IJPAM: Volume 49, No. 1 (2008)

PERIODIC APPROXIMATIONS BASED ON SINC

Frank Stenger$^1$, Brandon Baker$^2$, Carl Brewer$^3$, Geoffrey Hunter$^4$,
Sasha Kaputerko$^5$, Jason Shepherd$^6$
School of Computing
University of Utah
Salt lake City, UT, 84112, USA
$^{2,3}$Department of Electrical Engineering
University of Utah
Salt Lake City, UT, 84112, USA
$^4$Department of Mathematics
University of Utah
Salt Lake City, UT, 84112, USA
$^5$Department of Geology and Geophysics
University of Utah
Salt Lake City, UT, 84112, USA
$^6$Scientific Computing and Imaging Institute
University of Utah
Salt Lake City, UT, 84112, USA


Abstract.In this paper we derive some novel formulas for interpolating functions that are periodic with period $T$ on $\R = \{x : -\infty < x < \infty\}$. These formulas are all based on the Whittaker Cardinal series expansion. Let $N$ be a positive integer. If the spacing $h$ of this interpolatory expansion is defined by $h = T/(2\,N)$, then the infinite Cardinal series reduces to a Fourier interpolation polynomial, which is obtainable by interpolation with the Dirichlet kernel,

\begin{displaymath} D_e(N,T,x) = \dis \frac{\sin\{2\,N\,\pi \, x/T\}}{2\,N\,\tan\left\{\pi\,x/T\right\}}\,. \end{displaymath}

On the other hand, if the spacing $h$ of this interpolatory expansion is defined by $h = T/(2\,N+1)$, then the infinite Cardinal series reduces to a Fourier interpolation polynomial, which is obtainable by interpolation with the Dirichlet kernel,

\begin{displaymath} D_o(N,T,x) = \frac{\sin\{(2\,N+1)\,\pi \, x/T\}}{(2\,N+1)\,\sin\left\{\pi\,x/T\right\}}\,. \end{displaymath}

These results show that Fourier polynomials are a special case of Cardinal expansions.

Two standard families of approximations are thus obtainable, one, starting with Cardinal interpolation at the points $\{k\,h\,: k \in \Z\}$, and the other, starting with Cardinal interpolation at the points $\{(k + 1/2)\,h\,: k \in \Z\}$. In this way the well known formulas of e.g., the trapezoidal rule over the real line, reduce to the trapezoidal rule over $[0,T]$, and similarly for the midordinate rule.

The coefficients of each type of expansion are point evaluations of functions to be approximated, i.e., we differ from Fourier polynomial approximations in that no computations are required for obtaining the Fourier approximations.

We then also derive some relations with polynomials in $y$ via use of the transformation $y = \cos(2\,\pi\,x/T)$. It thus follows that algebraic polynomials are a special case of Fourier polynomials.

We give some comparative examples of approximations of smooth periodic functions and discontinuous functions via both our periodic basis as well as with corresponding polynomial approximations.

Received: August 19, 2008

AMS Subject Classification: 41A05

Key Words and Phrases: interpolating functions, algebraic polynomials, Fourier polynomials, Cardinal expansions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 49
Issue: 1