Related papers: Analyse et synth\`ese harmoniques
The product of any number of Legendre functions, under a restricted domain, can be expanded by the corresponding Legendre polynomials, with the coefficient being the sinc function. While an analogous expansion can be made for any number of…
We develop analytic tools for the asymptotics of general trie statistics, which are particularly advantageous for clarifying the asymptotic variance. Many concrete examples are discussed for which new Fourier expansions are given. The tools…
These lecture notes accompanied the course Time-Frequency Analysis given at the Faculty of Mathematics of the University of Vienna in the summer term 2021. The material is suitable for an advanced undergraduate course in mathematics or a…
Fourier analysis plays a major role in the analysis and understanding of many phenomena in physics and contemporary engineering. However, students, who have often discovered this notion through numerical tools, do not necessarily understand…
The Fueter-Sce mapping theorem stands as one of the most profound outcomes in complex and hypercomplex analysis, producing hypercomplex generalizations of holomorphic functions. In recent years, delving into the factorization of the second…
Dunkl theory is a far reaching generalization of Fourier analysis and special function theory related to root systems. During the sixties and seventies, it became gradually clear that radial Fourier analysis on rank one symmetric spaces was…
Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2,…
Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional…
We study the relationship between stable sampling sequences for bandlimited functions in $L^p(\R^n)$ and the Fourier multipliers in $L^p$. In the case that the sequence is a lattice and the spectrum is a fundamental domain for the lattice…
{\em Quantum Fourier analysis} is a new subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum…
The construction of a multiresolution analysis starts with specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of…
These informal notes deal with Fourier series in one or more variables, Fourier transforms in one variable, and related matters.
Positive time varying frequency representation for transient signals has been a hearty desire of signal analysts due to its theoretical and practical importance. During approximately the last two decades there has formulated a signal…
This is a survey of the use of Fourier analysis in additive combinatorics, with a particular focus on situations where it cannot be straightforwardly applied, but needs to be generalized first. Sometimes very satisfactory generalizations…
Taking the Fourier integral theorem as our starting point, in this paper we focus on natural Monte Carlo and fully nonparametric estimators of multivariate distributions and conditional distribution functions. We do this without the need…
We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(\mu)$ with measures $\mu$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics.…
This paper offers a comprehensive introduction to Bayesian inference, combining historical context, theoretical foundations, and core analytical examples. Beginning with Bayes' theorem and the philosophical distinctions between Bayesian and…
Reconstructing continuous signals from a small number of discrete samples is a fundamental problem across science and engineering. In practice, we are often interested in signals with 'simple' Fourier structure, such as bandlimited,…
A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be…
This paper gives an introduction to the theory of orthogonal projection of functions or signals. Several kinds of decomposition are explored: Fourier, Fourier-Legendre, Fourier-Bessel series for 1D signals, and Spherical Harmonic series for…