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Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

We show that, if the coefficients (an) in a series a 0 / 2 + n = 1 a n c o s ( n t ) tend to 0 as n → ∞ and satisfy the regularity condition that m = 0 j = 1 [ n = j 2 m ( j + 1 ) 2 m - 1 | a n - a n + 1 | ] ² 1 / 2 < , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series n = 1 b n s i n ( n t ) tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if n = 1 | b n | / n < . These conclusions were previously known to hold under stronger restrictions on the sizes of the differences...

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