions contain the membrane width as an additional parameter. In this paper it will be shown that the zero-order amplitude is directly obtained from the sampling- theorem expressions of the minus fluid model. We treat the special case of membrane-pair as- semblies which contain two fluid layers per unit cell. After physical or chemical treatment different physical states are recorded. It often happens that the indivi- dual membranes of the modified system are the same as the normal membranes: the only change in the altered structure is in the fluid widths. The basic problem in the structural analysis of the modified structures is to derive the Fourier transform of the individual membranes. Let mix) denote the electron density of each membrane with the origin at the center of re(x). The Fourier transform of m(x) is M( X ), where m(x).c~M(X). The straightforward calculation of M(X) from the computed electron density re(x) is feasible but lengthy and approximate. It will be shown that the Fourier transform MiX) can be directly evaluated using sampling-theorem expressions.