On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.     

Inverse polynomial expansions of Laurent series

An algorithm is introduced, and shown to lead to various unique series expansions of formal Laurent series, as the sums of reciprocals of polynomials. The degrees of approximation by the rational functions which are the partial sums of these series are investigated. The types of series corresponding to rational functions themselves are also characterized.     

Pointwise fourier inversion and related eigenfunction expansions

Necessary and sufficient conditions are found for the convergence at a pre-assigned point of the spherical partial sums (resp. integrals) of the Fourier series (resp. integral) in the class of piecewise smooth functions on Euclidean space. These results carry over unchanged to spherical harmonic expansions, Fourier transforms on hyperbolic space, and Dirichlet eigenfunction expansions with respect to the Laplace operator on a class of Riemannian manifolds. © 1994 John Wiley & Sons, Inc.     

On the dual nature of partial theta functions and Appell–Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell–Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell–Lerch sums. In this sense, Appell–Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers–Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral q-series with mixed mock modular behaviour. In subsequent work we see that our bilateral series are well-suited for computing radial limits of Ramanujan's mock theta functions.     

Fourier series and the δ process

We discuss the effect of a particular sequence acceleration method, the δ²${\delta }^2$ process, on the partial sums of Fourier series. We show that for a very general class of functions, this method fails on a dense set of points; not only does it not speed up convergence, it turns the sequence of partial sums into a sequence with multiple limit points.     

The estimations for maximal operators

In this paper we study the almost everywhere convergence of the spectral expansions related to the Laplace–Beltrami operator on the unit sphere. Using the spectral properties of the functions with logarithmic singularities, the estimate for maximal operator of the Riesz means of the partial sums of the Fourier–Laplace series is established. We have constructed a different method for investigating the summability problems of Fourier–Laplace series, which based on the theory of spectral decompositions of the self-adjoint Laplace–Beltrami operator.     

Trigonometric sums and fresnel-type integrals

If the partial sums of a trigonometric series are non-negative and two additional conditions are satisfied, then the given series is a Fourier series. One of these conditions is analysed here and necessary expansions and numerical values are given.     

Existence of Universal Taylor Series for Nonsimply Connected Domains

It is known that, for any simply connected proper subdomain Ω of the complex plane and any point ζ in Ω, there are holomorphic functions on Ω that possess “universal” Taylor series expansions about ζ; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in ℂ\Ω that have connected complement. This paper shows, for nonsimply connected domains Ω, how issues of capacity, thinness and topology affect the existence of holomorphic functions on Ω that have universal Taylor series expansions about a given point.     

Near-best L1 approximations on circular and elliptical contours

Polynomial approximations are obtained to analytic functions on circular and elliptical contours by forming partial sums of order n of their expansions in Taylor series and Chebyshev series of the second kind, respectively. It is proved that the resulting approximations converge in the L₁ norm as n → ∞, and that they are near-best L₁ approximations within relative distances of the order of log n. Practical implications of the results are discussed, and they are shown to provide a theoretical basis for polynomial approximation methods for the evaluation of indefinite integrals on contours.     

Gibbs Phenomenon of Framelet Expansions and Quasi-projection Approximation

Journal of Fourier Analysis and Applications - The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the nth Fourier partial sums...     

The use of partition polynomial series in Laplace inversion of composite functions with applications in fractional calculus

This paper presents an analytical method towards Laplace transform inversion of composite functions with the aid of Bell polynomial series. The presented results are used to derive the exact solution of fractional distributed order relaxation processes as well as time‐domain impulse response of fractional distributed order operators in new series forms. Evaluation of the obtained series expansions through computer simulations is also given. The results are then used to present novel series expansions for some special functions, including the one‐parameter Mittag‐Leffler function. It is shown that truncating these series expansions when combined with using potential partition polynomials provides efficient approximations for these functions. At the end, the results are shown to be also useful in studying asymptotical behavior of partial Bell polynomials. Numerical simulations as well as analytical examples are provided to verify the results of this paper.     

A Family of Approximation Formulas and some Applications

The general scheme, suggested in [1] using a basis of an infinite-dimensional space and allowing to construct finite-dimensional orthogonal systems and interpolation formulas, is improved in the paper. This results particularly in a generalization of the well-known scheme by which periodic interpolatory wavelets are constructed. A number of systems which do not satisfy all the conditions for multiresolution analysis but have some useful properties are introduced and investigated.

Starting with general constructions in Hilbert spaces, we give a more careful consideration to the case connected with the classic Fourier basis.

Convergence of expansions which are similar to partial sums of the summation method of Fourier series, as well as convergence of interpolation formulas are considered.

Some applications to fast calculation of Fourier coefficients and to solution of integrodifferential equations are given. The corresponding numerical results have been obtained by means of MATHEMATICA 3.0 system.     

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

Journal of Theoretical Probability - In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions....     

H_1(D)空间的Bessel级数

本文给出关于 Ｈ１（Ｄ）空间中函数的 Ｂｅｓｓｅｌ级数的部分和用幂级数的部分和表示的一个恒等式．基于它,可以得到Ｂｅｓｓｅｌ级数部分和偏差的诸多精确估计．     

H1（D）空间的Bessel级数

本文给出关于H1（D）空间中函数的Bessel级数的部分和用幂级的部分和表示的一个恒等式，基于它，可以得到Bessel 级数部分和偏差的诸多精确估计。     

A short note on the generalized Euler transform for the summation of power series

Applying the generalized Euler transform to the firstn partial sums of a power series results in a triangular array whose inferior diagonal givesn approximate values of the sum of this series. The aim of this short note is to estimate the best of thesen values and to compare it with the actual best one for a collection of test series.     

Convergence of Classical Cardinal Series and Band Limited Special Functions

We consider the behavior of symmetric partial sums of the classical cardinal series. Necessary and sufficient conditions for convergence are recorded and a bound on the asymptotic behavior of the limiting function is established. Questions concerning sampling, uniqueness, and the effect of index shifts are also addressed. Examples of certain band limited special functions that can be expressed as limits of symmetric partial sums of classical cardinal series are included.     

Arithmetical properties of some series with logarithmic coefficients

We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler’s constant γ, log and log σ, where σ is Somos’s quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.     

Ramanujan expansions of arithmetic functions of several variables

We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical and unitary Ramanujan sums, respectively, run parallel.     

Limit Curves for Zeros of Sections of Exponential Integrals

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of \(O(n)\) , and after rescaling, we explicitly calculate their limit curve. We find that the rate at which the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings, we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.