The sinc functions are defined as S0(x)=(sin(πx / h))/(πx / h), while Sj(x)=S0(x-j h) whenever h

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The sinc functions are defined as S0(x)=(sin(πx / h))/(πx /...

The sinc functions are defined as $S_0(x)=\frac{\sin (\pi x / h)}{\pi x / h}$, while $S_j(x)=S_0(x-j h)$ whenever $h>0$ and $j$ is an integer. We will interpolate a function $f(x)$ at the mesh points $x_j=j h, j=0, \ldots, n$, by a linear combination of sinc functions: $S(x)=$ $c_0 S_0(x)+\cdots+c_n S_n(x)$. What are the coefficients $c_j$ ? Graph and discuss the accuracy of the sinc interpolant for the functions $x^2$ and $\frac{1}{2}-\left|x-\frac{1}{2}\right|$ on the interval $[0,1]$ using $h=.25, .1$, and .025 .

Key Concepts

Sinc Function

The sinc function, defined as sin(?x/h) divided by (?x/h), is central to many interpolation and signal processing methods. Its most important property is that it equals 1 when x is 0 and 0 at nonzero integer multiples of the scaling factor, thus serving as an idealized interpolation kernel for bandlimited functions.

Cardinal (Shifted) Functions

By shifting the sinc function appropriately, a set of cardinal functions is obtained, where each function is centered at a mesh point x = jh. These functions have the property that each one takes the value 1 at its own center and 0 at all other mesh points. This unique feature significantly simplifies the interpolation process, as each coefficient can be associated directly with a function value at the corresponding grid point.

Interpolation Coefficients

Due to the cardinal property of the shifted sinc (or cardinal) functions, the coefficients in a sinc interpolant are obtained simply by evaluating the target function at the interpolation (mesh) points. In other words, the coefficient c_j is equal to f(x_j). This direct relationship is a hallmark of interpolation methods that use basis functions with the Kronecker delta property.

Mesh Points and Sampling Parameter

The placement of sample points, or mesh points, is crucial in interpolation. With a spacing h between them, the sampling parameter h controls the density of the points on the interval and influences the accuracy of the reconstructed function. A smaller h results in a denser grid and typically yields a more accurate interpolant, assuming the function is sufficiently smooth.

Accuracy and Error Analysis in Sinc Interpolation

The performance of the sinc interpolant depends on both the chosen mesh spacing and the nature of the function being approximated. For smooth or analytic functions, such as a quadratic, sinc interpolation tends to converge rapidly with decreasing h. However, for functions with lower smoothness or discontinuities in derivatives, like piecewise linear functions, the approximation may be less accurate or exhibit slower convergence. Graphical comparisons of the interpolant and the original function across various values of h help in assessing and discussing these error characteristics.

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