Riemann-Lebesgue Lemma Research Articles

From ABC to KPZ

AbstractWe study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $$N\in {\mathbb {N}}$$ N ∈ N points, denoted by $${\mathbb {T}}_N$$ T N , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $$N\rightarrow \infty $$ N → ∞ , to a system of stochastic partial differential equations, that can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann–Lebesgue lemma which is of independent interest.

Octonion quadratic-phase Fourier transform: inequalities, uncertainty principles, and examples

In this article, we define the octonion quadratic-phase Fourier transform (OQPFT) and derive its inversion formula, including its fundamental properties such as linearity, parity, modulation, and shifting. We also establish its relationship with the quaternion quadratic-phase Fourier transform (QQPFT). Further, we derive the Parseval formula and the Riemann–Lebesgue lemma using this transform. Furthermore, we formulate two important inequalities (sharp Pitt’s and sharp Hausdorff–Young’s inequalities) and three main uncertainty principles (logarithmic, Donoho–Stark’s, and Heisenberg’s uncertainty principles) for the OQPFT. To complete our investigation, we construct three elementary examples of signal theory with graphical interpretations to illustrate the use of OQPFT and discuss their particular cases.

Van der Corput lemmas for Mittag-Leffler functions. I

In this paper, we study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function to study oscillatory-type integrals appearing in the analysis of time-fractional partial differential equations. Several generalisations of the first and second van der Corput lemmas are proved. Optimal estimates on decay orders for particular cases of the Mittag-Leffler functions are also obtained. As an application of the above results, the generalised Riemann–Lebesgue lemma and the Cauchy problem for the time-fractional evolution equation are considered.

The double Fourier transform of non-Lebesgue integrable functions of bounded Hardy–Krause variation

AbstractIn the classical Fourier analysis, the representation of the double Fourier transform as the integral off(x,y)exp(−i⟨(x,y),(s1,s2)⟩f(x,y)\exp(-i\langle(x,y),(s_{1},s_{2})\rangleis usually defined from the Lebesgue integral. Using an improper Kurzweil–Henstock integral, we obtain a similar representation of the Fourier transform of non-Lebesgue integrable functions onR2\mathbb{R}^{2}. We prove the Riemann–Lebesgue lemma and the pointwise continuity for the classical Fourier transform on a subspace of non-Lebesgue integrable functions which is characterized by the bounded variation functions in the sense of Hardy–Krause. Moreover, we generalize some properties of the classical Fourier transform defined onLp⁢(R2)L^{p}(\mathbb{R}^{2}), where1<p≤21<p\leq 2, yielding a generalization of the results obtained by E. Hewitt and K. A. Ross. With our integral, we define the space of integrable functionsKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2})which contains a subspace whose completion is isometrically isomorphic to the space of integrable distributions on the plane as defined by E. Talvila [The continuous primitive integral in the plane,Real Anal. Exchange45(2020), 2, 283–326]. A question arises about the dual space of the new spaceKP⁢(R2)\mathrm{KP}(\mathbb{R}^{2}).

VAN DER CORPUT LEMMA WITH BESSEL FUNCTIONS

In this article, we study analogues of the van der Corput lemmas [19] involving Bessel functions. In harmonic analysis, one of the most important estimates is the van der Corput lemma, which is an estimate of the oscillatory integrals. This estimate was first obtained by the Dutch mathematician Johannes Gaultherus van der Corput. Van der Corput interested in the behavior for large positive λ of the oscillatory integral R b a e iλφ(x)ψ(x)dx, where φ is a real-valued smooth function (the phase) and ψ is complex valued smooth function (amplitude). In case a = −∞, b = +∞, it is assumed that ψ has a compact support in R. In our case we replace the exponential function with the Bessel functions, to study oscillatory integrals appearing in the analysis of wave equation with singular damping. More specifically, we study integral of the form I(λ) = R b a Jn(λφ(x))ψ(x)dx for the range n = 0, where ψ ∈ C and smooth, and λ is a positive real number that can vary.The generalisations of the van der Corput lemma is proved. As an application of the above results, the generalised Riemann-Lebesgue lemma is considered.

Success of digital adiabatic simulation with large Trotter step

The simulation of adiabatic evolution has deep connections with Adiabatic Quantum Computation, the Quantum Approximate Optimization Algorithm and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time $T$. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed as the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1). The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2). Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3). As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.

Investigation of infinitely rapidly oscillating distributions

We rigorously investigate the rapidly oscillating contributions in the sinc-function representation of the Dirac delta function and the Fourier transform of the Coulomb potential. Beginning with a derivation of the standard integral representation of the Heaviside step function, we examine the representation of the Dirac delta function that contains a rapidly oscillating sinc function. By contour integration, we prove that the representation satisfies the properties of the Dirac delta function, although it is a function divergent at nonzero points. This is a good pedagogical example demonstrating the difference between a function and a distribution. In most textbooks, the rapidly oscillating contribution in the Fourier transform of the Coulomb potential into the momentum space has been ignored by regulating the oscillatory divergence with the screened potential of Wentzel. By performing the inverse Fourier transform of the contribution rigorously, we demonstrate that the contribution is a well-defined distribution that is indeed zero, even if it is an ill-defined function. Proofs are extended to exhibit that the Riemann–Lebesgue lemma can hold for a sinc function, which is not absolutely integrable.

