Foundations of Discrete Harmonic Analysis by Vasily N. Malozemov & Sergey M. Masharsky

Author:Vasily N. Malozemov & Sergey M. Masharsky
Language: eng
Format: epub
ISBN: 9783030470487
Publisher: Springer International Publishing

3.12 Second Limit Theorem

3.12.1

Now we turn to examination of limit behavior of the interpolating spline in a case of (and fixed r and m). For this end we present in the following form:

(3.12.1)

where is a normalized B-spline defined by the formula

By virtue of Lemma 3.3.2,

Taking into account non-negativity of we conclude that the values of belong to the segment [0, 1].

Let us find out how the normalized B-spline behaves when n is growing.

3.12.2

We introduce a continuous periodic B-spline of the first order. We define it as an m-periodic function of a real argument that is determined on the main period [0, m] by the following formula:

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