Fourier Analysis Techniques Research Articles

Sharp Trace and Korn Inequalities for Differential Operators

AbstractWe establish sharp trace and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $$p=1$$ p = 1 , a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $$p=1$$ p = 1 and so despite the failure of the Calderón-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full k-th order gradient estimates. Moreover, for $$1<p<\infty $$ 1 < p < ∞ , where sharp trace estimates yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist – even though the reduction to global Calderón-Zygmund estimates by extension operators might not be possible.

Uniform Convergence of Fourier Series for Hölder Continuous Functions and Functions of Bounded Variation

This article discusses two significant results about the uniform convergence of Fourier series. One is the uniform convergence of the Fourier series for Hölder-α(α>0) continuous functions, while the other is the uniform convergence of the Fourier series for functions that can be expressed as the sum of finitely many continuous monotonic functions. The two results are particularly useful. Because Hölder continuity condition ensures a specific regularity that facilitates the analysis and provides a robust framework for proving uniform convergence and many practical functions can be decomposed into monotonic components, making the theorem broadly applicable. By providing detailed proofs and applications, this study demonstrates the significant implications of uniform convergence in both theoretical and practical contexts. Moreover, this work paves the way for future research in higher-dimensional Fourier analysis, non-periodic functions, and adaptive Fourier techniques, underscoring the ongoing relevance and versatility of Fourier series in modern mathematics. Several well-known results emerge as corollaries of these findings, highlighting the robustness and applicability of these theorems in Fourier analysis.

The Cauchy problem for a class of linear degenerate evolution equations on the torus

We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator P(t,Dx,Dt)=Dt−a2(t)Δx+∑j=1Na1,j(t)Dxj+a0(t), with T>0, t∈[0,T] and a2,a1,1,…,a1,N,a0∈C([0,T];C). Using Fourier analysis techniques, we obtain a complete characterization for the well-posedness of a class of degenerate initial-value problems in the Sobolev, Smooth, Gevrey and Real-Analytic frameworks.

A High-Order Numerical Method Based on a Spatial Compact Exponential Scheme for Solving the Time-Fractional Black–Scholes Model

This paper investigates a high-order numerical method based on a spatial compact exponential scheme for solving the time-fractional Black–Scholes model. Firstly, the original time-fractional Black–Scholes model is converted into an equivalent time-fractional advection–diffusion reaction model by means of a variable transformation technique. Secondly, a novel high-order numerical method is constructed with (2−α) accuracy in time and fourth-order accuracy in space based on a spatial compact exponential scheme, where α is the fractional derivative. The uniqueness of solvability of the derived numerical method is rigorously discussed. Thirdly, the unconditional stability and convergence of the derived numerical method are rigorously analyzed using the Fourier analysis technique. Finally, numerical examples are presented to test the effectiveness of the derived numerical method. The proposed numerical method is also applied to solve the time-fractional Black–Scholes model, whose exact analytical solution is unknown; numerical results are illustrated graphically.

Plasma kinetic simulation in Eulerian approach fully respecting f≥0 requirement and total particle number conservation

Abstract Plasma kinetic simulation in Eulerian approach (PKSiEA) can yield a dynamic description on plasmas and hence is a popular method in theoretical plasmas physics. Its mathematical model consists of 8 members: 5 partial differential equations (PDEs), a phase space boundary condition and an initial condition of the distribution function (PDF) f, and a mandatory non-negativity requirement on f. Because Vlasov equation (VE) is a PDE with variable coefficients and the non-negativity requirement is mandatory, when 5 PDEs (4 Maxwell equations and the VE) are treated as an initial-value problem, the temporal evolution of f is difficult to be attacked by existing popular computational mathematics techniques, such as Fourier analysis technique and finite-difference technique, and hence these techniques do not yield efficient and reliable schemes of the PKSiEA. In a universal scheme of the PKSiEA presented here, because the phase space boundary condition f∣ r=∞,∣υ∣=c = 0 causes the VE to yield a conservation of total particle number ∫fd 3 rd 3 υ, we propose an efficient and strict expression of the f-profile in terms of two classes of functions of phase space coordinates r , υ , and the VE displays a universal relation between two classes. By this new expression, we express the initial-value problem of 5 PDEs in terms of allowed deformation modes from an initial f-profile, and ensure the PKSiEA fully respecting the non-negativity requirement on f.

