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We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series. Multiply the Fourier coefficients of the latter by a power of 2 to get our modified Hurwitz class numbers. As its corollaries, the modified Hurwitz class numbers can be used to calculate the traces of Brandt matrices and the type number of the Eichler orders.

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers., 25 pages

The process of surface reconstruction has received considerable interest from researchers in recent years. Surface reconstruction plays a major role in many applications, such as visualization, geometric modeling and multiresolution analysis. In this paper, we present an approach that approximates a surface from a set of oriented points. Our algorithm combines the implicit surface and frequency-based frameworks to convert the indicator function of the surface into an implicit function from which we can extract the required surface. In contrast to traditional frequency-based approaches, our approach avoids voxelization of the input points and calculates the Fourier coefficients directly from the surface, which reduces the amount of memory required to settle the voxel grid and eliminates the mathematical errors corresponding to this voxelization. In addition, we exploit the recent advances of GPUs embedded in graphics cards to accelerate the calculation of the Fourier coefficients. Finally, some examples are given to demonstrate the validity of the proposed technique.

We study the normal number of prime factors of a f ( p ) + a g ( p ) with p prime and f , g distinct Hecke eigenforms of weight two. Assuming a quasi-generalized Riemann hypothesis, we show that the normal order is log log p . We also obtain an estimate for the number of primes p for which a f ( p ) + a g ( p ) = 0 .

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

A new approximation algorithm is developed to compute the electromagnetic (EM) tensor Green's function in the biaxial anisotropic media based on the Fourier series expansion. First, a rectangular region is chosen as the computation region, in which the EM field can be expressed as a 2-D Fourier series. The Fourier coefficients can be regarded as the EM field in discrete wavenumber domain. We further obtain the EM solution in the spatial domain in the form of Fourier series. Then, we use the finite terms of the Fourier series to approximate the EM field for enhancement of computation efficiency. Because the new method avoids the numerical integration in the infinite wavenumber domain, it is more convenient to implement than other methods based on integral transform. Finally, the spatial distribution of the tensor Green's function for a transversely isotropic medium is presented to verify the proposed approach. The agreements between the results obtained by the present method and the analytic solution demonstrate the validity and the robustness of our algorithm.

With few exceptions, simulation output analysis has focused on static characterizations, to determine a property of the steady-state distribution of a performance metric such as a mean, a quantile, or the distribution itself. Analyses often seek to overcome difficulties induced by autocorrelation of the output stream. But sample paths generated by stochastic simulation exhibit dynamic behaviour that is characteristic of system structure and associated distributions. In this paper, we explore these dynamic characteristics, as captured by the Fourier transform of a dynamic steady-state simulation trajectory. We find that Fourier coefficient magnitudes can have greater discriminatory power than the usual test statistics when two systems have different utilisations and/or dynamic behaviour, and with simpler analysis resulting from the statistical independence of coefficient estimates at different frequencies.

In this paper, Fourier series expansion of Tangent polynomials are derived and the integral representation and explicit formula at rational arguments of these polynomials are established.

We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on [Formula: see text] in the case where N is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O’Sullivan, Petridis, and Risager. In one case we find less cancelation in this sum than would be predicted by the common phenomenon of “square root cancelation”, while in another case we find more cancelation.

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients.

We prove power-saving bounds for general Kloosterman sums on $\operatorname{Sp}(4)$ associated to all Weyl elements via a stratification argument coupled with $p$-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of $\operatorname{Sp}(4)$ Poincar\'e series., Comment: Published version

We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin transforms and the $L$-functions related to the forms. Further we discuss the case of products of Dirichlet $L$-functions and their Mellin duals, which are convolution products of $\vartheta$-series. The latter are intriguingly similar to non-holomorphic Maass forms of weight zero as suggested by their Fourier coefficients., Comment: 1+23 pages, 1 appendix, v2: typos corrected, references added

