The article is devoted to frequency sampling filters (FSF's). FSF's are very efficient linear-phase digital filters that can be significantly more computationally efficient than Parks-McClellan FIR filters. With the help of FSF's, it is convenient to implement filter banks that can be used to create efficient high-performance spectrum analyzers. These filters have been known for a long time, but, despite their effectiveness, they are not widely used. The article presents the authors' point of view on the reasons for the weak use of FSF's. It is shown how their transfer function (TF) was obtained using an analog prototype. As an analog prototype, the TF of the filter built with the use of LC-resonators connected in parallel was used. An analysis of the provisions of the theory of synthesis of analog filters was carried out. The TF of the analog filter, which implements the requirements for absolute linearity of the phase response, is considered. The bilinear z-transform method was applied to this TF. An analysis of the frequency characteristics of the digital filter was carried out based on the obtained TF with real coefficients. Applying L'Hôpital's rule, an analytical expression was found for determining the weighting coefficients of the resulting TF. It was concluded that it is necessary to modify this TF in order to achieve frequency independence of the weighting coefficients. The following modification was carried out. The modified TF of the FSF with real coefficients simplifies the processes of approximation and implementation. Studying methods of approximation of an analog prototype facilitates the approach to solving the problem of FSF's approximation. The article provides examples of calculated FSF's and a filter bank based on them. Graphs of frequence magnitude responses and tables of calculated optimal weighting coefficients are shown. The FSF block diagram is shown.