Spectral Analysis of Laplacians on Certain Fractals

Harmonic analysis;Laplaicans;Pure Mathematics

Zhou, Denglin

University of Waterloo

关键词: Harmonic analysis;    Laplaicans;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/3435/1/uwthesis8.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

Surprisingly, Fourier series on certain fractals can have betterconvergence properties than classical Fourier series. This is aresult of the existence of gaps in the spectrum of the Laplacian. Inthis work we prove a general criterion for the existence of gaps.Most of the known examples on which the Laplacians admit spectraldecimation satisfy the criterion. Then we analyze the infinitefamily of Vicsek sets, finding an explicit formula for the spectraldecimation functions in terms of Chebyshev polynomials. TheLaplacians on this infinite family of fractals are also shown tosatisfy our criterion and thus have gaps in their spectrum.