In this paper, a coupled system of fractional differential equations along with integral boundary conditions is discussed by means of the iterative reproducing kernel algorithm. Towards this end, a recently advanced analytical approach is proposed to obtain approximate solutions of nonclassical-types boundary value problems of fractional derivatives in Caputo sense. This approach optimizes approximate solutions based on the Gram-Schmidt process on Sobolev spaces that execute to generate Fourier expansion within a fast convergence rate, whereby the constructed kernel function fulfills homogeneous integral boundary conditions. Moreover, the solution is presented in the form of a fractional series over the entire Hilbert spaces without unwarranted assumptions on the considered models. The validity of the present algorithm is illustrated by expounding and testing two numerical examples. The achieved results indicate that the proposed algorithm is systematic, feasibility, stability, and convenient for dealing with other fractional systems emerging in the physical, technology and engineering.

The goal of this paper is to investigate the method outlined by one of us (PR) in Cherubini et al. (2009) to compute option prices. We name it the SINC approach. While the COS method by Fang and Osterlee (2009) leverages the Fourier-cosine expansion of truncated densities, the SINC approach builds on the Shannon Sampling Theorem revisited for functions with bounded support. We provide several results which were missing in the early derivation: i) a rigorous proof of the convergence of the SINC formula to the correct option price when the support grows and the number of Fourier frequencies increases; ii) ready to implement formulas for put, Cash-or-Nothing, and Asset-or-Nothing options; iii) a systematic comparison with the COS formula for several log-price models; iv) a numerical challenge against alternative Fast Fourier specifications, such as Carr and Madan (1999) and Lewis (2000); v) an extensive pricing exercise under the rough Heston model of Jaisson and Rosenbaum (2015); vi) formulas to evaluate numerically the moments of a truncated density. The advantages of the SINC approach are numerous. When compared to benchmark methodologies, SINC provides the most accurate and fast pricing computation. The method naturally lends itself to price all options in a smile concurrently by means of Fast Fourier techniques, boosting fast calibration. Pricing requires to resort only to odd moments in the Fourier space. A previous version of this manuscript circulated with the title `Rough Heston: The SINC way'., Comment: 49 pages, 4 figures, 13 tables

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

The authors provide the main statements of the method of approximation of discontinuous functions of two variables that describe an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using the projections from a computer tomograph. The method is based on specially designed discontinuous two-variable splines and finite Fourier sums whose Fourier coefficients can be found using the projection data. The difference between the function being approximated and the specified discontinuous spline is a continuous function and can be approximated by finite Fourier sums without the Gibbs phenomenon. In the computing experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. Analysis of the calculation results confirmed the theoretical statements of the study. The method makes it possible to obtain a prescribed approximation accuracy with a smaller number of projections, i.e, with less irradiation.

Objective: To investigate the importance of membrane charge oscillations and redistribution in multi-compartmental ultrasonic neuromodulation (UNMOD) intramembrane cavitation models. Methods: The Neuronal Intramembrane Cavitation Excitation (NICE) model and multiScale Optimized model of Neuronal Intramembrane Cavitation (SONIC) of UNMOD are compared for a nanoscale multi-compartmental and point neuron approximation of the bilayer sonophore and surrounding proteins. The temporal dynamics of charge oscillations and their effect on the resulting voltage oscillations are investigated by fourier series analysis. Results: Comparison of excitation thresholds and neuronal response between nanoscale multi-compartmental and point models, implemented in the SONIC and NICE framework, demonstrates that the explicit modeling of fast spatial charge redistribution is critical for an accurate multi-compartmental UNMOD-model. Furthermore, the importance of modeling partial protein coverage is quantified by the excitability thresholds. Subsequently, we establish by fourier analysis that these charge oscillations are slowly changing in time. Conclusion: Fast charge redistribution significantly alters neuronal excitability in a multi-compartmental nanoscale UNMOD-model. Also the mutual exclusivity between protein and sonophore coverage should be taken into account, when simulating the dependency of neuronal excitability on coverage fractions. Charge oscillations are periodic and their fourier components change on a slow timescale. Furthermore, the resulting voltage oscillations decrease in energy with overtone number, implying that an extension of the existing multiscale model (SONIC) to multi-compartmental neurons is possible by taking into account a limited number of fourier components. Significance: First steps are taken towards a morphologically realistic and computationally efficient UNMOD-model, improving our understanding of the underlying ultrasonic neuromodulation mechanisms.

