# IJPAM: Volume 49, No. 1 (2008)

**PERIODIC APPROXIMATIONS BASED ON SINC**

Sasha Kaputerko, Jason Shepherd

School of Computing

University of Utah

Salt lake City, UT, 84112, USA

Department of Electrical Engineering

University of Utah

Salt Lake City, UT, 84112, USA

Department of Mathematics

University of Utah

Salt Lake City, UT, 84112, USA

Department of Geology and Geophysics

University of Utah

Salt Lake City, UT, 84112, USA

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 on
. These formulas are all based on the Whittaker Cardinal
series expansion. Let be a positive integer. If the spacing
of this interpolatory expansion is defined by , then the
infinite Cardinal series reduces to a Fourier interpolation polynomial,
which is obtainable by interpolation with the Dirichlet kernel,

On the other hand, if the spacing of this interpolatory expansion is defined by , then the infinite Cardinal series reduces to a Fourier interpolation polynomial, which is obtainable by interpolation with the Dirichlet kernel,

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 , and the other, starting with Cardinal interpolation at the points . In this way the well known formulas of e.g., the trapezoidal rule over the real line, reduce to the trapezoidal rule over , 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 via use of the transformation . 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