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This is a notation question: In R. Duffin, “The Reciprocal of a Fourier Series,” Proc. Am. Math. Soc., vol. 13, no. 6, pp. 965–970, 1962. After Eq. 1, the author says "The Fourier coefficients of a function $F\left(x\right) \in L$ are denoted by $f_j$..."

My interpretation of this statement was that $F\left(x\right)$ is continuous, and has a continuous first derivative (but may not necessarily have a continuous second derivative), is this correct? Or does it refer to the function $F\left(x\right)$ being absolutely integrable, i.e. $\int_{-\infty}^{\infty} \left|F\left(x\right)\right|\text{d}x<\infty$? Any help on interpreting this notation would be very helpful. Sorry that the original post was not very clear, and thanks for the encouragement to improve it.

Martin Sleziak
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okj
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    Read it out loud? It means $F(x)$ is an element of $L$. – M.B. Nov 19 '13 at 22:11
  • @M.B.: True, but not very helpful. What is $L$? Does it mean that $F\left(x\right)$ is continuous? Or has a continuous first derivative? – okj Nov 19 '13 at 22:17
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    Perhaps you should give some context... – Zhen Lin Nov 19 '13 at 22:17
  • @ZhenLin: It comes from a paper that says: "Suppose $F\left(x\right) \in L$ and $G\left(x\right) \in L^2$..." – okj Nov 19 '13 at 22:19
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    This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level. – Did Nov 19 '13 at 22:20
  • @All: See the improved and expanded version of the question, thank you. – okj Nov 19 '13 at 22:32
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    "Suppose $F(x) \in L$ and $G(x) \in L^2$" makes me suspect that $L = L^1$. But don't just take my word for it. – Daniel Fischer Nov 19 '13 at 22:37
  • @DanielFischer: Thank you, could you tell me the meaning of $L^1$ in your comment? – okj Nov 19 '13 at 22:39
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    $L^p$ is the space of $p$-integrable functions, measurable functions such that $\int \lvert f\rvert^p < \infty$. So $L^1$ is the space of absolutely integrable functions. – Daniel Fischer Nov 19 '13 at 22:41
  • @DanielFischer: Perfect, thank you. If you submit your comment as an answer I will gladly accept it. – okj Nov 19 '13 at 22:48
  • Well, the problem is that I don't know if that's the right interpretation. – Daniel Fischer Nov 19 '13 at 22:50
  • To those willing to read from outside of MIT: Read it at AMS – AlexR Nov 19 '13 at 23:17

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Looking a bit more downwards, at page 967 a proof is made, that $F\in L^1$ and the conclusion says $F\in L$ so this should definately mean $L=L^1$. Also, $L^1$ as a space of definition of the FT makes sense in itself.

AlexR
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