We present an analytical method for the design of narrow-band X-ray multilayer coatings having greatly reduced reflected side-lobe intensity, for the realization of X-ray mirrors that have improved spectral purity. The method uses a specific variation of the individual layer thicknesses as a function of depth in the multilayer stack, derived from Laplace transform analysis of the multilayer's reflectance profile. The design process and mathematical foundations are outlined. Pt/C multilayers with 5 nm d-spacing for hard X-rays are designed, fabricated and measured to demonstrate the validity and effectiveness of the method are presented. As an extrapolation, three additional side lobe suppressed multilayers for soft X-rays and EUVs are also designed and investigated: 1) Cr/Sc multilayer for soft X-rays (4.96 nm wavelength) at high grazing angle (30°), 2) Mo/Si multilayer for EUV (13.5 nm wavelength) at normal incidence angle and 3) SiC/Mg multilayer for EUV (30.4 nm wavelength) at normal incidence angle. The calculated reflectances demonstrate that the presented method is robust for the energy range from X-rays to EUVs.