## BV function and its property involves translation

**Theorem 1** * Suppose , then if and only if such that*

*Moreover, one can take . Here is the tanslation operator. *

*Proof:* Firstly suppose . Let us prove

To show that

where

it is easy to verify therefore (1) is proved. Next one can choose such with (it is easy to show by mollification). By dominating theorem, one get

The other direction need more analysis.

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