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| In [[complex analysis|analysis]], a '''lacunary function''', also known as a '''lacunary series''', is an [[analytic function]] that cannot be [[analytic continuation|analytically continued]] anywhere outside the [[radius of convergence]] within which it is defined by a [[power series]]. The word ''lacunary'' is derived from [[wiktionary:lacuna|lacuna]] (''pl.'' lacunae), meaning gap, or vacancy.
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| The first known examples of lacunary functions involved [[Taylor series]] with large gaps, or lacunae, between the non-zero coefficients of their expansions. More recent investigations have also focused attention on [[Fourier series]] with similar gaps between non-zero coefficients. There is a slight ambiguity in the modern usage of the term '''lacunary series''', which may be used to refer to either Taylor series or Fourier series.
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| == A simple example ==
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| Consider the lacunary function defined by a simple power series:
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| :<math>
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| f(z) = \sum_{n=0}^\infty z^{2^n} = z + z^2 + z^4 + z^8 + \cdots\,
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| </math>
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| The power series converges uniformly on any open domain |''z''| < 1. This can be proved by comparing ''f'' with the [[geometric series]], which is absolutely convergent when |''z''| < 1. So ''f'' is analytic on the open unit disk. Nevertheless ''f'' has a singularity at every point on the unit circle, and cannot be analytically continued outside of the open unit disk, as the following argument demonstrates.
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| Clearly ''f'' has a singularity at ''z'' = 1, because
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| :<math>
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| f(1) = 1 + 1 + 1 + \cdots\,
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| </math>
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| is a divergent series. But since
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| :<math>
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| f(z^2) = f(z) - z \qquad f(z^4) = f(z^2) - z^2 \qquad f(z^8) = f(z^4) - z^4 \cdots\,
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| </math>
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| we can see that ''f'' has a singularity at a point ''z'' when ''z''<sup>2</sup> = 1 (that is, when ''z'' = ±''1''), and also when ''z''<sup>4</sup> = 1 (that is, when ''z'' = ±''1'' or when ''z'' = ±''i''). By the induction suggested by the above equations, ''f'' must have a singularity at each of the 2<sup>''n''</sup>th [[root of unity|roots of unity]] for all natural numbers ''n''. The set of all such points is [[dense set|dense]] on the unit circle, hence by continuous extension every point on the unit circle must be a singularity of ''f''.<ref>(Whittaker and Watson, 1927, p. 98) This example apparently originated with Weierstrass.</ref>
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| == An elementary result ==
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| Evidently the argument advanced in the simple example can also be applied to show that series like
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| :<math>
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| f(z) = \sum_{n=0}^\infty z^{3^n} = z + z^3 + z^9 + z^{27} + \cdots \qquad
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| g(z) = \sum_{n=0}^\infty z^{4^n} = z + z^4 + z^{16} + z^{64} + \cdots\,
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| </math>
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| also define lacunary functions. What is not so evident is that the gaps between the powers of ''z'' can expand much more slowly, and the resulting series will still define a lacunary function. To make this notion more precise some additional notation is needed.
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| We write
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| :<math>
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| f(z) = \sum_{k=1}^\infty a_kz^{\lambda_k} = \sum_{n=1}^\infty b_n z^n\,
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| </math>
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| where ''b''<sub>''n''</sub> = ''a''<sub>''k''</sub> when ''n'' = λ<sub>''k''</sub>, and ''b''<sub>''n''</sub> = 0 otherwise. The stretches where the coefficients ''b''<sub>''n''</sub> in the second series are all zero are the ''lacunae'' in the coefficients. The monotonically increasing sequence of positive natural numbers {λ<sub>''k''</sub>} specifies the powers of ''z'' which are in the power series for ''f''(''z'').
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| Now a theorem of [[Jaques Hadamard|Hadamard]] can be stated.<ref>(Mandelbrojt and Miles, 1927)</ref> If
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| :<math>
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| \lim_{k\to\infty} \frac{\lambda_k}{\lambda_{k-1}} > 1 + \delta \,
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| </math>
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| where ''δ'' > 0 is an arbitrary positive constant, then ''f''(''z'') is a lacunary function that cannot be continued outside its circle of convergence. In other words, the sequence {λ<sub>''k''</sub>} doesn't have to grow as fast as 2<sup>''k''</sup> for ''f''(''z'') to be a lacunary function – it just has to grow as fast as some geometric progression (1 + δ)<sup>''k''</sup>. A series for which λ<sub>''k''</sub> grows this quickly is said to contain '''Hadamard gaps'''. See [[Ostrowski-Hadamard gap theorem]].
