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We present a complete formulation and an exact solution to the problem of designing systems for simultaneous sampling rate increase and fractional-sample delay in the Lagrangian sense. The problem may be regarded as that of a linear transformation, i.e., scaling, and/or shifting, of the uniform sampling grid of a discrete-time signal having a Newton series representation. It is proved that the solution forms a three-parameter family of maximally flat finite impulse response digital filters with a variable group-delay at the zero frequency. Various properties of the solution, including Nyquist properties and conditions for a linear phase response are analyzed. The solution, obtained in the closed form, is exact for polynomial inputs. We show that it is also suited for processing discrete-time versions of certain continuous-time bandlimited signals and signals having a rational Laplace transform. We then derive a generalization of the solution by augmenting the family with a fourth parameter that controls the number of multiple zeros at the roots of unity. This four-parameter family contains various types of maximally flat filters including those due to Herrmann and Baher. We list specific conditions on the four parameters to obtain many of the maximally flat filters reported in the literature. A significant part of the family of systems characterized by the solutions has been hitherto unknown. Examples are provided to elucidate this part as well.