Assessment of tidal current potential in the Pará River Estuary (Amazon Region – Brazil)

World energy consumption will grow by almost 50 % between 2018 and 2050, resulting in a 79 % increase in electricity generation over this period. In general, renewable energy has been an important source of electricity generation for several years due to its ability to produce unlimited energy. Among the renewable energies are the oceanic ones, which depend on the resources of the tidal waters and the oceans. Tides are a promising energy source, given their periodicity and predictability, using techniques of harmonic analysis or ocean modeling. The Amazon is a region with multiple sparsely distributed populations, with difficult access to energy. Renewable sources are an alternative to bring energy and development to these communities. This paper focuses on the assessment of the power of tidal currents in the Pará River Estuary (PRE, Brazilian Amazon) as an alternative source of electricity generation in the region. This estuary is characterized by particularities such as not having its own source, it has an extension of 300 km and a depth between 0 and 70 m. The methodology was based on the two-dimensional hydrodynamic model (2DH) of SisBahia. The results show the occurrence of five areas with potential for tidal energy exploitation. Power densities for these areas are in the range of 0.4–0.7 MWh/m2 in a spring and neap tidal cycle. This article demonstrates the relevance of choosing parameterized quantities that do not depend on specific equipment, as well as the importance of an adequate characterization of the hydrodynamic patterns in the PRE, necessary to optimize the use of hydrokinetic energy.

Spectrally accurate numerical quadrature formulas for a class of algebraically singular periodic Hadamard finite part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integrals Hσ(t;u)=⨎0T|sinπ(x−t)T|σu(x)dx,0<t<T,σ<−1,σ⁄∈Z,u(x) being sufficiently differentiable and T-periodic on R. Thus σ=−(m+δ), m∈{1,2,3,…}, 0<δ<1. For each such σ, we regularize Hσ(t;u), and show that Hσ(t;u)=Hσ+2r(t;Uσ),r=⌊(m+1)/2⌋, where Uσ(x)=∑k=0raku(2k)(x) for some constants ak, Hσ+2r(t;Uσ) being a regular integral. We then propose to approximate Hσ(t;u) by the quadrature formula Qσ,n(t;u)≡Hσ(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qσ,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qσ,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R).We also show that the theoretical developments and numerical quadrature formulas developed for the HFP integrals Hσ(t;u) with σ<−1 and σ⁄∈Z apply, with no changes, to the regular singular integrals Hσ(t;u) with σ>−1 and σ⁄∈Z.We illustrate the effectiveness of Qσ,n(t;u) with numerical examples both for σ<−1 and σ>−1.Finally, we show that the HFP or regular integral ⨎0Tf(x)dx of any T-periodic integrand f(x) that has algebraic singularities of the form |x−t+kT|σ, 0<t<T, k=0,±1,±2,…, with σ⁄∈Z, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed as Hσ(t;u), where u(x) is a T-periodic and sufficiently differentiable function of x on R that can be computed from f(x). Therefore, ⨎0Tf(x)dx can be computed efficiently using our new numerical quadrature formulas Qσ,n(t;u) on the individual Hσ(t;u). Again, only 2n function evaluations, namely, u(xn,k), k=0,1,…,2n−1, are needed for the whole process.

The initial value problem for the equations of motion of fractional compressible viscous fluids

In this paper we consider the initial value problem to the fractional compressible isentropic generalized Navier-Stokes equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations is described by the nonlocal fractional Laplace operator. We give one detailed spectrum analysis on a linearized operator and the decay law in time of the solution semigroup for the linearized fractional compressible isentropic generalized Navier-Stokes equations around a constant state by the Fourier analysis technique, which is shown that the order of the fractional derivatives plays a key role in the analysis so that the spectrum structure involved here is more complex than that of the classical compressible Navier-Stokes system. Based on this and the elaborate energy method, the global-in-time existence and one optimal decay rate in time of the smooth solution are obtained under the assumption that the initial data are given in a small neighborhood of a constant state.

A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model

This work is focused on the derivation and analysis of a novel numerical technique for solving time fractional mobile-immobile advection-dispersion equation which models many complex systems in engineering and science. The scheme is derived using the effective combination of Euler and Caputo numerical techniques for approximating the integer and fractional time derivatives respectively, and a fourth order exponential compact scheme for spatial derivatives. The Fourier analysis technique is used to prove that the proposed numerical scheme is unconditionally stable and perform convergence analysis. To assess the viability and accuracy of the proposed scheme, some numerical examples are demonstrated with constant as well as variable order time fractional derivatives for this model.

Learning pyramidal multi-scale harmonic wavelets for identifying the neuropathology propagation patterns of Alzheimer's disease.

