床関数と天井関数
⌊ x ⌋ = x − 1 2 + 1 π ∑ k = 1 ∞ sin  ( 2 π k x ) k . {\displaystyle \lfloor x\rfloor =x-{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.} ⌈ x ⌉ = x + 1 2 + 1 π ∑ k = 1 ∞ sin  ( 2 π k x ) k . {\displaystyle \lceil x\rceil =x+{\frac {1}{2}}+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.}
床関数と天井関数の平均は次のようにフーリエ級数展開できる。 1 2 ( ⌊ x ⌋ + ⌈ x ⌉ ) = x + 1 π ∑ k = 1 ∞ sin  ( 2 π k x ) k . {\displaystyle {\frac {1}{2}}\left(\lfloor x\rfloor +\lceil x\rceil \right)=x+{\frac {1}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin(2\pi kx)}{k}}.}