Lemma 3.7 . [042I]

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Lemma 3.7.

The series defining $\tilde{\beta}_{i}$ converge for $\eta\notin\mathbb{Z}$ , and $\tilde{\beta}_{i}$ are 1-periodic in $\eta$ . Morever if $|x|\leq\frac{1}{2}$ , then

|\tilde{\beta}_{i}-\beta_{i}|\leq C(|\eta|+\frac{|\mu_{1}|+|\mu_{2}|}{A^{1/4}}).