Functions Of Order Zero Research Articles

Theoretical Study on Non-Improvement of the Multi-Frequency Direct Sampling Method in Inverse Scattering Problems

Generally, it has been confirmed that applying multiple frequencies guarantees a successful imaging result for various non-iterative imaging algorithms in inverse scattering problems. However, the application of multiple frequencies does not yield good results for direct sampling methods (DSMs), which has been confirmed through simulation but not theoretically. This study proves this premise theoretically by showing that the indicator function with multi-frequency can be expressed by the Bessel and Struve functions and the propagation direction of the incident field. This is based on the fact that the indicator function with single frequency can be expressed by the exponential and Bessel function of order zero of the first kind. Various simulation outcomes are shown to support the theoretical result.

The Fourier transform of a function related to cylinder functions and asymptotic expansions with logarithmic terms

We consider a function that is obtained by the first kind Hankel function of order zero by removing its logarithmic singularity. This function, apart from a multiple of the zero-order Bessel function as an additive term, can be considered as the ‘entire part’ of the Hankel function. We constitute the Fourier transform of this function by considering a fifth-order ordinary differential equation which can be considered as a generalization of the differential equation for the zero-order cylinder functions. Since the asymptotics of our entire part of the Hankel function contains logarithmic terms, this new function can be used to derive asymptotic expansions of some other functions whose asymptotics contain logarithmic terms too. In the Fourier image it can be shown that these expansions are not merely asymptotic but actually convergent.

Entire Functions of Order Zero with Zeros on a Logarithmic Spiral

We prove the Valiron-type and Valiron–Titchmarsh-type theorems for entire functions of order zero with zeros on a logarithmic spiral.

Fast identification of small perfectly conducting cracks without diagonal elements of multi-static response matrix

ABSTRACTWe explore mathematical structures of indicator functions used in subspace migration to identify the location of a set of perfectly conducting cracks with small lengths when the diagonal elements of a Multi-Static Response (MSR) matrix cannot be obtained. By using the asymptotic formula for the presence of cracks and the structure of the singular vectors linked to the nonzero singular values of the MSR matrix, we prove that the indicator function can be represented by a Bessel function of order zero of the first kind and the total number of incident and observation directions in a full-view inverse scattering problem. We also present experimental results of numerical simulations with noisy data to validate our findings.

Sufficient Conditions for the Existence of the υ $$ \upsilon $$ -Density of Zeros for an Entire Function of Order Zero

We select subclasses of zero-order entire functions f for which we present sufficient conditions for the existence of the $$ \upsilon $$ -density of zeros of f in terms of the asymptotic behavior of the logarithmic derivative F and regular growth of the Fourier coefficients of F.

On value distribution and uniqueness of meromorphic function with finite logarithmic order concerning its derivative and q-shift difference

In this paper, we study the value distribution of a meromorphic function concerning its derivative and q-shift difference , where is of finite logarithmic order. We also investigate the uniqueness of differential-q-shift-difference polynomials with more general forms of entire functions of order zero.

The Nonconvergence of $$h$$ h -Refinement in Prolate Elements

Prolate elements are a "plug-compatible" modification of spectral elements in which Legendre polynomials are replaced by prolate spheroidal wave functions of order zero. Prolate functions contain a"bandwidth parameter" $$c \ge 0 $$ c ? 0 whose value is crucial to numerical performance; the prolate functions reduce to Legendre polynomials for $$c\,=\,0$$ c = 0 . We show that the optimal bandwidth parameter $$c$$ c not only depends on the number of prolate modes per element $$N$$ N , but also on the element widths $$h$$ h . We prove that prolate elements lack $$h$$ h -convergence for fixed $$c$$ c in the sense that the error does not go to zero as the element size $$h$$ h is made smaller and smaller. Furthermore, the theoretical predictions that Chebyshev and Legendre polynomials require $$\pi $$ ? degrees of freedom per wavelength to resolve sinusoidal functions while prolate series need only 2 degrees of freedom per wavelength are asymptotic limits as $$N \rightarrow \infty $$ N ? ? ; we investigate the rather different behavior when $$N \sim O(4-10)$$ N ~ O ( 4 ? 10 ) as appropriate for spectral elements and prolate elements. On the other hand, our investigations show that there are certain combinations of $$N,\,h$$ N , h and $$c>0$$ c > 0 where a prolate basis clearly outperforms the Legendre polynomial approximation.

