Inversion Formula Research Articles

On the Remainder Term in the Approximate Fourier Inversion Formula for Distribution Functions

On the Remainder Term in the Approximate Fourier Inversion Formula for Distribution Functions

INVERSE CONNECTION FORMULAE FOR GENERALISED BESSEL POLYNOMIALS

Abstract We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.

Discrete linear canonical shearlet transform

In this paper, we introduce the continuous and discrete version of the linear canonical shearlet transform (LCST). The authors begin with the definition of the LCST and then establish a relationship between the linear canonical transform (LCT) and the LCST. Next, the paper derives several basic properties like Parseval’s Formula, inversion formula, and the characterization of the transform’s range. In addition to the continuous version of the transform, the authors also present a discrete version of the LCST. This discrete version allows for practical implementation and efficient computation of the transform in digital signal processing systems. Lastly, the paper establishes a frame condition for the discrete LCST, which thereby helps in establishing the reconstruction formula for the discrete LCST. The paper ends with a conclusion.

A novel wavelet transform in the quaternion quadratic-phase domain

In this paper, we propose a novel integral transform coined as quaternion quadratic-phase wavelet transform (QQPWLT) by invoking the elegant convolution structure associated with the quaternion quadratic-phase Fourier transform. First, we explore some mathematical properties of the QQPWLT, including the orthogonality relation, inversion formula, reproducing kernel and some notable inequalities. Second, we study Heisenberg’s uncertainty principles and the logarithmic uncertainty principle associated with the quadratic-phase wavelet transform in quaternion domain. We culminate our investigation by presenting some illustrative examples.

A note on the paper “A note on inversion of Toeplitz matrices” by X.-G. Lv and T.-Z. Huang

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (2007), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have given, especially when the size of the Toeplitz matrix is very large. Numerical experiments illustrate the effectiveness of our theoretical results.

A new method for generating equidistant/equiratio line between two points on the earth ellipsoid

It is of very great significance to achieve accurate solution of equidistant points/equiratio line between two points at a large regional or global scale. At present, no effective method has been found for equidistant/equiratio line between two points on the Earth ellipsoid. In large scale space, the solution plane of equidistant/equiratio points between two points on the map cannot meet the requirements of high-precision space analysis, due to the existence of map projection deformation. The key to solving the above problems is to establish a method generating the equidistant/equiratio line based on solution to geodesic problem, and implement the entire calculation process on the Earth ellipsoid. Based on the Karney's direct and inverse formulas, the paper uses the pendulum method to design a cohesive method for calculating the equidistant/equiratio line between two points on the Earth ellipsoid. Several specific examples are selected for experimental verification, which proves that this algorithm can solve the problem of calculating equidistant/equiratio line of two points of different scale line in short distance, medium distance and long distance, and the precision of solving coordinate space is less than 0.00001 m, which can meet the requirements of long distance, wide range and high-precision spatial analysis.

Signed graphs and inverses of their incidence matrices

The Laplacian matrix L of a signed graph G may or may not be invertible. We present a combinatorial formula of the Moore-Penrose inverse of L. This is achieved by finding a combinatorial formula for the Moore-Penrose inverse of an incidence matrix of G. This work generalizes related known results on incidence and Laplacian matrices of an unsigned graph. Several examples are provided to show the usefulness of these combinatorial formulas.

Some Orthogonal Combinations of Legendre Polynomials

The principle objective of this article is to introduce and investigate a type of orthogonal polynomials that are written as combinations of Legendre polynomials. This kind of polynomials can be viewed as generalized Jacobi polynomials (GJPs), since they are written in terms of Jacobi polynomials (JPs) with certain negative parameters. The analytic and inversion formulas of these polynomials are established. New derivative expressions for these polynomials are derived in detail in terms of their original polynomials. Other derivative expressions for these polynomials are found but in terms of some orthogonal and non-orthogonal polynomials. Some product formulas with some other polynomials are also obtained. Certain definite and weighted definite integrals are obtained using the newly introduced connection and product formulas.

Eigenvalue estimates for Fourier concentration operators on two domains

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. The spectral profiles of these operators describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers, for the first time, non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel’s work on one dimensional intervals arXiv:1502.04404v1. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

Sectorial Equidistribution of the Roots of x2 + 1 Modulo Primes

Abstract The equation $x^2 + 1 = 0\mod p$ has solutions whenever p = 2 or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of $SL_2(\mathbb{R})$. Since our Poincare series have a nontrivial dependence on their Iwasawa θ-coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work.

Analytic and numerical bootstrap for the long-range Ising model

We combine perturbation theory with analytic and numerical bootstrap techniques to study the critical point of the long-range Ising (LRI) model in two and three dimensions. This model interpolates between short-range Ising (SRI) and mean-field behaviour. We use the Lorentzian inversion formula to compute infinitely many three-loop corrections in the two-dimensional LRI near the mean-field end. We further exploit the exact OPE relations that follow from bulk locality of the LRI to compute infinitely many two-loop corrections near the mean-field end, as well as some one-loop corrections near SRI. By including such exact OPE relations in the crossing equations for LRI we set up a very constrained bootstrap problem, which we solve numerically using SDPB. We find a family of sharp kinks for two- and three-dimensional theories which compare favourably to perturbative predictions, as well as some Monte Carlo simulations for the two-dimensional LRI.

Inversion formulas for Toeplitz-plus-Hankel matrices

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix An of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2×2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix An−2 of order n−2 in the center of An is established and sufficient invertibility criteria for both An and An−2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself.

