Prolate spheroidal wave functions,Sonine spaces,and the Riemann zeta function

In this paper we study underlying spaces associated with A. Connes? trace formula (see Connes (1999) [3], Li (2010) [14]). In particular the explicit formula in the theory of prime numbers is expressed as the trace of an operator acting on a Hilbert space, which is a direct sum of a Sonine space, the space of prolate spheroidal wave functions, and a variant of the space of prolate spheroidal wave functions. A formula is obtained for the orthogonal projection of the Hilbert space onto the Sonine space. A base is given for the variant space of the space of prolate spheroidal wave functions.     

Asymptotic Approximations for Prolate Spheroidal Wave Functions

Uniformly valid (with respect to the independent variable) asymptotic approximations to the radial, prolate spheroidal wave functions are constructed from Bessel-function and Coulomb-wave-function models for large values of the wave number c. The prolate angular functions also are considered, but more briefly. The emphasis is on qualitative accuracy (such as might be useful to the physicist), rather than on efficient algorithms for very accurate numerical computation, and the error factor for most of the approximations is 1 + O (1/c) as c↑∞.     

Constructing prolate spheroidal quaternion wave functions on the sphere

Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three‐dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

On convergence rates of prolate interpolation and differentiation

Prolate spheroidal wave functions of order zero (PSWFs) are widely used in scientific computation. There are few results about the error bounds of the prolate interpolation and differentiation. In this paper, based on the Cauchy’s residue theorem and asymptotics of PSWFs, the convergence rates are derived. To get stable approximation, the first barycentric formula is applied. These theoretical results and high accuracy are illustrated by numerical examples.     

Prolate spheroidal wave functions on a disc—Integration and approximation of two-dimensional bandlimited functions

We consider the problem of integrating and approximating 2D bandlimited functions restricted to a disc by using 2D prolate spheroidal wave functions (PSWFs). We derive a numerical scheme for the evaluation of the 2D PSWFs on a disc, which is the basis for the numerical implementation of the presented quadrature and approximation schemes. Next, we derive a quadrature formula for bandlimited functions restricted to a disc and give a bound on the integration error. We apply this quadrature to derive an approximation scheme for such functions. We prove a bound on the approximation error and present numerical results that demonstrate the effectiveness of the quadrature and approximation schemes.     

Further spectral properties of the weighted finite Fourier transform operator and approximation in weighted Sobolev spaces

In this work, we first give some mathematical preliminaries concerning the generalized prolate spheroidal wave function (GPSWF). This set of special functions have been defined as the infinite and countable set of the eigenfunctions of a weighted finite Fourier transform operator. Then, we show that the set of the singular values of this operator has a super‐exponential decay rate. We also give some local estimates and bounds of these GPSWFs. As an application of the spectral properties of the GPSWFs and their associated eigenvalues, we give their quality of approximation in a weighted Sobolev space. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.     

Periodic Prolate Spheroidal Wavelets

Prolate spheroidal wavelets (PS wavelets) were recently introduced by the authors. They were based on the first prolate spheroidal wave function (PSWF) and had many desirable properties lacking in other wavelets. In particular, the subspaces belonging to the associated multiresolution analysis (MRA) were shown to be closed under differentiation and translation. In this paper, we introduce periodic prolate spheroidal wavelets. These periodic wavelets are shown to possess properties inherited from PS wavelets such as differentiation and translation. They have the potential for applications in modeling periodic phenomena as an alternative to the usual periodic wavelets as well as the Fourier basis.     

Wavelets Based on Prolate Spheroidal Wave Functions

The article is concerned with a particular multiresolution analysis (MRA) composed of Paley–Wiener spaces. Their usual wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions (PSWFs) which have much better time localization than the sinc function. The new wavelets preserve the high energy concentration in both the time and frequency domain inherited from PSWFs. Since the size of the energy concentration interval of PSWFs is one of the most important parameters in some applications, we modify the wavelets at different scales to retain a constant energy concentration interval. This requires a slight modification of the dilation relations, but leads to locally positive kernels. Convergence and other related properties, such as Gibbs phenomenon, of the associated approximations are discussed. A computationally friendly sampling technique is exploited to calculate the expansion coefficients. Several numerical examples are provided to illustrate the theory.     

Numerical investigation of the acoustic scattering problem from penetrable prolate spheroidal structures using the Vekua transformation and arbitrary precision arithmetic

A complete set of radiating “outwards” eigensolutions of the Helmholtz equation, obtained by transforming appropriately through the Vekua mapping the kernel of Laplace equation, is applied to the investigation of the acoustic scattering by penetrable prolate spheroidal scatterers. The scattered field is expanded in terms of the aforementioned set, detouring so the standard spheroidal wave functions along with their inherent numerical deficiencies. The coefficients of the expansion are provided by the solution of linear systems, the conditioning of which calls for arbitrary precision arithmetic. Its integration enables the polyparametric investigation of the convergence of the current approach to the solution of the direct scattering problem. Finally, far‐field pattern visualization in the 3D space clarifies the preferred scattering directions for several frequencies of the incident wave, ranging from the “low” to the “resonance” region.     

