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Some useful properties of the Chebyshev polynomials are derived. By virtue of their discrete orthogonality, a truncated Chebyshev polynomials series is used to approximate a function whose discrete samples are the only available data. If minimization of the sum of the discrete squared error is used as the criterion, subject to some constraints on initial conditions and/or terminal conditions, the coefficients of the polynomials are easy to obtain. The simplicity of computing the coefficients of the polynomials from the discrete values of the function to be approximated is utilized to the approximate determination of optimal trajectories via dynamic programming using the technique of polynomial approximation. This allows use of the functional equation approach to solve multi-dimensional variational problems.