Pipeline thickness estimation using the dispersion of higher-order SH guided waves

Abstract Thickness measurement plays an important role in the monitoring of pipeline corrosion damage. However, the requirement for prior knowledge of the shear wave velocity in the pipeline material for popular ultrasonic thickness measurement limits its widespread application. This paper proposes a method that utilizes cylindrical shear horizontal (SH) guided waves to estimate pipeline thickness without prior knowledge of shear wave velocity. The inversion formulas are derived from the dispersion of higher-order modes with the high-frequency approximation. The waveform of the example problems is simulated using the real-axis integral method. The data points on the dispersion curves are processed in the frequency domain using the wave-number method. These extracted data are then substituted into the derived formulas. The results verify that employing higher-order SH guided waves for the evaluation of thickness and shear wave velocity yields less than 1% error. This method can be applied to both metallic and non-metallic pipelines, thus opening new possibilities for health monitoring of pipeline structures.

Solutions of certain initial-boundary value problems via a new extended Laplace transform

Abstract In this article, we present a novel extended exponential kernel Laplace-type integral transform. The Laplace, natural, and Sumudu transforms are all included in the suggested transform. The existence theorem, Parseval-type identity, inversion formula, and other fundamental aspects of the new integral transform are examined in this article. Integral identities define the connections between the new transforms and the established transforms. In order to solve specific initial-boundary value problems, the new transforms are used.

Analysis of the weighted conical Radon transform

In this article, we consider the weighted conical Radon transform—the transform is motivated by Compton camera imaging as well as optical tomography. Our contribution involves introducing new inversion formulas for the weighted conical Radon transform, including explicit formulas and properties associated with convolution frames. Furthermore, we propose reconstruction formulas that solve for variety weighted parameters in the two-dimensional space.

Higher water ecological service values have better network connectivity in the middle Yellow River basin

There are numerous ecological challenges that are of considerable concern caused by urbanization and natural calamities. The spatial pattern of the landscape is of great significance in maintaining species diversity and regulating material cycles and individual ecophysiological processes as a medium for ecosystem service functions. Resolving the conflicting relationships between people and land necessitates the precise identification of essential ecological sources. Therefore, there is an urgent need to investigate the relationship between ecological sources and ecosystem service values (ESVs) and to optimize the matter and energy.connectivity network. The study optimizes ecological network simulation using a penetration threshold, population density inverse distance formula, and geodetector weight analysis, constructs ecological spatial networks and evaluates their structure using the minimum cumulative resistance (MCR) model and complex network analysis, and discusses the relationship between network topological characteristics (NTCs) and the spatial pattern of the various ESVs. The results show that the major sources and corridors are primarily along the banks of the Wei River and the Fen River. The clustering coefficient has a significant negative correlation with both water supply and water regulation service values. It is found that the sources with higher ecological service values related to water resources have better network connectivity. This result plays a major reference role for the identification of several crucial ecological components, such as sources, nodes and stepping stones, in the middle Yellow River basin (MYRB). Ultimately, we propose strengthening the ecological protection of the eastern part of the Qinling Mountains and increasing the ecological communication capacity of the forestland on both sides of the river valley to create an optimized ecological pattern.

Uncertainty principles for windowed coupled fractional Fourier transform

The windowed coupled fractional Fourier transform was recently proposed in the literature. It may be considered as a generalized version of the windowed fractional Fourier transform. In this study, we first present various basic properties of the windowed coupled fractional Fourier transform including linearity, shifting, modulation, parity, orthogonality relation, and inversion formula. Further, the close relation of the windowed coupled fractional Fourier transform with the two‐dimensional Fourier transform and the windowed fractional Fourier transform is studied. By combining the properties and relation, we derive several versions of the uncertainty inequalities related to the windowed coupled fractional Fourier transform.

Linear canonical Fourier–Bessel wavelet transform: properties and inequalities

The purpose of this paper is to introduce and study the linear canonical Fourier–Bessel wavelet transform. We prove an orthogonality relation, inversion formula and some inequality for linear canonical Fourier–Bessel wavelet transform. We first present a direct relationship between the linear canonical Bessel wavelet transform and ordinary Bessel wavelet transform. Based on this relation, we provide an alternative proof of the orthogonality relation for the linear canonical Bessel wavelet transform. Some of its essential properties are also studied in detail. Finally, we explicitly derive several versions of inequalities associated with the linear canonical Bessel wavelet transform.

Robust sequential fusion Kalman estimators with asymptotic equivalence and stability for networked uncertain sensor systems

For networked uncertain sensor systems (NUSSs) with an improved unified sensor model with “hard” and “soft” sensors including five uncertainties plus two-step random measurement delays and colored measurement noises, applying the block matrix inversion formula, a new sequential inverse covariance fusion (SICF) approach is presented. It can sequentially compute the global inverse covariance according to the time order of the received data, but need not wait until all data are received. The corresponding minimax robust time-varying and steady-state sequential fusion (SF) Kalman estimators (predictor, filter, and smoother) are presented in the sense that their actual error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. They have lower computation complexities and higher accuracies equivalent to ones of the robust optimal batch-fusion (BF) Kalman estimators, and are suitable for real-time applications. Their asymptotic equivalence, absolute asymptotic stability, and accuracy relations are proved. Their strict complexity analysis is given. They constitute a novel universal robust SF estimation, convergence and stability theory for NUSSs. One simulation example applied to the uncertain ARMA signal processing verifies the effectiveness of the proposed approach and theory.

Application of Post–Widder inversion formula to the calculation of relaxation spectrum from relaxation modulus

Application of Post–Widder inversion formula to the calculation of relaxation spectrum from relaxation modulus