## Displaying similar documents to “Some theorems on orthogonal systems”

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We show that, if the coefficients (an) in a series ${a}_{0}/2+{\sum }_{n=1}^{\infty }{a}_{n}cos\left(nt\right)$ tend to 0 as n → ∞ and satisfy the regularity condition that ${\sum }_{m=0}^{\infty }{{\sum }_{j=1}^{\infty }\left[{\sum }_{n=j{2}^{m}}^{\left(j+1\right){2}^{m}-1}|{a}_{n}-{a}_{n+1}|\right]²}^{1/2}<\infty$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series ${\sum }_{n=1}^{\infty }{b}_{n}sin\left(nt\right)$ 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...

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