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Pietro Majer
Pietro Majer
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"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an examplean example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.

"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.

"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.

Source Link
Pietro Majer
Pietro Majer
  • 60.5k
  • 4
  • 122
  • 269

"Objection, the question assumes facts not in evidence!"

Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. Maybe the point is just that a lacunary form makes it easier to construct badly behaved series, which is slightly different. An example: we know that a real entire function $f$, say with real coefficients, may grow as fast as any given increasing function on $g:\mathbb{R}\to\mathbb{R}$, and building an example is easy by means of lacunary series. But $f(z+1)$ grows even faster, although the translation destroys the lacunary form.