Previous studies have established that neurodegenerative disease such as Alzheimer's disease (AD) is a disconnection syndrome, where the neuropathological burdens often propagate across the brain network to interfere with the structural and functional connections. In this context, identifying the propagation patterns of neuropathological burdens sheds new light on understanding the pathophysiological mechanism of AD progression. However, little attention has been paid to propagation pattern identification by fully considering the intrinsic properties of brain-network organization, which plays an important role in improving the interpretability of the identified propagation pathways. To this end, we propose a novel harmonic wavelet analysis approach to construct a set of region-specific pyramidal multi-scale harmonic wavelets, it allows us to characterize the propagation patterns of neuropathological burdens from multiple hierarchical modules across the brain network. Specifically, we first extract underlying hub nodes through a series of network centrality measurements on the common brain network reference generated from a population of minimum spanning tree (MST) brain networks. Then, we propose a manifold learning method to identify the region-specific pyramidal multi-scale harmonic wavelets corresponding to hub nodes by seamlessly integrating the hierarchically modular property of the brain network. We estimate the statistical power of our proposed harmonic wavelet analysis approach on synthetic data and large-scale neuroimaging data from ADNI. Compared with the other harmonic analysis techniques, our proposed method not only effectively predicts the early stage of AD but also provides a new window to capture the underlying hub nodes and the propagation pathways of neuropathological burdens in AD.

On fractional semidiscrete Dirac operators of Lévy–Leblond type

AbstractIn this paper, we introduce a wide class of space‐fractional and time‐fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space‐time lattice (), resembling to fractional semidiscrete counterparts of the so‐called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup , carrying the parameter constraints and . The results obtained involve the study of Cauchy problems on .

Regularity of semigroups for exponentially tempered stable processes with drift

In this paper, we study the regularity of semigroups for the exponentially tempered 2β-stable processes. First, we use the Fourier analysis technique to prove that for 1≤p<∞, the semigroup for the tempered stable process without drift is analytic in Lp(Rn). Next, it is shown that the semigroup for the tempered stable process with drift is an analytic semigroup if 1/2≤β<1 and the semigroup is a Gevrey semigroup of order δ with δ>1/(2β) if 0<β<1/2. Finally, we find that the result on Gevrey regularity is optimal in some sense.

Statistical Assessment of Biogenic Risk for the Human Population from New Viral Infections Based on COVID-19

Introduction. Understanding the epidemic curve and spatiotemporal dynamics of SARS-CoV-2 virus is of fundamental importance for the work of the health system during epidemic and pandemic periods. Firstly, the data obtained allow us to assess the epidemiological characteristics of the virus. Secondly, it becomes possible to develop and coordinate measures to counter the spread of COVID-19, to allocate resources reasonably. The work objective is to create and initialize a mathematical model of the epidemic process, which makes it possible to explain the observed dynamics, to predict its development and to assess the reliability of such forecasts. Materials and Methods. Scientific research was based on the statistical data analysis. A hierarchy of mathematical models describing the dynamics of the spread of a new coronavirus infection (COVID-19) and the mortality of COVID-positive patients from 12.02.2020 to 22.09.2021 has been constructed. The incidence submodel reflects regular (aperiodic and periodic), as well as random components. To study and predict the processes, the classical technique of time series research, correlation and Fourier analysis were used. This approach allowed using the method of moments to identify the statistical properties of the scientific research object, and then visualize the stages and algorithm of work. Results. An optimistic, pessimistic and intermediate scenario of infection spread has been mathematically investigated. Their strengths and weaknesses are noted. Numerical characteristics of the trend model and the model of fluctuations in the incidence of COVID-19 are systematized in the form of tables. Based on these data, a conclusion is formulated about the optimality of the pessimistic model: after the highest possible indicators, the infection curve reaches a plateau, and the virus remains in the population. It has been established that the spread of a new coronavirus infection has a pronounced seasonal character with a period of 1/3 of the year. Mathematical analysis and modeling of the mortality dynamics of COVID-positive patients revealed weekly fluctuations in the level of deaths. At the same time, it turned out that the maximum risk corresponds to the 15th and 22nd day of infection. According to the hypothesis proposed by the authors, this virus will be characteristic of the human population. The mortality rate is expected to be 1.75 %. The calculations have shown that the influence of random components of morbidity and mortality will correspond to seasonal fluctuations. Discussion and Conclusion. The probable frequency of the epidemic has been established — three times a year. The potential mortality rate is determined as constant. It is caused by epidemiological and organizational reasons, i. e. the work of medical institutions and authorities. Taking into account the features of the new coronavirus strain (omicron), it is possible to predict the further dynamics of the pandemic and make recommendations regarding its prevention. The authors believe that vaccination should be carried out three times a year. Optimal periods of vaccination campaigns:05. 02–15. 02, 17. 05–28. 05, 24. 09–5. 10.

