A Note on Trigonometric Interpolation and the Discrete Fourier Transform

The concept of Fourier Series is widely used in several Engineering problems like Wave Equations, Heat Equations, Laplace Equations, Signal Processing and much more. The concept of Discrete Fourier Transforms is the equivalent of continuous Fourier Transforms (DFT) for signals transmitted at finite number of points. The interpolation process allows us to manipulate the values of a discrete data in between the given input values. The process of interpolation is usually done with finite differences methods using forward, backward, central operators. We can also make interpolation process by using trigonometric functions. The Fourier series is a classic example for trigonometric interpolation where we use sine and cosine functions with different harmonics and try to express the given function as a linear combination of such harmonics. In this paper by considering equidistant data points, we show how the DFT allows us to construct the corresponding trigonometric interpolation.