2025/04/03 by Felipe Gonçalves, Gonçalves, Felipe, Guilherme Vedana +1 · 1 citation
Computer Science · Mathematics · #30D10 #52C23 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms #Metric Geometry (math.MG) #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2504.02741
openalex publication_date 2025/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We completely classify Fourier summation formulas of the form ∫ℝ \widehatφ(t) dμ(t)=∑n=0∞ a(λn)φ(λn), that hold for any test function φ, where \widehatφ is the Fourier transform of φ, μ is a fixed complex measure on ℝ and a:\λn\n≥ 0→ℂ is a fixed function. We only assume the decay condition ∫ℝ \fracd |μ|(t)(1+t2)c1 + ∑n≥ 0 |a(λn)|e-c2 |λn|lt;∞, for some c1,c2>0. This completes the work initiated by the first author previously, where the condition c1≤ 1 was required. We prove that any such pair (μ,a) can be uniquely associated with a holomorphic map F(z) in the upper-half space that is both almost periodic and belongs to a certain higher index Nevanlinna class. The converse is also true: For any such function F it is possible to generate a Fourier summation pair (μ,a). We provide important examples of such summation formulas not contemplated by the previous results, such as Selberg's trace formula.