Sharp Pointwise Weyl Laws for Schrödinger Operators with Singular Potentials on Flat Tori

The Weyl law of the Laplacian on the flat torus $${\mathbb {T}}^n$$ is concerning the number of eigenvalues $$\le \lambda ^2$$ , which is equivalent to counting the lattice points inside the ball of radius $$\lambda $$ in $${\mathbb {R}}^n$$ . The leading term in the Weyl law is $$c_n\lambda ^n$$ , while the sharp error term $$O(\lambda ^{n-2})$$ is only known in dimension $$n\ge 5$$ . Determining the sharp error term in lower dimensions is a famous open problem (e.g. Gauss circle problem). In this paper, we show that under a type of singular perturbations one can obtain the pointwise Weyl law with a sharp error term in any dimensions. This result establishes the sharpness of the general theorems for the Schrödinger operators $$H_V=-\Delta _{g}+V$$ in the previous work (Huang and Zhang (Adv Math, arXiv:2103.05531 )) of the authors, and extends the 3-dimensional results of Frank and Sabin (Sharp Weyl laws with singular potentials. arXiv:2007.04284 ) to any dimensions by using a different approach. Our approach is a combination of Fourier analysis techniques on the flat torus, Li–Yau’s heat kernel estimates, Blair–Sire–Sogge’s eigenfunction estimates, and Duhamel’s principle for the wave equation.

Low regularity well–posedness for dispersive equations on semi–infinite intervals

We summarize and report recent advancements in the theory of dispersive equations posed on semi–infinite intervals. By employing modern Fourier analysis techniques we show how these initial-boundary value problems can be resolved for initial data in Sobolev spaces of low regularity. In almost all the cases, we obtain sharp results that match the regularity of their full line analogues. We especially address the issues of local well–posedness, nonlinear smoothing, and uniqueness of solutions. In the second part of the paper we apply some of the tools we discussed in the first part, to prove existence and uniqueness of $ L^2 $ solutions for the biharmonic equation, extending the result in [10].

A constructive low-regularity integrator for the one-dimensional cubic nonlinear Schrödinger equation under Neumann boundary condition

Abstract A new harmonic analysis technique using the Littlewood–Paley dyadic decomposition is developed for constructing low-regularity integrators for the one-dimensional cubic nonlinear Schrödinger equation in a bounded domain under Neumann boundary condition, when the frequency analysis based on the Fourier series cannot be used. In particular, a low-regularity integrator is constructively designed through the consistency analysis by the Littlewood–Paley decomposition of the solution, in order to have almost first-order convergence (up to a logarithmic factor) in the $L^{2}$ norm for $H^{1}$ initial data. A spectral method in space, using fast Fourier transforms with $\mathcal{O}(N\ln N)$ operations at every time level, is constructed without requiring any Courant-Friedrichs-Lewy (CFL) condition, where $N$ is the degrees of freedom in the spatial discretization. The proposed fully discrete method is proved to have an $L^{2}$-norm error bound of $\mathcal{O}(\tau [\ln (1/\tau )]^{2}+ N^{-1})$ for $H^{1}$ initial data, where $\tau $ is the time-step size.

Edge effect removal in Fourier ptychographic microscopy via periodic plus smooth image decomposition

Fourier ptychographic microscopy (FPM) is a promising computational imaging technique with high resolution, wide field-of-view (FOV) and quantitative phase recovery. So far, a series of system errors that may corrupt the image quality of FPM has been reported. However, an imperceptible artifact caused by edge effect caught our attention and may also degrade the precision of phase imaging in FPM with a cross-shape artifact in the Fourier space. We found that the precision of reconstructed phase at the same subregion depends on the different sizes of block processing as a result of different edge conditions, which limits the quantitative phase measurements via FPM. And this artifact is caused by the aperiodic image extension of fast Fourier transform (FFT). Herein, to remove the edge effect and improve the accuracy, two classes of opposite algorithms termed discrete cosine transform (DCT) and periodic plus smooth image decomposition (PPSID) were reported respectively and discussed systematically. Although both approaches can remove the artifacts in FPM and may be extended to other Fourier analysis techniques, PPSID-FPM has a comparable efficiency to conventional FPM algorithm. The PPSID-FPM algorithm improves the standard deviation of phase accuracy as a factor of 4 from 0.08 radians to 0.02 radians. Finally, we summarized and discussed all the reported system errors of FPM within a generalized model.

