# Basic Properties and Concept of Selected Subsequence of Zero Based Finite Sequences

Yatsuka Nakamura 1  and Hisashi Ito 1
• 1 Shinshu University, Nagano, Japan

## Basic Properties and Concept of Selected Subsequence of Zero Based Finite Sequences

Here, we develop the theory of zero based finite sequences, which are sometimes, more useful in applications than normal one based finite sequences. The fundamental function Sgm is introduced as well as in case of normal finite sequences and other notions are also introduced. However, many theorems are a modification of old theorems of normal finite sequences, they are basically important and are necessary for applications. A new concept of selected subsequence is introduced. This concept came from the individual Ergodic theorem (see ) and it is the preparation for its proof.

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