Lacunary system

of order $ p> 2 $, $ S _ {p} $- system

An orthonormal system of functions $ \{ \phi _ {n} \} _ {n=} 1 ^ \infty $ of the space $ L _ {p} $ such that if the series

$$ \tag{* } \sum _ { n= } 1 ^ \infty a _ {n} \phi _ {n} $$

converges in $ L _ {2} $, then its sum belongs to $ L _ {p} $. If the system of functions $ \{ \phi _ {n} \} $ is an $ S _ {p} $- system for any $ p > 2 $, it is called an $ S _ \infty $- system. S. Banach proved (see [2]) that from any sequence of functions bounded in $ L _ {p} $ and orthonormal in $ L _ {2} $ one can extract an $ S _ {p} $- system. For an orthonormal system of functions $ \{ \phi _ {n} \} $ to be an $ S _ {p} $- system it is necessary and sufficient that there is a constant $ \mu _ {p} $ depending only on $ p $ and such that

$$ \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {p} } \leq \ \mu _ {p} \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {2} } $$

for all $ N $ and $ \{ a _ {n} \} $. If $ \{ \phi _ {n} \} $ is an $ S _ {p} $- system for some $ p > 2 $, then there is a constant $ m $ such that

$$ \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {2} } \leq \ m \left \| \sum _ { n= } 1 ^ { N } a _ {n} \phi _ {n} \right \| _ {L _ {p} } $$

for all $ N $ and $ \{ a _ {n} \} $. A system of functions with this property is called a Banach system. These definitions extend to non-orthogonal systems of functions (see [3], for example). Sometimes a lacunary system of functions is understood to be a system of functions whose series have one or several properties of lacunary trigonometric series, in dependence on which they take different names. For example, with the theory of uniqueness for lacunary trigonometric series there is associated the concept of a lacunary system of $ \epsilon $- uniqueness. A system $ \{ \phi _ {n} \} _ {n=} 1 ^ \infty $ is called a system of $ \epsilon $- uniqueness if there is a number $ \epsilon > 0 $ such that the convergence of the series (*) to zero everywhere, except possibly on a set of measure less than $ \epsilon $, implies that all its coefficients are zero.

References

[1]	S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2]	G. Alexits, "Konvergenzprobleme der Orthogonalreihen" , Deutsch. Verlag Wissenschaft. (1960)
[3]	V.F. Gaposhkin, "Lacunary series and independent functions" Russian Math. Surveys , 21 : 6 (1966) pp. 1–82 Uspekhi Mat. Nauk , 21 : 6 (1966) pp. 3–82