Let λf (n) be the normalized n-th Fourier coefficient of holomorphic eigenform f for the full modular group and ℙc(x): = {p ≤ x ∣ [pc]prime}, c ∈ ℝ+. In this paper, we show that for all 0 < c < 1 the mean value of λf(n) in ℙc(x) is ≪x log−Ax assuming the Riemann Hypothesis. Unconditionally, in the sense of Lebesgue measure, it holds for almost all c ∈ (e, 1 − e).

In this paper, a rose-flower variety classification scheme, using color and shape features is presented. The first three statistical moments of the R, G, and B planes of the image were calculated to describe the color, while Fourier coefficients are used to describe the shape. For shape description, signatures (wave-forms) of the boundary contour of the binary images were extracted. Fourier coefficients that are used to describe the shape were estimated using the signatures generated. Depending on the Fourier coefficients, a representation of sums of angles formed along boundaries of the flowers was defined. Using these sums and the color features as input to an artificial neural network (ANN), the flowers were classified into their respective target classes. The eighteen flower varieties considered in this study were classified with an accuracy of 95.6%, 98.9%, and 100% using their shape, color, and combination of both shape and color features, respectively. Comparing these results, it was found that the combination of the two features is an efficient criterion for rose flower variety discrimination and classification.

Asymptotic approximations of the Apostol-Genocchi numbers andpolynomials are derived using Fourier series and ordering of poles ofthe generating function. Asymptotic formulas for the Apostol-Eulernumbers and polynomials are obtained as consequence. Asymptoticformulas for special cases which include the Genocchi numbers andpolynomials are also explicitly stated.

New relations between the Banach algebras of absolutely convergent Fourier integrals of complex-valued measures of Wiener and various issues of trigonometric Fourier series (see classical monographs by A. Zygmund [1] and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new properties of the Fourier series from the known properties of the Wiener algebras, as well as new results to be obtained for those algebras from the known properties of Fourier series. For example, criteria, i.e. simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to be a Fourier series, or the Fourier series of a function of bounded variation, and so forth. Approximation properties of various linear summability methods of Fourier series (comparison, approximation of function classes and single functions) and summability almost everywhere (often with the set indication) are considered. The presented material was reported by the author on 12.02.2021 at the Zoom-seminar on the theory of real variable functions at the Moscow State University.

For a partial differential equation of order 2n with constant coefficients in the domain G := {x = (x1,…, xm) : 0 < xj < 1 < ∞, j = 1,…, m, m ϵ ℕ} , we study the problem with conditions that are multipoint perturbations of the Dirichlet boundary conditions by using the Fourier method. To investigate the spectral properties of a multipoint problem, we use the operator of transformation R: L2 (G) → L2 (G) that establishes the relationship RL0 = LR between the self-adjoint operator L0 of the Dirichlet problem and the operator L of multipoint problem. The solution of the problem with homogeneous multipoint conditions is constructed in the form of Fourier series in the system of eigenfunctions of the operator of the problem. Moreover, the conditions for its existence and uniqueness are established.

The Fourier-series-based insulated-gate bipolar transistor (IGBT) model has been very effective in simulating its switching behaviors and meanwhile, reflecting carrier dynamics. However, in turn-on process, such a model has difficulty in distinguishing the boundary between undepleted area and depletion layer of the drift region due to Gibbs phenomenon and therefore, cannot accurately describe the carrier behavior of carrier storage region (CSR). In this study, the Fourier-series-based model is modified to determine the more precise depletion layer edge for turn on. This is achieved by utilizing the actual characteristics of the carrier concentration gradient at the edge of CSR in turn-on process. Such a modified model is implemented in MATLAB/Simulink and it can better reflect the depletion layer and simulate the distribution of both the carrier concentration and electric field during turn-on process. It corrects the unreasonable voltage tail caused by the error of drift region voltage calculation in previous models. The results of the proposed model under both the resistive load and inductive load are in good agreement with TCAD. Such a modified model can more accurately simulate both carrier dynamics and switching behaviors in the turn-on process.

