{"id":388,"date":"2013-10-15T01:00:30","date_gmt":"2013-10-15T01:00:30","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=388"},"modified":"2015-07-29T00:54:30","modified_gmt":"2015-07-29T00:54:30","slug":"atomic-singular-inner-function","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/10\/15\/atomic-singular-inner-function\/","title":{"rendered":"Atomic Singular Inner Function"},"content":{"rendered":"
\"Atomic<\/a>

Atomic Singular Inner Function with Atoms at Fifth Roots of Unity –
Elias Wegert, www.visual.wegert.com<\/p><\/div>\n

This picture, drawn by Elias Wegert<\/a>, uses colors to show the phase $f\/|f|$ of the complex function<\/p>\n

$$f(z) = \\prod_{k=1}^5 \\exp\\left(\\frac{z+\\omega^k}{z-\\omega^k}\\right)$$<\/p>\n

where $\\omega$ is a nontrivial fifth root of unity. This is an ‘atomic singular inner function’. To understand what that means, it helps to start with some complex analysis.<\/p>\n

The Hardy space<\/b><\/a> $H^\\infty$ is the space of bounded analytic functions in the complex unit disk $D$. Any function $f$ in the Hardy space $H^\\infty$ can be written as a product of an ‘inner function’ $g$ and an ‘outer function’ $h$:
\n$$ f = gh $$
\nThe inner function<\/b> $g$ has modulus $1$ almost everywhere on $S^1$. The outer function<\/b> $h$ has no zeros in $D$ and is completely determined by the boundary values of $|f|$ on the unit circle $S^1$.<\/p>\n

The inner function can be further split into two factors, $g=bs$, where $b$ is a ‘Blaschke product’ and $s$ is a so-called ‘singular’ innner function. The Blaschke product<\/b><\/a> is completely determined by the zeros of $f$. The singular inner function<\/b> is an inner function with no zeros in $D$ and modulus 1 almost everywhere on $S^1$.<\/p>\n

So, the singular inner function coming from $f$ is a kind of ‘remainder’ which can neither be determined from the zeros of $f$ in the unit disk nor from $|f|$ on the unit circle!<\/p>\n

The prototypical example of a singular inner function is
\n$$ \\exp\\left(\\frac{z+t}{z-t}\\right)$$
\nfor any point $t$ on the unit circle. This has an essential singularity at $t$. Starting from this, one gets the most general singular inner function as follows:
\n$$ s(z)=c\\,\\exp \\int_{S^1}\\frac{z+t}{z-t}\\,d\\mu(t)$$
\nwhere $\\mu$ is a normalized singular measure on $S^1$ and $|c|=1$. Remember, a measure on the circle is singular<\/b><\/a> if it is supported on a set of Lebesgue measure zero. The easiest singular measures to understand are the atomic<\/b><\/a> ones, which are (possibly infinite) linear combinations of Dirac delta measures.<\/p>\n

In the example at hand we have taken a linear combination of Dirac deltas supported at the fifth roots of unity. So, the picture uses colors to show the phase $f\/|f|$ of the function<\/p>\n

$$f(z) = \\prod_{k=1}^5 \\exp\\left(\\frac{z+\\omega^k}{z-\\omega^k}\\right)$$<\/p>\n

where $\\omega=\\exp(2\\pi i\/5)$. This function has essential singularities at the fifth roots of unity, no zeros in $D$, and modulus 1 on $S^1$.<\/p>\n

You can learn a lot about a function by looking at a color picture of its phase. For more on this, with lots of wonderful pictures, explore Wegert’s website:<\/p>\n

\u2022 Elias Wegert, Visualizing complex functions using phase portraits<\/a>.<\/p>\n

and read this paper:<\/p>\n

\u2022 Elias Wegert, Phase plots of complex functions: a journey in illustration<\/a>.<\/p>\n

The factorization of functions in $H^\\infty$ into inner and outer functions is due to Beurling, and it has various generalizations:<\/p>\n

\u2022 Beurling-Lax theorem<\/a>, Encyclopedia of Mathematics<\/i>, Springer.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This picture, drawn by Elias Wegert<\/a>, uses colors to show the phase $f\/|f|$ of the complex function<\/p>\n

$$f(z) = \\prod_{k=1}^5 \\exp\\left(\\frac{z+\\omega^k}{z-\\omega^k}\\right)$$<\/p>\n

where $\\omega$ is a nontrivial fifth root of unity. This is an ‘atomic singular inner function’. To understand what that means, it helps to start with some complex analysis.<\/p>\n

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