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Convergence of series of translations

Authors:	Joseph Rosenblatt

Institution:	(1) Mathematics Department, Ohio State University, 43210 Columbus, Ohio, USA

Abstract:	For a mean zero norm one sequence (f _n)L ₂0, 1], the sequence (f _n{nx+y}) is an orthonormal sequence inL ₂(0, 1]²); so if $\sum\limits_{n = 1}^\infty {\left\| {c_n } \right\|^2 \log ^2 } n< \infty$ , then $\sum\limits_{n = 1}^\infty {c_n f_n \{ nx + y\} }$ converges for a.e. (x, y)0, 1]² and has a maximal function inL ₂(0, 1]²). But for a mean zerofL ₂0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of $\sum\limits_{n = 1}^\infty {c_n f_n \{ nx\} }$ . Ifc _n0 and $\sum\limits_{n = 1}^\infty {c_n = \infty }$ , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂0, 1]. Also, there are mean zerofL0, 1] such that $\sum\limits_{n = 1}^\infty {({1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n})} f\{ nx\}$ never converges and there is a mean zero continuous functionf with $\mathop {sup}\limits_N \left\| {\sum\limits_{n = 1}^N {(\log {n \mathord{\left/ {\vphantom {n n}} \right. \kern-\nulldelimiterspace} n})} f \{ nx\} } \right\| = \infty$ a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if \|c _n\| = 0(n ^–) for >1/2, then $\sum\limits_{n = 1}^\infty {c_n } f\{ nx\}$ converges a.e. and unconditionally inL ₂0, 1]. In addition, for any mean zerof of bounded variation, the series $\sum\limits_{n = 1}^\infty {({1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n})} f\{ nx\}$ has its maximal function in allL _p0, 1] with 1p<. Finally, if (f _n)L 0, 1] is a uniformly bounded mean zero sequence, then $\sum\limits_{n = 1}^\infty {\left\\| {f_n } \right\\|_2^2 }< \infty$ is a necessary and sufficient condition for $\sum\limits_{n = 1}^\infty {f_n \{ x_n + y\} }$ to converge for a.e.y and a.e. (x _n)0, 1]. Moreover, iffL 0, 1] is mean zero and $\sum\limits_{n = 1}^\infty {\left\| {c_n } \right\|^2 }< \infty$ , then for a.e. (x _n)0, 1], $\sum\limits_{n = 1}^\infty {c_n f} \{ x_n + y\}$ converges for a.e.y and in allL _p0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides 0, 1] under addition modulo one.

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