Based on the morphological analysis techniques developed under the guidance of Yu.P. Pyt’ev, a method for filtering time series is proposed that is capable of detecting an almost cyclic component with a varying cycle length and varying series members within cycles. The effectiveness of the approach is illustrated as applied to decomposition of time series of atmospheric СО2 concentrations. After filtering out the series component responsible for diurnal variability, the series residual becomes stationary, so mathematical statistical methods and Fourier analysis can be used for its further investigation. The results are verified by comparing them with Fourier analysis data. A cyclicity with a period longer than one day is studied using Fourier expansion and wavelet analysis of the original series.

Let $$\lambda _\pi (1,\ldots ,1,n)$$ be the normalized Fourier coefficients of an even Hecke–Maass form $$\pi $$ for $$SL(m, {\mathbb {Z}})$$ with $$m\ge 3$$ , and $$r_{3}(n)=\#\{(n_1,n_2,n_3)\in {\mathbb {Z}}^3:n=n_1^2+n_2^2+n_3^2\}$$ . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients $$\lambda _\pi (1,\ldots ,1 ,n)$$ and $$r_{3}(n)$$ , which improves previous results (even under the generalized Ramanujan conjecture).

A complete description of multipliers of Fourier–Haar series in Lorentz spaces is given. Necessary and sufficient conditions for sequences of complex numbers to belong to the class $$m(L_{p,r}\to L_{q,s})$$ of multipliers of Fourier–Haar series are obtained.

In this paper, we consider an inverse boundary value problem for a mixed type partial differential equation with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The differential equation depends from another positive parameter in mixed derivatives. With respect to first variable this equation is a fractional-order nonhomogeneous differential equation in the positive part of the considering segment, and with respect to second variable is a second-order differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of direct and inverse boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. It is proved the stability of the solution with respect to redefinition functions, and with respect to parameter given in mixed derivatives. For irregular values of the spectral parameter, an infinite number of solutions in the form of a Fourier series are constructed.

A broadband constant beam pattern (CBP) array is an acoustic array whose beam patterns are independent of operating frequencies. In this article, the theoretical far-field acoustic beam pattern analytic formulations have been mathematically derived for a cylindrical array on an infinite rigid cylinder, after applying the Fourier series expansion, the Fourier sine or cosine transform, and the stationary phase method to the Helmholtz equation with boundary conditions. When the ratio of the array radius over the operating frequency wavelength was large, the horizontal (xoy) plane beam patterns demonstrated broadband CBP performances under the far-field conditions. The vertical (xoz) plane acoustic beam patterns are determined by the array vertical aperture shading's Fourier spectrum coupled with the residuals from the asymptotic approximations of the Hankel function and its derivative. The theoretical results matched with the results by Kirchhoff's integral numerical simulation with various horizontal shading examples for directional acoustic beams by the broadband CBP approach.

Numerical solution of the wave parabolic equation on a rectangular grid is analyzed when the discrete split-step Fourier transform (FT) method is used to calculate the field values in an inhomogeneous medium on the next step in the range. The goal is to find out the limiting possibilities of the FT method itself, so studies have been carried out for radio wave propagation in free space only. Two related problems have been solved. First, the minimum value of the root-mean-square error (RMSE) of the calculated field has been estimated. Second, the same has been done for the transmission coefficients of the Fourier series harmonics and of the coefficients of the artificial absorbing layer (AL), which are dependent on the parameters of the computational scheme. It is shown that the dependence of the RMSE value on the distance to the source always has a maximum. The forms of the optimal ALs differ from those used conventionally primarily by the presence of a significant imaginary component.

