On Fourier coefficients of functions continued evenly or oddly

Consideration is given to the question of how the orders of decrease of the Fourier coefficients of the two functions $F$ and $G$ are related, where $F$ and $G$ are, respectively, the odd and the even continuations onto the interval $[-\pi,0]$ of the function $f(x)\in L(0,\pi)$. It is shown that in passing from the one continuation to the other, an arbitrary rate of decrease of the Fourier coefficients is worsened, the degree of worsening depending on their initial rate of decrease.