Mixing
characterizing complex functions whose Laurent series coefficients have bounded 1-norm
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Suppose $h:mathbbTtomathbbC$ is a complex function on the unit circle with a valid Laurent series about zero:
$$
h(z) = sum_k=-infty^infty c_k z^kqquadtextfor |z|=1
$$
The coefficients $c_k$ satisfy certain constraints, and I would like to translate them to equivalent constraints on the function $h(z)$.
First question: is it true that the following statements are equivalent?
The coefficients satisfy $sum_k=-infty^infty |c_k| le 1$
The function satisfies $|h(z)| le 1$ for all $|z| = 1$.
It's clear that 1 implies 2 because $|h(z)| le sum_k|c_k||z^k| = sum_k|c_k|$ by the triangle inequality. Is the converse even true? How to prove it?
Second question: In addition to item 1 above, further assume that the coefficients $c_k$ are all real and nonnegative. Is there a nice way to represent this more complicated constraint on the coefficients as a constraint on $h$ ?
complex-analysis laurent-series
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
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up vote
1
down vote
favorite
Suppose $h:mathbbTtomathbbC$ is a complex function on the unit circle with a valid Laurent series about zero:
$$
h(z) = sum_k=-infty^infty c_k z^kqquadtextfor |z|=1
$$
The coefficients $c_k$ satisfy certain constraints, and I would like to translate them to equivalent constraints on the function $h(z)$.
First question: is it true that the following statements are equivalent?
The coefficients satisfy $sum_k=-infty^infty |c_k| le 1$
The function satisfies $|h(z)| le 1$ for all $|z| = 1$.
It's clear that 1 implies 2 because $|h(z)| le sum_k|c_k||z^k| = sum_k|c_k|$ by the triangle inequality. Is the converse even true? How to prove it?
Second question: In addition to item 1 above, further assume that the coefficients $c_k$ are all real and nonnegative. Is there a nice way to represent this more complicated constraint on the coefficients as a constraint on $h$ ?
complex-analysis laurent-series
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose $h:mathbbTtomathbbC$ is a complex function on the unit circle with a valid Laurent series about zero:
$$
h(z) = sum_k=-infty^infty c_k z^kqquadtextfor |z|=1
$$
The coefficients $c_k$ satisfy certain constraints, and I would like to translate them to equivalent constraints on the function $h(z)$.
First question: is it true that the following statements are equivalent?
The coefficients satisfy $sum_k=-infty^infty |c_k| le 1$
The function satisfies $|h(z)| le 1$ for all $|z| = 1$.
It's clear that 1 implies 2 because $|h(z)| le sum_k|c_k||z^k| = sum_k|c_k|$ by the triangle inequality. Is the converse even true? How to prove it?
Second question: In addition to item 1 above, further assume that the coefficients $c_k$ are all real and nonnegative. Is there a nice way to represent this more complicated constraint on the coefficients as a constraint on $h$ ?
complex-analysis laurent-series
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
Suppose $h:mathbbTtomathbbC$ is a complex function on the unit circle with a valid Laurent series about zero:
$$
h(z) = sum_k=-infty^infty c_k z^kqquadtextfor |z|=1
$$
The coefficients $c_k$ satisfy certain constraints, and I would like to translate them to equivalent constraints on the function $h(z)$.
First question: is it true that the following statements are equivalent?
The coefficients satisfy $sum_k=-infty^infty |c_k| le 1$
The function satisfies $|h(z)| le 1$ for all $|z| = 1$.
It's clear that 1 implies 2 because $|h(z)| le sum_k|c_k||z^k| = sum_k|c_k|$ by the triangle inequality. Is the converse even true? How to prove it?
Second question: In addition to item 1 above, further assume that the coefficients $c_k$ are all real and nonnegative. Is there a nice way to represent this more complicated constraint on the coefficients as a constraint on $h$ ?
complex-analysis laurent-series
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
edited Jul 26 at 15:46
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
asked Jul 26 at 6:59
Laurent Lessard
1,4211211
1,4211211
add a comment |Â
add a comment |Â
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