Find (i) the sample values, and (ii) the trigonometric...

Find (i) the sample values, and (ii) the trigonometric interpolant corresponding to the following discrete Fourier coefficients:
(a) $c_{-1}=c_1=1, c_0=0$,
(b) $c_{-2}=c_0=c_2=1, c_{-1}=c_1=-1$,
(d) $c_0=c_2=c_4=1, c_1=c_3=c_5=-1$.
(c) $c_{-2}=c_0=c_1=2, c_{-1}=c_2=0$,

Key Concepts

Evaluation at Sample Points

In the context of the inverse Fourier transform, sample values are obtained by evaluating the sum of the complex exponentials associated with the Fourier coefficients at specific, evenly spaced points. This process is fundamental in numerical analysis and signal processing, as it bridges the gap between the abstract frequency representation and the concrete time or spatial domain representation of a periodic function.

Discrete Fourier Transform

This concept relates to representing a periodic function in terms of a finite sum of complex exponentials. The discrete Fourier coefficients capture the amplitude and phase information for each harmonic of the function, and the DFT provides a framework for analyzing and processing periodic signals in both the time and frequency domains.

Inverse Discrete Fourier Transform

The inverse process reconstructs the original sampled function data from its Fourier coefficients. By summing the corresponding exponentials with the given coefficients, one obtains the sample values and ultimately the trigonometric interpolant, which is essential for understanding how the frequency components combine to form the original signal.

Trigonometric Interpolation

This is the process of constructing a trigonometric polynomial that exactly matches the given sample values of a periodic function at equidistant points. The interpolation is performed using the Fourier coefficients, and the resulting trigonometric interpolant provides an explicit form of the function in the spatial domain, revealing its structure and periodic behavior.

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