Spectrally accurate numerical quadrature formulas for a class of periodic Hadamard Finite Part integrals by regularization

We consider the numerical computation of Hadamard Finite Part (HFP) integralsKm(t;u)=⨎0TSm(π(x−t)T)u(x)dx,0<t<T,m∈{1,2,…}, where u(x) is T-periodic and sufficiently differentiable andS2r−1(y)=cos⁡ysin2r−1⁡y,S2r(y)=1sin2r⁡y,r=1,2,3,…. For each m, we regularize the HFP integral Km(t;u) and show thatKm(t;u)=K0(t;Um)≡∫0T(log⁡|sin⁡π(x−t)T|)Um(x)dx,Um(x) being some linear combination of the first m derivatives of u(x). We then propose to approximate Km(t;u) by the quadrature formula Qm,n(t;u)≡Km(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qm,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qm,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R). We illustrate the effectiveness of Qm,n(t;u) with numerical examples for m=0,1,…,5.We also show that the HFP integral ⨎0Tf(x,t)dx of any T-periodic integrand f(x,t) that has mth order poles at x=t+kT, k=0,±1,±2,…, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed in terms of the Ks(t;u(⋅,t)), where u(x,t) is a T-periodic and sufficiently differentiable function in x on R that can be computed from f(x,t). Therefore, ⨎0Tf(x,t)dx can be computed efficiently using our new numerical quadrature formulas Qs,n(t;u(⋅,t)) on the individual Ks(t;u(⋅,t)). Again, only 2n function evaluations, namely, u(xn,k,t), k=0,1,…,2n−1, are needed for the whole process.

Asymptotic study of supercritical surface Quasi-Geostrophic equation in critical space

In this paper we prove, if θ∈C([0,∞),H2−2α(R2)) is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then ‖θ(t)‖H2−2α decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.

Confounding Effect of Undergraduate Semester-Driven "Academic" Internet Searches on the Ability to Detect True Disease Seasonality in Google Trends Data: Fourier Filter Method Development and Demonstration.

Internet search volume for medical information, as tracked by Google Trends, has been used to demonstrate unexpected seasonality in the symptom burden of a variety of medical conditions. However, when more technical medical language is used (eg, diagnoses), we believe that this technique is confounded by the cyclic, school year-driven internet search patterns of health care students. This study aimed to (1) demonstrate that artificial "academic cycling" of Google Trends' search volume is present in many health care terms, (2) demonstrate how signal processing techniques can be used to filter academic cycling out of Google Trends data, and (3) apply this filtering technique to some clinically relevant examples. We obtained the Google Trends search volume data for a variety of academic terms demonstrating strong academic cycling and used a Fourier analysis technique to (1) identify the frequency domain fingerprint of this modulating pattern in one particularly strong example, and (2) filter that pattern out of the original data. After this illustrative example, we then applied the same filtering technique to internet searches for information on 3 medical conditions believed to have true seasonal modulation (myocardial infarction, hypertension, and depression), and all bacterial genus terms within a common medical microbiology textbook. Academic cycling explains much of the seasonal variation in internet search volume for many technically oriented search terms, including the bacterial genus term ["Staphylococcus"], for which academic cycling explained 73.8% of the variability in search volume (using the squared Spearman rank correlation coefficient, P<.001). Of the 56 bacterial genus terms examined, 6 displayed sufficiently strong seasonality to warrant further examination post filtering. This included (1) ["Aeromonas" + "Plesiomonas"] (nosocomial infections that were searched for more frequently during the summer), (2) ["Ehrlichia"] (a tick-borne pathogen that was searched for more frequently during late spring), (3) ["Moraxella"] and ["Haemophilus"] (respiratory infections that were searched for more frequently during late winter), (4) ["Legionella"] (searched for more frequently during midsummer), and (5) ["Vibrio"] (which spiked for 2 months during midsummer). The terms ["myocardial infarction"] and ["hypertension"] lacked any obvious seasonal cycling after filtering, whereas ["depression"] maintained an annual cycling pattern. Although it is reasonable to search for seasonal modulation of medical conditions using Google Trends' internet search volume and lay-appropriate search terms, the variation in more technical search terms may be driven by health care students whose search frequency varies with the academic school year. When this is the case, using Fourier analysis to filter out academic cycling is a potential means to establish whether additional seasonality is present.