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I have a trigonometric function series

$$f(x)={1 \over 2}{\Lambda _0} + \sum\limits_{l = 1}^\infty {{\Lambda _l}\cos \left( {lx} \right)} $$

with the normalization condition

$$\Lambda_0 + 2\sum\limits_{l = 1}^\infty {{\Lambda _l} = 1} $$

and ##\Lambda_l## being monotonic decrescent weights, i.e. ##\Lambda_0>\Lambda_1>\Lambda_2...##

Clearly from these two latter conditions one can prove ##f(x)## exists, but my problem is to characterise this function. Which theorems do you suggest to be helpful to this aim?