The octonion linear canonical transform: Definition and properties

The linear canonical transform (LCT) is a kind of integral transforms with wide applications in signal analysis. There have been numerous studies in the literature to generalize the LCT by making use of the quaternion algebra. In this paper, we first define the octonion linear canonical transform (OCLCT). Based on the definition of OCLCT, we extend the relationship between the LCT and the Fourier transform (FT) to the OCLCT and the octonion Fourier transform (OFT). Then explore related properties for the OCLCT such as shift property, inversion formula, isometry and Riemann-Lebesgue lemma. The relation between OCLCT and 3-D LCT is also builded. Moreover, based on these properties, we obtain Heisenberg’s uncertainty principle and Donoho-Stark’s uncertainty principle associated with the OCLCT. Finally, some potential applications are presented to show the effectiveness of the OCLCT.

Van der Corput lemmas for Mittag-Leffler functions. II. α–directions

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Ergodicity and stability for (b, l)-regularized resolvent operator families

This paper is concerned with the ergodicity and strong stability for (b, l)-regularized resolvent operator families. First, we study Abel ergodicity and Cesaro ergodicity of (b, l)-regularized resolvent operator families by using the methods of operator theory and complex Tauberian theorem. And then, by constructing a new operator-valued function, we obtain some sufficient conditions on the strong stability of bounded (b, l)-regularized resolvent operator families by means of Cauchy theorem and Riemann–Lebesgue lemma. Our results generalize the related conclusions for operator semigroups and resolvent operator families.

Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation

The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra $$Cl_{(3,1)}$$ of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties such as continuity, Plancherel identity, Riemann–Lebesgue lemma, and to establish the associated Hausdorff–Young inequality. Moreover, using the observer related space-time split several uncertainty inequalities are established, including Heisenberg’s uncertainty principle and Hardy’s theorem.

Harmonic analysis associated to the canonical Fourier Bessel transform

ABSTRACT The aim of this paper is to develop a new harmonic analysis related to a Bessel type operator on the real line: We define the canonical Fourier Bessel transform and study some of its important properties. We prove a Riemann–Lebesgue lemma, inversion formula and operational formulas for this transformation. We derive Plancherel theorem and Babenko inequality for In the present paper, several uncertainty inequalities and theorems for the canonical Fourier Bessel transform are given, including the Heisenberg inequality, Hardy theorem, Nash-type inequality, Carlson-type inequality, global uncertainty principle, local uncertainty principle, logarithmic uncertainty principle in terms of entropy and Miyachi uncertainty principle.

Order of magnitude of multiple Walsh–Fourier coefficients of functions of bounded $$\phi$$-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus $$\mathbb {I}^n:=[0,1)^n$$ $$(n\in \mathbb N)$$ , let $$\hat{f}(\mathbf{k})$$ denote the multiple Walsh–Fourier coefficient of f, where $$\mathbf{k}=(k_1,\dots ,k_n)\in (\mathbb {Z}^+)^n$$ , $$\mathbb {Z^+}=\mathbb {N}\cup \{0\}$$ . The Riemann–Lebesgue lemma shows that $$\hat{f}(\mathbf{k})=o(1)$$ as $$|\mathbf{k}|\rightarrow \infty$$ for any $$f\in L^1(\mathbb I^n)$$ . However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. The definitive result is due to Ghodadra for functions of bounded p-variation. In this paper, similar definite results are proved for functions of bounded $$\phi$$ -variation which generalize the known results for functions of bounded p-variation. The techniques used are some key inequalities for convex functions (including the Jensen’s inequality) and the Taibleson-like technique for Walsh-Fourier coefficients.

Spectral Multipliers on 2-Step Stratified Groups, I

Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L}, -i \mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i \mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L},-i\mathcal{T})$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

On values of the Riemann zeta function at positive integers

We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2n) = ηnπ2n, we obtain the generating functions of the sequences ηn and (−1)nηn. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2n) and ζ(2n + 1). Furthermore, we prove some series equations for $$ {\sum}_{k=1}^{\infty }{\left(-1\right)}^{k-1}\upzeta \left(p+k\right)/k. $$

English

For a Lebesgue integrable complex-valued function $f$ defined on $\mathbb R^+:=[0,\infty )$ let $\hat f$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat f(y)\to 0$ as $y\to \infty $. But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^1(\mathbb R^+)$ there is a definite rate at which the Walsh-Fourier transform tends to zero. We determine this rate for functions of bounded variation on $\mathbb R^+$. We also determine such rate of Walsh-Fourier transform for functions of bounded variation in the sense of Vitali defined on $(\mathbb R^+)^N$, $N\in \mathbb N$.

Spectral Multipliers on 2-Step Stratified Groups, II

Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ is a Schwartz function, then $m$ equals almost everywhere a Schwartz function.

A note on measures vanishing at infinity

In this paper, we review the basic properties of measures vanishing at infinity and prove a version of the Riemann–Lebesgue lemma for Fourier transformable measures.

Inequalities in the Setting of Clifford Analysis

In this paper, we prove the Riemann-Lebesgue lemma for the Clifford-Fourier transform. We give an alternative proof of Heisenberg’s inequality. Furthermore, we provide a version of Young inequalities in the framework of Clifford analysis.

101.33 The application of the Weierstrass approximation theorem in the Riemann-Lebesgue lemma

101.33 The application of the Weierstrass approximation theorem in the Riemann-Lebesgue lemma