ISSN:
1432-1807
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract For a mean zero norm one sequence (f n )⊂L 2[0, 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 zerof∈L 2[0, 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 n ≧0 and $$\sum\limits_{n = 1}^\infty {c_n = \infty } $$ , then this series will not converge inL 2-norm on a denseG δ subset of the mean zero functions inL 2[0, 1]. Also, there are mean zerof∈L∞[0, 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(α) with 1/2〈α≦1, and if |c n | = 0(n −δ) for δ〉1/2, then $$\sum\limits_{n = 1}^\infty {c_n } f\{ nx\} $$ converges a.e. and unconditionally inL 2[0, 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 p[0, 1] with 1≦p〈∞. Finally, if (f n )⊂L δ[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, iff∈L δ[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 p [0, 1] with 1≦p〈∞. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01367579
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