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New research defines function smoothness using group theory and Fourier analysis

By PulseAugur Editorial · [2 sources] ·

A new research paper explores the concept of "smoothness" for functions within the context of group theory, moving beyond its traditional definition in real-valued functions. The study proposes that smoothness in $L^2(G)$ for a group $G$ relates to the concentration of Fourier coefficients at lower frequencies. This requires an ordering of irreducible representations, which is non-canonical for non-abelian groups. The paper introduces a method to order these representations using the Laplacian of a Cayley graph, defining an ordering function that depends on the group and its generating set. AI

IMPACT This research could inform future AI developments by providing new mathematical frameworks for understanding complex data representations.

RANK_REASON The cluster contains an academic paper published on arXiv discussing mathematical concepts related to group theory and Fourier analysis. [lever_c_demoted from research: ic=2 ai=0.4]

Read on arXiv cs.LG →

AI-generated summary · Google Gemini · from 2 sources. How we write summaries →

New research defines function smoothness using group theory and Fourier analysis

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The cluster contains an academic paper published on arXiv discussing mathematical concepts related to group theory and Fourier analysis. [lever_c_demoted from research: ic=2 ai=0.4]
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COVERAGE [2]

  1. arXiv cs.LG TIER_1 English(EN) · Zachary P Bradshaw ·

    What is Smoothness?

    arXiv:2609.03246v1 Announce Type: cross Abstract: Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low fr…

  2. Hugging Face Daily Papers TIER_1 English(EN) ·

    What is Smoothness?

    Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of…

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