Two algorithms for periodic extension on uniform grids

Abstract

Given function values on a uniform grid in a domain Ω in $\mathbb {R}^{d}$, one is often interested in extending the values to a larger grid on a box B containing Ω. In particular, we are interested in “periodic extensions.” For such extensions the discrete Fourier transform (DFT) of the resulting grid values on B is expected to provide good efficient approximation to the underlying function on Ω. This paper presents two different extension algorithms. The first method is a natural approach to this problem, aiming at achieving the fastest decay of the DFT coefficients of the extended data.The second is a fast algorithm which is appropriate for the univariate case and for limited cases of multivariate scenarios. It is shown that if a “good” periodic extension exists, the proposed method will find an extension with similar properties.

Appendix

1.1 A.1 Decay rate of discrete Fourier coefficients for periodic C ^M functions

In the theory of classical Fourier series, it is known that the Fourier coefficients {c_k} of a C^M periodic function decay as follows:

$$c_{k}=O(k^{-M})\ \ \ \ \text{as}\ k \to \infty. $$

Using the equality

$$\big(\frac{-T}{2\pi \text{ik}}e^{-2\pi i\frac{k}{T}x}\big)'=e^{-2\pi i\frac{k}{T}x}\ ,$$

the above result is derived by applying M-times integration by part to the integral

$$c_{k}=\frac{1}{T}{\int}_{-\frac{T}{2}}^{\frac{T}{2}}f(x)e^{-2\pi i\frac{k}{T}x}\text{dx}\ .$$

Considering discrete fourier transform, we use the less known “summation by parts” formula:

$$ \sum\limits_{n=m_{1}}^{m_{2}}f_{n}{\Delta} g_{n}=(f_{m_{2}}g_{m_{2}+1}-f_{m_{1}}g_{m_{1}})-\sum\limits_{n=m_{1}}^{m_{2}-1}g_{n+1}{\Delta} f_{n}. $$

(A.1)

In our case m₁ = 0 and m₂ = N − 1, and assuming periodicity, f_N = f₀ and g_N = g₀, we have as follows:

$$ \sum\limits_{n=0}^{N-1}f_{n}{\Delta} g_{n}=-\sum\limits_{n=0}^{N-1}g_{n+1}{\Delta} f_{n}. $$

(A.2)

Using the above equality for the univariate discrete Fourier transform, we get, with f_n = f(2nπ/N), n ∈N,

$$ \begin{array}{@{}rcl@{}} c_{k} &=& \sum\limits_{n=0}^{N-1}f_{n}e^{-\frac{2\pi i}{N}kn} \end{array} $$

(A.3)

$$ \begin{array}{@{}rcl@{}} & =& \frac{1}{e^{-\frac{2\pi ik}{N}}-1} \sum\limits_{n=0}^{N-1}f_{n}{\Delta} e^{-\frac{2\pi i}{N}kn} \end{array} $$

(A.4)

$$ \begin{array}{@{}rcl@{}} & =& -\frac{1}{e^{-\frac{2\pi ik}{N}}-1} \sum\limits_{n=0}^{N-1}{\Delta} f_{n} e^{-\frac{2\pi i}{N}k(n+1)}. \end{array} $$

(A.5)

Assuming f ∈ C¹, we have |Δf_n| ≤ C₁N^− 1, and thus, we get the following:

$$ |c_{k}|\le C_{1} |e^{-\frac{2\pi ik}{N}}-1|^{-1}. $$

(A.6)

Assuming f ∈ C^M, we have |Δ^Mf_n| ≤ C_MN^−M, and by applying the summation by parts M times, we obtain the following:

$$ \begin{array}{@{}rcl@{}} |c_{k}| &=& |e^{-\frac{2\pi ik}{N}}-1|^{-M}\sum\limits_{n=0}^{N-1}{\Delta}^{M}f_{n}\ e^{-\frac{2\pi i}{N}kn} \end{array} $$

(A.7)

$$ \begin{array}{@{}rcl@{}} & \le & C_{M} N^{-M+1}|e^{-\frac{2\pi ik}{N}}-1|^{-M}. \end{array} $$

(A.8)

In the d-dimensional case, by performing summation by parts through each of the d-directions, we obtain the following:

$$ |c_{\mathbf{k}}|\le C N^{-(M-1)d}\prod\limits_{j=1}^{d}|e^{-\frac{2\pi ik_{j}}{N}}-1|^{-M}. $$

(A.9)