Construction of Periodic Prolate Spheroidal Wavelets Using Interpolation

Periodic prolate spheroidal wavelets (periodic PS wavelets), based on the periodizaton of the first prolate spheroidal wave function (PSWF), were recently introduced by the authors. Because of localization and other properties, these periodic PS wavelets could serve as an alternative to Fourier series for applications in modeling periodic signals. In this paper, we continue our work with periodic PS wavelets and direct our attention to their construction via interpolation. We show that they have a representation in terms of interpolation with the modified Dirichlet kernel. We then derive a group of formulas of interpolation type based on this representation. These formulas enable one to obtain a simple procedure for the calculation of the periodic PS wavelets and finding expansion coefficients. In particular, they are used to compute filter coefficients for the periodic PS wavelets. This is done for a number of concrete cases.     

Nonlinear approximation using Gaussian kernels

It is well known that nonlinear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron (in press) [2] for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore-Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.     

Prolate spheroidal wave functions, an introduction to the Slepian series and its properties

For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as Fourier, Legendre, and Chebyschev to solve a variety of problems. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions (PSWF) in the form of a new orthogonal expansion which we have named the Slepian series. We empirically show that the Slepian series is potentially optimal over more conventional orthogonal expansions for discontinuous functions such as the square wave among others. With regards to interpolation, we explore the connections the Slepian series has to the Shannon sampling theorem. By utilizing Euler's equation, a relationship between the even and odd ordered PSWFs is investigated. We also establish several other key advantages the Slepian series has such as the presence of a free tunable bandwidth parameter.     

On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter

We provide a rapid and accurate method for calculating the prolate and oblate spheroidal wave functions (PSWFs and OSWFs),   S_mn ( c , η)  , and their eigenvalues,  λ _mn   , for arbitrary complex size parameter c in the asymptotic regime of large  | c |  , m and n fixed. The ability to calculate these SWFs for large and complex size parameters is important for many applications in mathematics, engineering, and physics. For arbitrary  arg( c )  , the PSWFs and their eigenvalues are accurately expressed by established prolate -type or oblate -type asymptotic expansions. However, determining the proper expansion type is dependent upon finding spheroidal branch points,   c ^mn _{○; r}   , in the complex c -plane where the PSWF alternates expansion type due to analytic continuation. We implement a numerical search method for tabulating these branch points as a function of spheroidal parameters m , n , and  arg( c )  . The resulting table allows rapid determination of the appropriate asymptotic expansion type of the SWFs. Normalizations, which are dependent on c , are derived for both the prolate - and oblate -type asymptotic expansions and for both  ( n − m )  even and odd. The ordering for these expansions is different from the original ordering of the SWFs and is dictated by the location of   c ^mn _{○; r}   . We document this ordering for the specific case of  arg( c ) =π/4  , which occurs for the diffusion equation in spheroidal coordinates. Some representative values of  λ _mn   and   S_mn ( c , η)  for large, complex c are also given.     

The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non‐strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions.     

A generalization of the prolate spheroidal wave functions

Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of and and the Jacobi polynomials which are an orthogonal basis of a weighted The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of

The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both and a subspace of known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property.

The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.

     

Asymptotic Approximations for Radial Spheroidal Wavefunctions with Complex Size Parameter

Radial spheroidal wavefunctions are functions of four variables, usually denoted by m , n , x , and γ, the last of which is known as the size parameter. This parameter becomes complex when the problem of scattering of a sound pulse by a spheroid is treated using a Laplace transform with respect to time together with the method of separation of variables. Several asymptotic approximations, involving modified Bessel functions, are developed and analyzed.     

Towards a quaternionic function theory linked with the Lamé's wave functions

Over the past few years, considerable attention has been given to the role played by the Lamé's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lamé's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non‐commutative framework. We show that the theory of the LQWFs is determined by the Moisil‐Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lamé's wave equation, on one hand, and the quaternionic hyperholomorphic and anti‐hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel‐Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski‐Plemelj formulae, the ‐hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs. Copyright © 2015 John Wiley & Sons, Ltd.     

Generalized Prolate Spheroidal Wave Functions: Spectral Analysis and Approximation of Almost Band-Limited Functions

In this work, we first give various explicit and local estimates of the eigenfunctions of a perturbed Jacobi differential operator. These eigenfunctions generalize the famous classical prolate spheroidal wave functions (PSWFs), founded in 1960s by Slepian and his co-authors and corresponding to the case \(\alpha =\beta =0.\) They also generalize the new PSWFs introduced and studied recently in Wang and Zhang (Appl Comput Harmon Anal 29:303–329, 2010), denoted by GPSWFs and corresponding to the case \(\alpha =\beta .\) The main content of this work is devoted to the previous interesting special case \(\alpha =\beta >- 1.\) In particular, we give further computational improvements, as well as some useful explicit and local estimates of the GPSWFs. More importantly, by using the concept of a restricted Paley–Wiener space, we relate the GPSWFs to the solutions of a generalized energy maximisation problem. As a consequence, many desirable spectral properties of the self-adjoint compact integral operator associated with the GPSWFs are deduced from the rich literature of the PSWFs. In particular, we show that the GPSWFs are well adapted for the spectral approximation of the classical c-band-limited as well as almost c-band-limited functions. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.