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| == Lacunary trigonometric series ==
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| Mathematicians have also investigated the properties of lacunary trigonometric series
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| :<math>
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| S(\lambda_k,\theta) = \sum_{k=1}^\infty a_k \cos(\lambda_k\theta) \qquad
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| S(\lambda_k,\theta,\omega) = \sum_{k=1}^\infty a_k \cos(\lambda_k\theta + \omega) \,
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| </math>
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| for which the λ<sub>''k''</sub> are far apart. Here the coefficients ''a''<sub>''k''</sub> are real numbers. In this context, attention has been focused on criteria sufficient to guarantee convergence of the trigonometric series [[almost everywhere]] (that is, for almost every value of the angle ''θ'' and of the distortion factor ''ω'').
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| *[[Andrey Kolmogorov|Kolmogorov]] showed that if the sequence {λ<sub>''k''</sub>} contains Hadamard gaps, then the series ''S''(λ<sub>''k''</sub>, ''θ'', ''ω'') converges (diverges) almost everywhere when
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| ::<math>
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| \sum_{k=1}^\infty a_k^2\,
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| </math>
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| :converges (diverges).
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| *[[Antoni Zygmund|Zygmund]] showed under the same condition that ''S''(λ<sub>''k''</sub>, ''θ'', ''ω'') is not a Fourier series representing an [[integrable function]] when this sum of squares of the ''a''<sub>''k''</sub> is a divergent series.<ref>(Fukuyama and Takahashi, 1999)</ref>
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| == A unified view ==
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| Greater insight into the underlying question that motivates the investigation of lacunary power series and lacunary trigonometric series can be gained by re-examining the simple example above. In that example we used the geometric series
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| :<math>
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| g(z) = \sum_{n=1}^\infty z^n \,
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| </math>
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| and the [[Weierstrass M-test]] to demonstrate that the simple example defines an analytic function on the open unit disk.
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| The geometric series itself defines an analytic function that converges everywhere on the ''closed'' unit disk except when ''z'' = 1, where ''g''(''z'') has a simple pole.<ref>This can be shown by applying [[Abel's test]] to the geometric series ''g''(''z''). It can also be understood directly, by recognizing that the geometric series is the [[Maclaurin series]] for ''g''(''z'') = ''z''/(1−''z'').</ref> And, since ''z'' = ''e''<sup>''iθ''</sup> for points on the unit circle, the geometric series becomes
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| :<math>
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| g(z) = \sum_{n=1}^\infty e^{in\theta} = \sum_{n=1}^\infty \left(\cos n\theta + i\sin n\theta\right) \,
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| </math> | |
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| at a particular ''z'', |''z''| = 1. From this perspective, then, mathematicians who investigate lacunary series are asking the question: How much does the geometric series have to be distorted – by chopping big sections out, and by introducing coefficients ''a''<sub>''k''</sub> ≠ 1 – before the resulting mathematical object is transformed from a nice smooth [[meromorphic function]] into something that exhibits a primitive form of [[chaos theory|chaotic]] behavior?
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| == See also ==
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| *[[Analytic continuation]]
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| *[[Szolem Mandelbrojt]]
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| *[[Benoit Mandelbrot]]
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| *[[Mandelbrot set]]
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| == Notes ==
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| <references/>
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| == References ==
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| *Katusi Fukuyama and Shigeru Takahashi, ''Proceedings of the American Mathematical Society'', vol. 127 #2 pp.599-608 (1999), "The Central Limit Theorem for Lacunary Series".
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| *[[Szolem Mandelbrojt]] and Edward Roy Cecil Miles, ''The Rice Institute Pamphlet'', vol. 14 #4 pp.261-284 (1927), "Lacunary Functions".
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| *[[E. T. Whittaker]] and [[G. N. Watson]], ''A Course in Modern Analysis'', fourth edition, Cambridge University Press, 1927.
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| == External links ==
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| *[http://www.ams.org/proc/1999-127-02/S0002-9939-99-04541-4/S0002-9939-99-04541-4.pdf Fukuyama and Takahashi, 1999] A paper (PDF) entitled ''The Central Limit Theorem for Lacunary Series'', from the AMS.
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| *[http://hdl.handle.net/1911/8511 Mandelbrojt and Miles, 1927] A paper (PDF) entitled ''Lacunary Functions'', from Rice University.
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| *[http://mathworld.wolfram.com/LacunaryFunction.html MathWorld article on Lacunary Functions]
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| [[Category:Analytic functions]]
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