Lacunary trigonometric series

A series of the form

$$ \tag{1 } a _ {0} + \sum_{k=1} ^ \infty a _ {k} \cos n _ {k} x + b _ {k} \sin \ n _ {k} x , $$

with

$$ \lim\limits _ {k \rightarrow \infty } \inf \frac{n _ {k+} 1 }{n _ {k} } = \ \lambda > 1 . $$

In 1872 K. Weierstrass presented a continuous nowhere-differentiable function by means of a series of type (1). In 1892 J. Hadamard applied the series (1), calling them lacunary, to the study of analytic continuation of functions. A systematic study of lacunary trigonometric series began with a paper by P. Fatou in 1906, in which he proved that everywhere convergence of a lacunary trigonometric series for $ \lambda > 3 $ implies that

$$ \tag{2 } \sum_{k=1} ^ \infty | a _ {k} | + | b _ {k} | < + \infty . $$

Lacunary trigonometric series have properties essentially different from those of general trigonometric series. For example, A.N. Kolmogorov, having constructed the first example (1923) of a summable function with Fourier series diverging almost-everywhere, proved in 1924 that a lacunary Fourier series converges almost-everywhere; A. Zygmund proved (1948) that if the sums of two lacunary trigonometric series coincide on a set of positive measure, then these series are identical. For many applications of lacunary trigonometric series the dependence of properties of the series (1) on its coefficients, discovered by Zygmund in the 1930's, is important. Thus, if

$$ \tag{3 } \sum_{k=1} ^ \infty a _ {k} ^ {2} + b _ {k} ^ {2} < + \infty , $$

then the series (1) is the Fourier series of a function $ f $ belonging to all spaces $ L _ {p} ( 0 , 2 \pi ) $, $ 1 \leq p < + \infty $, hence it converges almost-everywhere. There are constants $ A _ {p} , B _ {p} > 0 $ depending only on $ p $ and $ \lambda $ such that

$$ A _ {p} \left ( \sum_{k=1} ^ \infty a _ {k} ^ {2} + b _ {k} ^ {2} \right ) ^ {1/2} \leq \ \left ( \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } | f | ^ {p} dx \right ) ^ {1/p\ } \leq $$

$$ \leq \ B _ {p} \left ( \sum_{k=1} ^ \infty a _ {k} ^ {2} + b _ {k} ^ {2} \right ) ^ {1/2} . $$

If (3) is not satisfied, then the series (1) diverges almost-everywhere, and, moreover, almost-everywhere it cannot be summed by any Toeplitz method (cf. Toeplitz matrix) (and is therefore not a Fourier series). If the series (1) converges at every point of some interval, then (2) is satisfied. If the coefficients of the series (1) are $ o ( 1 / n _ {k} ) $, then its sum is a continuous smooth function, differentiable at just those points at which the formally differentiated series of (1) converges.

References

[1]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)
[2]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[3]	P.L. Ul'yanov, "Solved and unsolved problems in the theory of trigonometric and orthogonal series" Russian Math. Surveys , 19 : 1 (1964) pp. 1–62 Uspekhi Mat. Nauk , 19 : 1 (1964) pp. 3–69
[4]	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
[5]	J.-P. Kahane, "Some random series of functions" , Cambridge Univ. Press (1985)
[6]	J.-P. Kahane, "Séries de Fourier absolument convergentes" , Springer (1970)
[7]	W. Rudin, "Fourier analysis on groups" , Interscience (1962)