On the Fourier coefficients of modular forms of half-integral weight 2025

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The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, shell theory, etc.
The Fourier Coefficients provide essential information about the original signal, including amplitude, phase, and frequency.
To find the coefficients a0, an and bn we use these formulas: a0 = 12L. L. L. f(x) dx. an = 1L. L. L. f(x) cos(nxL) dx. bn = 1L. L. L. f(x) sin(nxL) dx.
Fourier series are a powerful tool that can help us break down complex signals into their constituent parts. By using some basic mathematics, we can deconstruct signals into simple sine waves, making them much easier to understand and analyze.
With that in mind, how is half-wave symmetry going to affect the Fourier coefficients of the signal? A signal with half-wave symmetry consists of identical half-cycles with opposite polarities. Due to this alternating feature, we can conclude that the average value of the signal is zero.
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Fourier analysis allows one to identify, quantify, and remove the time-based cycles in data if necessary. The amplitudes, phases, and frequencies of data are evaluated by use of the Fourier transform.

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