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This paper aims to construct a Fourier framework within proportional arithmetic on the positive real line.

August 7, 2026AxiomsOpen Access

A Proportional-Arithmetic Framework for Fourier Analysis on the Positive Real Line

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Development of a Fourier framework using proportional arithmetic, demonstrating deep connections to classical harmonic analysis.

This paper aims to construct a Fourier framework within proportional arithmetic on the positive real line.
Constructed complex scalar field and corresponding operators for Fourier analysis.
Developed a correspondence theorem linking proportional and classical Fourier transforms.
Illustrated operational calculus using model resolvent and heat equations.
Established a systematic relation between proportional transform and classical Fourier transform.
Demonstrated connections with Mellin transform on the imaginary axis.
Showed how the spectral modulus encodes information about log-scale width and preferred scale.

Cruz-Rodas et al. (2026) studied this question.

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