On some properties of subharmonic functions and entire functions of order zero

In this paper, we study properties of subharmonic functions and entire functions of order zero. We also propose a complete description for closed submodules of some modules of entire functions of order zero.

Application of Coulomb wave functions to an orthogonal series associated with steady axisymmetric Euler flows

The regular Coulomb wave function of order zero is applied to an orthogonal series which is associated with the steady axisymmetric Euler equations. The series expansion of a function of bounded variation is proved to converge to the mean value of the left- and the right-hand limits of the original function at each point. In this proof, some complex functions related to the regular and the irregular Coulomb wave functions of orders zero and one are used.

Pseudospectral method based on prolate spheroidal wave functions for frequency-domain electromagnetic simulations

We apply prolate spheroidal wave functions of order zero as basis functions in the pseudospectral method for frequency-domain electromagnetic simulation problems. Like the traditional pseudospectral frequency-domain (PSFD) methods based on Chebyshev and Legendre polynomial series, the prolate PSFD method yields exponential order of accuracy. In terms of the number of samples utilized per wavelength, the prolate expansion is superior to the Chebyshev and Legendre polynomial series by a factor of /spl pi//2. In addition, the prolate PSFD method employs a more uniform spatial grid, achieving better resolution near the center of the domain.

Chebyshev approximation of the second kind of modified Bessel function of order zero

The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J. P. Boyd’s rational Chebyshev basis.

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.

Existence of an Angular Density of Roots of Entire Functions of Order Zero

We obtain sufficient conditions for the set of roots of an entire function of order zero of strongly regular growth to possess an angular density.

Binding energies of on-axis hydrogenic and nonhydrogenic donors in GaAs/Ga1?x Al x As

In a previous work [Phys. Rev. B43(1991)9262], the binding energies of hydrogenic and nonhydrogenic on-axis donors in GaAs/Ga1−xA1 X As quantum-well wires of circular cross section have been calculated as functions of the radius of the quantum-well wire. In both the hydrogenic and nonhydrogenic cases, a variational trial wave function was chosen that could be written as the product of an “envelope function”, and a function containing the variational parameter. It was assumed in these calculations that the potential barrier that exists at the surface of the GaAs cylinder and the surrounding Ga1−xA1 X As matrix is infinite. For the envelope function, an ordinary Bessel function of the first kind and of order zero was chosen. This envelope function satisfies the boundary condition, the vanishing of the trial function at the surface of the quantum-well wire. The question arises: how sensitive are the calculated binding energies to the choice of the envelope function? In the present work, we attempt to provide a partial answer to this question by choosing another envelope function, a spherical Bessel function of order zero. This function also satisfies the boundary condition and makes the trial wave function vanish at the interface between the GaAs cylinder and the Ga1−xA1 X As matrix. Our calculations show that the binding energies of both the hydrogenic and the nonhydrogenic on-axis donors depend on the choice of the envelope function.

Generalization of spherical Bessel's differential equation to functions of order zero

The generalization of Bessel's equation to functions of order zero was investigated by Sharpe and applied in the theory of the reflection of sound by a paraboloid. Here, a modification of Sharpe's differential equation to express both the spherical Bessel's equation and Bessel's (cylindrical) equation for functions of order zero is introduced. The generalization of the modified spherical Bessel's. equation to functions of order zero is also discussed.

Some remarks on subharmonic functions of order zero in the plane

Some remarks on subharmonic functions of order zero in the plane

Partial solutions of finite elasticity-axially symmetric deformations

For axially symmetric deformations of isotropic incompressible hyperelastic materials new reduced forms of the equilibrium equations are given. For the neo-Hookean and extreme-Mooney materials these equations are shown to admit exact solutions expressible in terms of Bessel functions of order zero. These solutions can be rendered as approximate solutions of problems involving the large axially symmetric deformations of circular discs containing a concentric circular hole.

On p-Close-to-Star Functions of Order &#945

In a recent paper M. Finkelstein has used an iterated form of Schwarz's lemma to obtain growth estimates for convex functions and starlike functions of order zero. These estimates involve the second coefficient. W. E. Chase has announced results about the growth estimates and also about a radius of univalence and starlikeness for p-close-to-star functions of order zero. The author generalizes the above results to functions of order a, O<a:1.

On a Class of Subharmonic Functions of Order Zero

On a Class of Subharmonic Functions of Order Zero

Comparative Cost of Two Life Test Procedures

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