On convergence of Fourier series of functions of generalized bounded variation

Daniel Waterman

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