Fourier Analysis: A New Result

TL;DR

The paper develops a jump-detection mechanism for Fourier-series representations of BV-type functions. By introducing a normalized conjugate-sum $Y_n(x)$ and leveraging Lukács-type asymptotics, it shows pointwise convergence to a multiple of the jump $J(x)$ and a corresponding variation limit that counts jump magnitudes on intervals. The higher-dimensional generalization extends these ideas to $d$-dimensional tori, where each coordinate direction yields a separate jump-detection functional $Y_{j,n}(x)$ whose limits reveal jumps on coordinate hyperplanes and the total variation captures the sum of jump magnitudes over rectangles. Together, the results provide a precise, variation-based method to locate and quantify jump discontinuities via conjugate-Fourier-Dirichlet kernels.

Abstract

This article contains a new result in Fourier analysis concerning jump type discontinuities.

Theorems & Definitions (8)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1: Pointwise convergence at jump hyperplanes
• proof
• Lemma 3.2: Variation detects multidimensional jumps
• proof