Pointwise and Uniform Convergence of Fourier Extensions

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 neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Existence of Ψ-bounded Solutions for System of Linear Dynamic equations on Time Scales

In this work, we develop the criteria for existence of Ψ- bounded solutions of system of linear dynamic equations on time scales. The advantage of results in this dynamical system is it unifies discrete as well as continuous systems. Initially, we develop if and only if conditions for the existence of at least one Ψ-bounded solution for linear dynamic equation y∆(τ ) = P (τ )y +g(τ ), for each Ψ- delta integrable Lebesgue function g, on time scale T +. Later, we obtain asymptotic nature of Ψ-bounded solutions of dynamical system. Also we provided the examples for supporting the results.AMS Subject Classification: 74H20, 34N05, 34C11

Lebesgue Constants and Optimal Node Systems via Symbolic Computations

Polynomial interpolation is a classical method to approximatecontinuous functions by polynomials. To measure the correctness of theapproximation, Lebesgue constants are introduced. For a given node system $X^{(n+1)}=\{x_1<\ldots<x_{n+1}\}\, (x_j\in [a,b])$, the Lebesgue function $\lambda_n(x)$ is the sum of the modulus of the Lagrange basis polynomials built on $X^{(n+1)}$. The Lebesgue constant $\Lambda_n$ assigned to the function $\lambda_n(x)$ is its maximum over $[a,b]$. The Lebesgue constant bounds the interpolation error, i.e., the interpolation polynomial is at most $(1+\Lambda_n)$ times worse then the best approximation.The minimum of the $\Lambda_n$'s for fixed $n$ and interval $[a,b]$ is called the optimal Lebesgue constant $\Lambda_n^*$.For specific interpolation node systems such as the equidistant system, numerical results for the Lebesgue constants $\Lambda_n$ and their asymptoticbehavior are known \cite{3,7}. However, to give explicit symbolic expression for the minimal Lebesgue constant $\Lambda_n^*$ is computationally difficult. In this work, motivated by Rack \cite{5,6}, we are interested for expressing the minimalLebesgue constants symbolically on $[-1,1]$ and we are also looking for thecharacterization of the those node systems which realize theminimal Lebesgue constants. We exploited the equioscillation property of the Lebesgue function \cite{4} andused quantifier elimination and Groebner Basis as tools \cite{1,2}. Most of the computation is done in Mathematica \cite{8}.