Related papers: Eliminating oscillation in partial sum approximati…
One of the major challenges in finite element methods is the mitigation of spurious oscillations near sharp layers and discontinuities known as the Gibbs phenomenon. In this article, we propose a set of functionals to identify spurious…
Functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on…
We introduce a small change in the definition of the Fourier series so that we can guarantee the coincidence with the given function at the endpoints of the interval even if the function does not assume the same value at the endpoints. This…
The suitable basis functions for approximating periodic function are periodic, trigonometric functions. When the function is not periodic, a viable alternative is to consider polynomials as basis functions. In this paper we will point out…
The error term in the approximate functional equation for exponential sums involving the divisor function will be improved under certain conditions for the parameters of the approximate functional equation.
Approximate $p$-point Leibniz derivation formulas as well as interpolatory Simpson quadrature sums adapted to oscillatory functions are discussed. Both theoretical considerations and numerical evidence concerning the dependence of the…
We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to…
The fractional Fourier series generalizes the classical Fourier series by introducing a rotation angle $\alpha$ in the time-frequency plane, but inherits the Gibbs phenomenon for piecewise smooth functions. Unlike the classical setting, the…
Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighbourhoods of the endpoints. Fourier extensions circumvent…
We consider the class GM(2b) in pointwise estimate of the deviations in strong mean of almost periodic functions from matrix means of partial sums of their Fourier series.
In statistical mechanics, the generally called Stirling approximation is actually an approximation of Stirling's formula. In this article, it is shown that the term that is dropped is in fact the one that takes fluctuations into account.…
One reason why standard formulations of the central limit theorems are not applicable in high-dimensional and non-stationary regimes is the lack of a suitable limit object. Instead, suitable distributional approximations can be used, where…
Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier…
We propose a general method for optimization with semi-infinite constraints that involve a linear combination of functions, focusing on the case of the exponential function. Each function is lower and upper bounded on sub-intervals by…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
If a function $f:\mathbb{R}\to\mathbb{R}$ can be represented as the sum of $n$ periodic functions as $f=f_1+\dots+f_n$ with $f(x+\alpha_j)=f(x)$ ($j=1,\dots,n$), then it also satisfies a corresponding $n$-order difference equation…
The main purpose of this work is to ascertain when arithmetic operations with periodic functions whose domains may not coincide with the whole real line preserve periodicity.
The Gibbs phenomenon is widely known for Fourier expansions of periodic functions and refers to the phenomenon that the $n$th Fourier partial sums overshoot a target function at jump discontinuities in such a way that such overshoots do not…
We derive and prove an explicit formula for the sum of the fractional parts of certain geometric series. Although the proof is straightforward, we have been unable to locate any reference to this result. This summation formula allows us to…
The Fourier-based analysis customarily employed to analyze the dynamics of a simple pendulum is here revisited to propose an elementary iterative scheme aimed at generating a sequence of analytical approximants of the exact law of motion.…