then converges absolutely for every t and on the other hand, it is enough that converges absolutely for even one t, then this condition holds. In other words, for absolute convergence there is no issue of where the sum converges absolutely — if it converges absolutely at one point then it does so everywhere. The family of all functions with absolutely converging Fourier series is a Banach algebra (the operation of multiplication in the algebra is a simple multiplication of functions). It is called the Wiener algebra, after Norbert Wiener, who proved that if ƒ has absolutely converging Fourier series and is never zero, then 1/ƒ has absolutely converging Fourier series. The original proof of Wiener's theorem was difficult; a simplification using the theory of Banach algebras was given by Israel Gelfand. Finally, a short elementary proof was given by Donald J. Newman in 1975. If belongs to a α-Hölder class for α > 1/2 then for the constant in the Hölder condition, a constant only dependent on ; is the norm of the Krein algebra. Notice that the 1/2 here is essential—there are 1/2-Hölder functions, which do not belong to the Wiener algebra. Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only and then not summable. If ƒ is of bounded variation and belongs to a α-Hölder class for some α > 0, it belongs to the Wiener algebra. Norm convergence The simplest case is that of L2, which is a direct transcription of general Hilbert space results. According to the Riesz–Fischer theorem, if ƒ is square-integrable then i.e., converges to ƒ in the norm of L2. It is easy to see that the converse is also true: if the limit above is zero, ƒ must be in L2. So this is an if and only if condition. If 2 in the exponents above is replaced with some p, the question becomes much harder. It turns out that the convergence still holds if 1 < p < ∞. In other words, for ƒ in Lp, converges to ƒ in the Lp norm. The original proof uses properties of holomorphic functions and Hardy spaces, and another proof, due to Salomon Bochner relies upon the Riesz–Thorin interpolation theorem. For p = 1 and infinity, the result is not true. The construction of an example of divergence in L1