The pseudoinverse of the Laplacian matrix: Asymptotic behavior of its trace
Analysis Mathematica, cilt.49, sa.2, ss.409-442, 2023 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 49 Sayı: 2
- Basım Tarihi: 2023
- Doi Numarası: 10.1007/s10476-023-0216-4
- Dergi Adı: Analysis Mathematica
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, zbMATH, Civil Engineering Abstracts
- Sayfa Sayıları: ss.409-442
- Anahtar Kelimeler: asymptotic approximation, elementary classical function, lattice sum
- Boğaziçi Üniversitesi Adresli: Evet
Özet
In this paper we are concerned with the asymptotic behavior of tr(ℒsq+)=14∑j,k=0(j,k)≠(0,0)n−111−12(cos2πjn+cos2πkn), the trace of the pseudoinverse of the Laplacian matrix related with the square lattice, as n → ∞. The method we developed for such sums in former papers depends on the use of Taylor approximations for the summands. It was shown that the error term depends on whether the Taylor polynomial used is of degree two or higher. Here we carry this out for the square lattice with a fourth degree Taylor polynomial and thereby obtain a result with an improved error term which is perhaps the most precise one can hope for.