(OST property 2) Let be OST. Then iff The closure is taken in .

of . Suppose takes place. According to the proposition ( Main property of Fourier decomposition ), We take Fourier transform of both sides: and use the formula ( Property of scale and transport 4 ): Thus we have with . We have by the proposition ( Parseval equality ).

of . Suppose takes place. Then we perform the inverse Fourier transform and the above calculations performed in reverse order lead to .