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Wiener then asked if the analogue of his theorem were true for H1 [l]. (That is, whether the square in the integrals of Theorem A may be replaced by the first power.) This question has added interest now because Boas [2 ], [3 ] has shown that if a function with positive Fourier series coefficients is "good" in a neighborhood of zero, then the function is "good" everywhere for several senses of the word "good." In this note we point out that a result of Hardy implies a negative answer to Wiener's question, and at the same time gives an example of a trigonometric series with positive coefficients whose behaviour is good near the origin but not everywhere. Theorem B. Let 0<p<2. Then there exists

A problem of Wiener and the failure of a principle for Fourier series with positive coefficients