This paper studies signal concentration in the time and frequency domains using the general constrained variational method of Franks. The minimum k th (k = 0, 2, 4,...) moment time-duration measure for band-limited signals is formulated. A complete, orthonormal set of band-limited functions in L²([-W,W]) with the minimum fourth-moment time-duration measure is obtained. Numerical investigations of our optimal 4th moment functions show: 1) less energy concentration in the main lobe than the prolate spheroidal wave functions (PSWF); 2) higher energy concentration in the main lobe than Gabor's function; and 3) a larger main lobe than both PSWF and Gabor. Applications for our basis functions include: 1) radar systems and high resolution communication systems, and 2) representation and approximation to any band-limited signal in a given time interval.