Harmonic analysis

Summary

Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic Analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word harmonikos, meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier

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Harmonic analysis and mean field theory

Denis Karateev

We review some aspects of harmonic analysis for the Euclidean conformal group, including conformally-invariant pairings, the Plancherel measure, and the shadow transform. We introduce two efficient methods for computing these quantities: one based on weight-shifting operators, and another based on Fourier space. As an application, we give a general formula for OPE coefficients in Mean Field Theory (MFT) for arbitrary spinning operators. We apply this formula to several examples, including MFT for fermions and "seed" operators in 4d, and MFT for currents and stress-tensors in 3d.

Springer2019

3D Spectral Nonrigid Registration of Facial Expression Scans

Gabriel Louis Cuendet, Christophe René Joseph Ecabert, Hazim Kemal Ekenel, Jean-Philippe Thiran, Marina Zimmermann

In this paper, we introduce a new template-based spectral nonrigid registration method in which the target is represented using multilevel partition of unity (MPU) implicit surfaces and the template is embedded in a discrete Laplace-Beltrami based spectral representation using the manifold harmonics transform (MHT). The implicit surface parametrization of the target allows us to avoid computing correspondences during the registration as in classical nonrigid iterative closest point (ICP) techniques. It also allows us to denoise the 3D scans and fill the holes by interpolating the noisy 3D data and to incorporate different types of 3D surfaces into our model, independently of their original parametrization. We take advantage of spectral geometry processing methods to compute a spectral embedding of the template and use it as a parametric surface deformation model. We optimize the nonrigid deformation directly in the spectral domain, thus effectively reducing the size of the parameter space as compared with the classical per vertex affine transformation deformation model. In addition, we introduce a new 3D facial expressions database, EPFL3DFace, on which we apply the proposed method to nonrigidly register 3D face scans that contain different expressions. This database consists of 3D scans of 120 subjects posing 35 different facial expressions. These include various standard prototypical facial expressions as well as individual action units, visemes, and the facial movement of biting one’s own upper lip, which are suitable for a large variety of applications.

Institute of Electrical and Electronics Engineers

We consider a divergence-form elliptic difference operator on the lattice Zd, with a coefficient matrix that is an i.i.d. perturbation of the identity matrix. Recently, Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series related to the averaged Green’s function of this model. Our main contribution is a refinement of Bourgain’s approach which improves the key decay rate from −2d+ϵ to −3d+ϵ. (The optimal decay rate is conjectured to be −3d.) As an application, we derive estimates on higher derivatives of the averaged Green’s function which go beyond the second derivatives considered by Delmotte–Deuschel and related works.

2019