Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on d-variate cubes. Such approximations offer a number of advantages, including uniform convergence. However, like Fourier series, the rate of convergence is typically slow. In this paper we extend Eckhoff's method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions. In the univariate setting it is known that Eckhoff's method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set.

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Convergence acceleration of modified Fourier series in one or more dimensions

The current paper considers the problem of recovering a function using a lim- ited number of its Fourier coefficients. Specifically, a method based on Bernoulli-like poly- nomials suggested and developed by Krylov, Lanczos, Gottlieb and Eckhoff is examined. Asymptotic behavior of approximate calculation of the so-called "jumps" is studied and asymptotic L2 constants of the rate of convergenceof the method are computed.

Asymptotic behavior of eckhoff's method for fourier series convergence acceleration

Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of 1/n

Modified Fourier expansions present an alternative to more standard algorithms for the approximation of nonperiodic functions in bounded domains. This thesis addresses the theory of such expansions, their effective construction and computation, and their application to the numerical solution of partial differential equations. As the name indicates, modified Fourier expansions are closely related to classical Fourier series. The latter are naturally defined in the d-variate cube, and, in an analogous fashion, we primarily study modified Fourier expansions in this domain. However, whilst Fourier coefficients are commonly computed with the Fast Fourier Transform (FFT), we use modern numerical quadratures instead. In contrast to the FFT, such schemes are adaptive, leading to great potential savings in computational cost. Standard algorithms for the approximation of nonperiodic functions in d-variate cubes exhibit complexities that grow exponentially with dimension. The aforementioned quadratures permit the design of approximations based on modified Fourier expansions that do not possess this feature. Consequently, such schemes are increasingly effective in higher dimensions. When applied to the numerical solution of boundary value problems, such savings in computational cost impart benefits over more commonly used polynomial-based methods. Moreover, regardless of the dimensionality of the problem, modified Fourier methods lead to well-conditioned matrices and corresponding linear systems that can be solved cheaply with standard iterative techniques. The theoretical component of this thesis furnishes modified Fourier expansions with a convergence analysis in arbitrary dimensions. In particular, we prove uniform convergence of modified Fourier expansions under rather general conditions. Furthermore, it is known that the notion of modified Fourier expansions can be effectively generalised, resulting in a family of approximation bases sharing many of the features of the modified Fourier case. The purpose of such a generalisation is to obtain both faster rates and higher degrees of convergence. Having detailed the approximation-theoretic properties of modified Fourier expansions, we extend this analysis to the general case and thereby verify this improvement. A central drawback of these expansions is that their convergence rate is both fixed and typically slow. This makes the construction of effective convergence acceleration techniques imperative. In the final part of this thesis, we design and analyse a robust method, applicable in arbitrary numbers of dimensions, for accelerating convergence of modified Fourier expansions. When employed in the approximation of multivariate functions, this culminates in efficient, high-order approximants comprising relatively small numbers of terms.

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Modified Fourier expansions: theory, construction and applications

We will apply a fixed point method for proving the Hyers-Ulam stability of the functional equationf(x +y) = f(x)f(y) f(x)+f(y) .

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The Australian Journal of Mathematical Analysis and Applications

The Fourier series of a smooth function on a compact interval usually has slow convergence due to the Gibbs phenomena. A class of Fourier-Pade approximations is introduced and studied for performing a boundary correction. Additional acceleration is achieved by applying Fourier---Bernoulli scheme.

On a Rational Linear Approximation of Fourier Series for Smooth Functions

A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.

Accelerating the convergence of trigonometric series

Hereby, the convergence acceleration of two-dimensional trigonometric inter- polation for a smooth functions on a uniform mesh is considered. Together with theoretical estimates some numerical results are presented and discussed that reveal the potential of this method for application in image processing. Experiments show that suggested al- gorithm allows acceleration of conventional Fourier interpolation even for sparse meshes that can lead to an e-cient image compression/decompression algorithms and also to applications in image zooming procedures.

The Convergence Acceleration of Two-Dimensional Fourier Interpolation

This paper is devoted to the acceleration of the convergence of the partial sums of the classical Fourier series for the sufficiently smooth functions. 

Some universal and adaptive algorithms are constructed and studied. It is shown that the use of a finite number of Fourier coefficients makes it possible exact approximation of a given function from an infinite-dimensional set of quasi-polynomials. In this sense, we call the corresponding essentially nonlinear algorithms as over-convergent. 

The proposed algorithms are implemented using Wolfram Mathematica system. Numerical results demonstrate their effectiveness.

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On an Over-Convergence Phenomenon for Fourier series. Basic Approach.

In the proposed paper, some last autor’s results of studies devoted to the acceleration of the convergence of truncated Fourier series is presented. The corresponding universal (traditional) and special adaptive algorithms are constructed. The main result (the phenomenon of over-convergence for an non-linear adaptive algorithm) states that the use of finite Fourier coefficients leads to an exact approximation for functions from certain infinite-dimensional spaces of quasipolynomials. The corresponding summation formula of truncated Fourier series for smooth functions has unprecedented accuracy.

Anry Nersessian

Papers

On a Rational Linear Approximation of Fourier Series for Smooth Functions

Accelerating the convergence of trigonometric series

The Convergence Acceleration of Two-Dimensional Fourier Interpolation

On an Over-Convergence Phenomenon for Fourier series. Basic Approach.

Fourier Tools are Much More Powerful than Commonly Thought