On Periodic Hermite-Birkhoff Interpolation by Translation

Abstract

The equidistant periodic Lagrange interpolation in the space

$$V=span\{{{T}_{{{X}_{j}}}}\left. g \right|0\le j\angle N\}$$

spanned by the translates

$${{T}_{{{x}_{j}}}}g(x):=g(x-{{x}_{j}})$$

of a function g E C_Zn with the N knots

$${{x}_{j}}:=2\pi j/N,0\le j\angle N$$

as considered in the papers of DELVOS [1], KNAUFF and KRESS [2], LOCHER [3] and PRAGER [4], can be solved with the help of a very easy algorithm: The functions ${{B}_{t}}(x)/{{B}_{t}}(0)\in V$ with ${{B}_{t}}(x):=\sum\limits_{j=0}^{N-1}{{{e}_{t}}}({{x}_{j}}){{T}_{{{X}_{j}}}}g(x)$ ,0 ≤ t < N , Interpolate the exponential functions

$${{e}_{t}}(x):={{e}^{itx}}$$

at the N knots x_m = 2πm/N, 0 ≤ m < N. The simple summation

$${{s}_{0}}={{N}^{-1}}\sum\limits_{t=0}^{N-1}{{{B}_{t}}}(x)/{{B}_{t}}(0)\in V$$

of these interpolating functions builds the first fundamental function,

$${{s}_{0}}({{x}_{m}})={{N}^{-1}}\sum\limits_{t=0}^{N-1}{{{e}^{it{{x}_{m}}}}}={{\delta }_{m0}}$$

, 0 ≤ m < N , whose translates

$${{s}_{j}}(x):={{T}_{{{x}_{j}}}}{{s}_{0}}(x):={{s}_{0}}(x-{{x}_{j}})$$

do the rest of the work:

$${{s}_{j}}({{x}_{m}}):={{s}_{0}}({{x}_{m}}-{{x}_{j}})={{\delta }_{j,m}}$$