Let \(W_0(\mathbb {R})\) be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proved in the paper that, in particular, a trigonometric series \(\sum \nolimits _{k=-\infty }^\infty c_k e^{ikt}\) is the Fourier series of an integrable function if and only if there exists a \(\phi \in W_0(\mathbb {R})\) such that \(\phi (k)=c_k\), \(k\in \mathbb {Z}\). If \(f\in W_0(\mathbb {R})\), then the piecewise linear continuous function \(\ell _f\) defined by \(\ell _f(k)=f(k)\), \(k\in \mathbb {Z}\), belongs to \(W_0(\mathbb {R})\) as well. Moreover, \(\Vert \ell _f\Vert _{W_0}\le \Vert f\Vert _{W_0}\). Similar relations are established for more advanced Wiener algebras. These results are supplemented by numerous applications. In particular, new necessary and sufficient conditions are proved for a trigonometric series to be a Fourier series and new properties of \(W_0\) are established.

令\（W_0（\ mathbb {R}）\）为可由Lebesgue可积函数的Fourier积分表示的函数的Wiener Banach代数。在本文中证明，尤其是，当且仅当一个三角函数\（\ sum \ nolimits _ {k =-\ infty} ^ \ infty c_k e ^ {ikt} \）的傅里叶级数如果存在\（W_0（\ mathbb {R}）中的\ phi \），使得\（\ phi（k）= c_k \），\（\ mathbb {Z}中的k \）。如果\（f在W_0（\ mathbb {R}）\）中，则分段线性连续函数\（\ ell _f \）由\（\ ell _f（k）= f（k）\）定义，\（ k \ in \ mathbb {Z} \）属于\（W_0（\ mathbb {R}）\）也一样 此外，\（\ Vert \ ell _f \ Vert _ {W_0} \ le \ Vert f \ Vert _ {W_0} \）。对于更高级的维纳代数也建立了类似的关系。这些结果得到了众多应用的补充。特别地，证明了三角级数成为傅立叶级数的新的必要条件和充分条件，并建立了\（W_0 \）的新性质。