Function Approximation 1 Interpolation

Interpolation is a form of function approximation in which the approximating function (interpolant) and the underlying function must agree at a finite number of points. In some cases additional restrictions may be imposed on the interpolant. For example its first derivative evaluated at a finite number of points may have to agree with that of the underlying function. Other examples include additional constraints imposed on the interpolant such as monotonicity, convexity, or smoothness requirements. A choice must be made about which family of functions the interpolant is a member of. Some families of functions commonly used for interpolation include: • Polynomials (Polynomial Interpolation) • Trigonometric functions (Fourier Approximation) • Rational functions (Pade Approximation) Suppose we are interested in approximating some real-valued function f . The interpolant, f̃ , is chosen to be a linear combination of some set of basis functions, φ1(x), . . . , φn(x). The basis functions are linearly independent and span the family of functions chosen for the interpolation (any function in the family can be written as a linear combination of basis functions). Thus we have, f̃(x) ≡ n ∑