Surface electromyogram (sEMG) provides a non-invasive way to collect EMG signals. The sEMG signals acquired from the muscles of the forearm can be used to recognize the hand grasps and gestures. In this work, an automatic recognition algorithm to identify hand movements using sEMG signals has been proposed. The signals are decomposed into Fourier intrinsic band functions (FIBFs) using the Fourier decomposition method (FDM). The features like entropy, kurtosis, and L 1 norm are computed for each FIBF. Statistically relevant features are determined using the Kruskal Wallis test and used to train machine learning-based classifiers like support vector machine, k-nearest neighbor, ensemble bagged trees, and ensemble subspace discriminant. Two publicly available datasets are used to test the efficacy of the proposed algorithm. With an average accuracy of 99.49 % on the UCI dataset and 93.53 % on NinaPro DB5, the proposed method performs superior than the state-of-the-art algorithms. The performance of the proposed algorithm has also been analyzed in the presence of noise. The proposed method is based on Fourier theory, which makes it suitable for real-time implementation due to low computational complexity. It would help in the design of efficient and easy-to-use prosthetic hands.

In this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

Epilepsy is a disease recognized as the chronic neurological dysfunction of the human brain which is described by the sudden and excessive electrical discharges of the brain cells. Electroencephalogram (EEG) is a prime tool applied for the diagnosis of epilepsy. In this study, a novel and effective approach is introduced to decompose the non-stationary EEG signals using the Fourier decomposition method. The concept of position, velocity, and acceleration has been employed on the EEG signals for feature extraction using $$L^p$$ norms computed from Fourier intrinsic band functions (FIBFs). The proposed scheme comprises three main sections. In the first section, the EEG signal is decomposed into a finite number of FIBFs. In the second stage, the features are extracted from FIBFs and relevant features are selected by using the Kruskal–Wallis test. In the last stage, the significant features are passed on to the support vector machine (SVM) classifier. By applying 10-fold cross-validation, the proposed method provides better results in comparison to the state-of-the-art methods discussed in the literature, with an average classification accuracy of 99.96% and 99.94% for classification of EEG signals from the BONN dataset and the CHB-MIT dataset, respectively. It can be implemented using the computationally efficient fast Fourier transform (FFT) algorithm.

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

Let $$W_0(\mathbb {R})$$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series $$\sum \nolimits _{k=-\infty }^\infty c_k e^{ikt}$$ is the Fourier series of an integrable function if and only if there exists a $$\phi \in W_0(\mathbb {R})$$ such that $$\phi (k)=c_k$$ , $$k\in \mathbb {Z}$$ . If $$f\in W_0(\mathbb {R})$$ , then the piecewise linear continuous function $$\ell _f$$ defined by $$\ell _f(k)=f(k)$$ , $$k\in \mathbb {Z}$$ , belongs to $$W_0(\mathbb {R})$$ as well. Moreover, $$\Vert \ell _f\Vert _{W_0}\le \Vert f\Vert _{W_0}$$ . Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of $$W_0$$ are established.

Using the repeated de la Vallee Poussin sums, we have proved four theorems which show the upper bound of the repeated de la Vallee Poussin kernel, the convergence of Fourier series, the deviation between a continuous function and their repeated de la Vallee Poussin sums of partial sums of their Fourier series, and the degree of approximation of functions belonging to ordinary Lipschitz class.

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions $n-h, n, n+h$ such that $a(n-h)a(n)a(n+h) \neq 0$., 17 pages (now with consistent author naming)

Robust signal processing in any application depends on the representation and the understanding of the signals in hand. There is a plethora of signal processing methods, wherein choosing a method over others would depend on the attributes of the signals and their intended application. As of now, only spectral stationarity is discussed for the deterministic signals. This letter presents a novel amplitude-modulation-frequency-modulation (AM-FM) based measure of non-stationarity of the deterministic signals by extracting AM-FM components using the Fourier decomposition method (FDM). FDM generates analytic signal representation, where the real and imaginary parts are the Hilbert transform pairs. Simple and intuitive measures of amplitude and frequency non-stationarity are presented to quantitatively assess the extent of non-stationarity. Both these measures are zero for stationary signals, while at least one of them is positive for the non-stationary signals. Important observations regarding the stationarity of a signal are made with the help of some examples. The utility of the concept is demonstrated via an application of EEG signal classification.

