Is there a good introduction to stationary phase analysis (in context of getting bounds on an exponential sum of $f(x,y,z) $ over a region where the Hessian is non-singular. #math #harmonicanalysis

Local Band–Variation (LBVT) + Carleson absorption with explicit constants for xi(s).

Prototype bound:
V_on(M; T(I)) <= C*(1 + log M)*N_T(I)

Fejer-type energies and orthogonalized lifts give bandwise variation and depth control;
the scheme aims at localized zero-density estimates. I invite independent checks of the
inequalities and the constant bookkeeping; any challenges or pointers appreciated.
DOI: https://doi.org/10.5281/zenodo.17257870
#math #NumberTheory #HarmonicAnalysis #Zeta #RiemannHypothesis #preprint

The Fourier transform:
#fouriertransform #harmonicanalysis #functions #mathematics #scienceandtechnology
🌀

https://www.quantamagazine.org/what-is-the-fourier-transform-20250903/

"After finding the homeschooling life confining, the teen petitioned her way into a graduate class at Berkeley, where she ended up disproving a 40-year-old conjecture."

🌍 https://www.quantamagazine.org/at-17-hannah-cairo-solved-a-major-math-mystery-20250801/
📝 https://arxiv.org/abs/2502.06137

#HannahCairo #MizohataTakeuchiConjecture #HarmonicAnalysis #FourierRestrictionTheory #Mathematics https://mathstodon.xyz/@jcponcemath/115051499516743809

Just learned about Hannah Cairo, 17-year-old prodigy who provided a counterexample to a conjecture in harmonic analysis. https://www.youtube.com/watch?v=3ZeH_8sTyKA

The paper: https://arxiv.org/pdf/2502.06137

#HarmonicAnalysis #MizohotaTakeuchi #HannahCairo

Very glad of this new collaborative work "Herglotz-NET: Implicit Neural Representation of Spherical Data with Harmonic Positional Encoding" with Théo Hanon, Nicolas Mil-Homens Cavaco, John Kiely, Laurent Jacques https://arxiv.org/html/2502.13777v1

In this work, we propose Herglotz-NET (HNET), a novel INR architecture that employs a harmonic positional encoding based on complex Herglotz mappings. This encoding yields a well-posed representation on the sphere with interpretable and robust spectral properties. Moreover, we present a unified expressivity analysis showing that any spherical-based INR satisfying a mild condition exhibits a predictable spectral expansion that scales with network depth. Our results establish HNET as a scalable and flexible framework for accurate modeling of spherical data.

#INR #SphericalHarmonics #HarmonicAnalysis #Sphere

in many ways, #mathematical #physics is based on the #smoothing properties of the #integral

#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein

“(...) The advent of #DeepLearning started and affected my research area significantly. I decided to embrace this paradigm shift and delve research-wise into #ArtificialIntelligence. Looking back, this was one of the best decisions in my life.” - Gitta Kutyniok

➡️ https://hermathsstory.eu/gitta-kutyniok/

#Academia #Professor #PhD #AppliedMathematics #HarmonicAnalysis #ComputerScience #DecisionMaking #WomenInMaths #WomenInSTEM #HerMathsStory