Approximation and reconstruction of continuous signals using differential Chebyshev transformation

A method for physical process approximation using the differential Taylor transformation is substantiated. The power basis is transformed to bases of orthogonal Chebyshev polynomials. It is shown that the convergence of series is substantially increased by transition to expansion in Chebyshev polynomials of first kind and shifted Chebyshev polynomials. An algorithm for calculating differential spectrum discretes is formulated. It is observed that in the Chebyshev bases, the value of spectrum discretes decreases constantly as their number grows. In this case, it is possible to stop computing the discretes as they achieve the required small value, but this cannot be done in the power basis. Numerical examples illustrate the advantage of the proposed approach.