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New discrete-time filtering methods for the computation of interpolating function coefficients from input signal samples are presented. The techniques are based upon known properties of observers and form a link between bodies of knowledge commonly associated with signal processing and with control systems. Each iteration of the filter incorporates a new signal sample and discards the oldest sample in the previous iteration. Application is made to interpolating polynomials, including power series, Legendre polynomials and Chebyshev polynomials, to harmonic trigonometric series and to real exponential interpolating functions. In the harmonic trigonometric case, the filter performs recursive discrete Fourier transformation.