An Interpolation Method That Minimizes an Energy Integral of Fractional Order

Abstract

An interpolation method that minimizes an energy integral will be discussed. To be precise, given N + 1 points (x ₀,c ₀), (x ₁,c ₁),..., (x _N ,c _N ) with 0 = x ₀ < x ₁ < ⋯ < x _N  = 1 and c ₀ = c _N  = 0, we shall be interested in finding a sufficiently smooth function u on [0,1] that passes through these N + 1 points and minimizes the energy integral $E_\alpha(u) := \int_0^1 |u^{(\alpha)}(x)|^2 dx$, where u ^(α) denotes the fractional derivative of u of order α. As suggested in [1], a Fourier series approach as well as functional analysis arguments can be used to show that such a function exists and is unique. An iterative procedure to obtain the function will be presented and some examples will be given here.