We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance $\Delta$ between the two closest point sources is less than $1/M$. We introduce a clumps model where the point sources are closely spaced within several clumps. Under this assumption, we derive a non-asymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. The main novelty is that our lower bound obtains an exact dependence on the {\it Super-Resolution Factor} $SRF=(M\Delta)^{-1}$. As noise level increases, the {\it sensitivity of the noise-space correlation function in the MUSIC algorithm} degrades according to a power law in $SRF$ where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max error of super-resolution for the grid model, which in turn is closely related to the minimum singular value of Vandermonde matrices., Comment: fixed some typos; 40 pages, 5 figures; Applied and Computational Harmonic Analysis 2021

We prove that given any $$\epsilon >0$$ , a non-zero adelic Hilbert cusp form $${\mathbf {f}}$$ of weight $$k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n$$ and square-free level $$\mathfrak {n}$$ with Fourier coefficients $$C_{{\mathbf {f}}}(\mathfrak {m})$$ , there exists a square-free integral ideal $$\mathfrak {m}$$ with $$N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }$$ such that $$C_{{\mathbf {f}}}(\mathfrak {m})\ne 0$$ . The implied constant depends on $$\epsilon , F$$ .

Extremal quasimodular forms have been introduced by M.~Kaneko and M.Koike as as quasimodular forms which have maximal possible order of vanishing at $i\infty$. We show an asymptotic formula for the Fourier coefficients of such forms. This formula is then used to show that all but finitely many Fourier coefficients of such forms of depth $\leq4$ are positive, which partially solves a conjecture stated by M.~Kaneko and M.Koike. Numerical experiments based on constructive estimates confirm the conjecture for weights $\leq200$ and depths between $1$ and $4$., Comment: arXiv admin note: text overlap with arXiv:2002.02736

In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the number of representations of a positive integer n as a sum of $$s\;k$$ -th positive integral powers, based on the recently proved Main Conjecture in Vinogradov’s Mean Value Method.

This work introduces a new method for simulating facies distribution with two categories based on Fourier analysis of Boolean functions. According to this method, two categories of facies distributed along vertical wells are encoded as Boolean functions taking two values. The subsequent simulation process is divided into three consecutive steps. First, Boolean functions of the well data are decomposed into a binary version of a Fourier series. Decomposition coefficients are then simulated over the two-dimensional area as stationary random fields. Finally, synthetic data in the interwell space are reconstructed from simulated coefficients. The described method was implemented experimentally in software and tested on a case of a real oil field and on a case of a synthetic oil field model. Simulations on the synthetic model were used to test the performance of the method for two different bases in the Fourier expansion (Walsh functions and Haar wavelets). The simulation results were compared to those obtained on the same synthetic model via the classical sequential indicator simulation. It was shown that, for both bases, the new method reproduces statistical parameters of the well data better than sequential indicator simulation.

While broadening the applicability of (φ/ψ)-dependent target values for the bond angles in the peptide backbone, sequence/conformation categories with too few residues to analyze via previous methods were encountered. Here, a method of describing a conformation-dependent library (CDL) using two-dimensional Fourier coefficients is reported where the number of coefficients for individual categories is determined via complete cross-validation. Sample sizes are increased further by selective blending of categories with similar patterns of conformational dependence. An additional advantage of the Fourier-synthesis-based CDL is that it uses continuous functions and has no artifactual steps near the edges of populated regions of φ/ψ space. A set of libraries for the seven main-chain bond angles, along with the ω and ζ angles, was created based on a set of Fourier analyses of 48 368 residues selected from high-resolution models in the wwPDB. This new library encompasses both trans- and cis-peptide bonds and outperforms currently used discrete CDLs.

