Related papers: Winding numbers and Fourier series
We construct an example of two continuous maps f and g of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis.
The formulation of classical mechanics applicable to fermionic degrees of freedom is presented in mathematically rigorous terms, including a description of how the mathematical structure relates to the quantization of the theory. Canonical…
This paper consider the functional mechanics as one of modern approaches to a problem of the correspondence between classical mechanics and the statistical physics. Deviations from classical trajectories are calculated and evolution of the…
We reconsider quantum mechanical systems based on the classical action being the period of a one form over a cycle and elucidate three main points. First we show that the prepotenial V is no longer completely arbitrary but obeys a…
In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…
We introduce a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm…
In this paper, we propose a new algebraic winding number and prove that it computes the number of complex roots of a polynomial in a rectangle, including roots on edges or vertices with appropriate counting. The definition makes sense for…
In this expository article, we provide a self-contained overview of the notion of convolution embedded in different theories: from the classical Fourier theory to the theory of algebraic signal processing. We discuss their relations and…
We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal…
We show how an object from the combinatorially geometric version of the analytical number theory, namely geometric continued fractions, appears in the classical smooth dynamics, namely in the problem on the topological classification of…
This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have…
We describe fermions in terms of a classical statistical ensemble. The states $\tau$ of this ensemble are characterized by a sequence of values one or zero or a corresponding set of two-level observables. Every classical probability…
The import of the magnitude of fourier coefficients of a pitch class set is fairly well known. This paper deals with the angular component of these compelx numbers, the phase. It enables to shed new light on triads, the Tonnetz, and…
The nonlinear Fourier transform discussed in these notes is the map from the potential of a one dimensional discrete Dirac operator to the transmission and reflection coefficients thereof. Emphasis is on this being a nonlinear variant of…
Fourier Series is the second of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such as partial…
In the main part of the paper, on the basis of contour integration of complex meromorphic functions whose singularities lie onto an integration contour, in the first step, a concept of improper integrals absolute existence of meromorphic…
Floating point arithmetic allows us to use a finite machine, the digital computer, to reach conclusions about models based on continuous mathematics. In this article we work in the other direction, that is, we present examples in which…
Issues relevant to the flow chirality and structure are focused, while the new theoretical results, including even a distinctive theory, are introduced. However, it is hope that the presentation, with a low starting point but a steep rise,…
Fourier series with absolutely summable coefficients provide a classical example of a commutative Banach algebra, and these notes are concerned with this and related matters.
These notes present a first graduate course in harmonic analysis. The first part emphasizes Fourier series, since so many aspects of harmonic analysis arise already in that classical context. The Hilbert transform is treated on the circle,…