Fourier analysis

The purpose of spectral density estimation is to measure the amplitude of a signal or a time series according to its frequency [3:4]. The objective is to estimate the spectral density by detecting periodicities in the dataset. A scientist can better understand a signal or time series by analyzing its harmonics.

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The spectral theory

Spectral analysis for a time series should not be confused with the spectral theory, a subset of linear algebra that studies eigenfunctions on Hilbert and Banach spaces. In fact, harmonic analysis and Fourier analysis are regarded as subsets of the spectral theory.

Let's explore the concept behind the discrete Fourier series as well as its benefits as applied to financial markets. The Fourier analysis approximates any generic function as the sum of trigonometric functions, sine and cosine.

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Complex Fourier transform

This section focuses on the discrete Fourier series for real values. The generic Fourier transform applies to complex values...