In 1929, Fisher proposed a test for periodicity based on the largest periodogram ordinate. If the true frequency lies between two consecutive Fourier frequencies and the signal to noise ratio is low, the test may conclude that there is no periodicity. This loss of power was noted by Whittle in 1952, as well as the necessary assumption that the noise be white. Whittle and subsequent authors suggested remedies for the white noise assumption. This paper proposes simple tests, based on the Fourier coefficients, that is, the Fourier transforms at the Fourier frequencies, that have good power properties at all frequencies.

In the actual production, it is expected to find several sensitivity geometric errors which have great influence on machining accuracy, so as to provide references for the manufacturing, assembly, and other links of the machine tool, and fundamentally improve the working performance. To study this problem, a novel global sensitivity analysis (GSA) method is proposed. Based on the MBS theory, the spatial error model is established to analyze the local influence of geometric error on machining accuracy. An improved second-order partial correlation coefficient based on the Pearson product moment is proposed to analyze the correlation between geometric errors. In addition, the error parameters will change greatly in the stroke of the moving parts. However, the rapid fluctuation of geometric error value will have a dynamic impact on the machining accuracy of machine tools, which is rarely noticed in the previous sensitivity analysis. This change is defined as the fluctuation of error. The high fitting degree function of error-displacement is obtained by using the high order Fourier series and sine series. Then, the fluctuation of the error in the machining stroke is analyzed by using the derivative of the function to the displacement. Through geometric error characteristics (including the local influence, correlation, and fluctuation) are studied comprehensively, the GSA of error is carried out. Finally, taking a machining center as an example and combining the sensitivity analysis results, the improvement measures are proposed to verify the correctness of the method.

Discrete analogs of the Lebedev–Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels R e K α + i n ( x ) , I m K α + i n ( x ) , x ...

Pendekatan regresi nonparametrik digunakan apabila hubungan antara variabel prediktor dan variabel respon tidak diketahui polanya. Spline truncated dan deret Fourier merupakan estimator dalam pendekatan nonparametrik yang terkenal, karena memiliki fleksibilitas yang tinggi dan mampu menyesuaikan terhadap sifat lokal data secara efektif. Penelitian ini bertujuan untuk mendapatkan estimator model regresi nonparametrik terbaik menggunakan spline truncated dan deret Fourier. Metode estimasi kurva regresi nonparametrik dilakukan dengan menyelesaikan optimasi Ordinary Least Squares (OLS). Kriteria kebaikan model menggunakan GCV, R 2 dan MSE. Pemodelan regresi nonparametrik diterapkan pada data Case Fatality Rate (CFR) akibat Demam Berdarah Dengue (DBD) di Indonesia. Berdasarkan hasil analisis, hasil estimasi dari pemodelan regresi nonparametrik menunjukkan bahwa estimator spline truncated memberikan performa yang lebih baik dibandingkan estimator deret Fourier. Hal ini ditunjukkan dengan nilai R 2 dari estimator spline truncated yaitu sebesar 91,80% dan MSE sebesar 0,04, sedangkan dengan estimator deret Fourier diperoleh nilai R 2 sebesar 65,44% dan MSE sebesar 0,19. ABSTRACT The nonparametric regression approach is used when the relationship between the predictor variable and the response variable is unknown. Spline truncated and Fourier series are well-known estimators in the nonparametric approach because they have high flexibility and are able to adjust to the local properties of the data effectively. This study aims to obtain the best nonparametric regression model estimator using the truncated spline and the Fourier series. The nonparametric regression curve estimation method is done by completing the Ordinary Least Squares (OLS) optimization. The criteria for the goodness of the model use GCV, R 2 , and MSE. Nonparametric regression modeling is applied to Case Fatality Rate (CFR) modeling due to Dengue Hemorrhagic Fever (DBD) in Indonesia. Based on the analysis, the estimation results from the nonparametric regression modeling show that the truncated spline estimator provides better performance than the Fourier series estimator. This is shown by the R 2 value of the truncated spline estimator which is 91.80% and the MSE is 0.04, while the Fourier series estimator obtained an R 2 value of 65.44% and MSE of 0.19.