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Interpolation filters are used to calculate new samples at arbitrary time instants in between existing discrete-time samples. Polynomial-based interpolation filters can be efficiently implemented using the Farrow structure. Lagrange coefficients are often used to describe such classical polynomial interpolators. Previous references have concluded that there must be an even number of samples in the basepoint set to perform interpolation in order to satisfy linear phase requirement. This paper introduces a new method to construct a linear phase Lagrange interpolator using an odd number of basepoints. Although the conceptual analog reconstruction filter does not have a time-continuous impulse response, it can be proved that the interpolation results are time-continuous within the approximation error of polynomial-based interpolation.