Upper bounds for certain finite sums involving restriction $sumlimits_{n=1}^{N}{{{left| {{a}_{n}}...
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Let us suppose that ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ are $N$ complex numbers such that $sumlimits_{n=1}^{N}{{{left|a_nright|}^{2}}}=1$, and define finite sums ${{S}_{1}}={{left| sumlimits_{n=1}^{N}{{{a}_{n}}} right|}^{2}}$ and ${{S}_{2}}=underset{k=1,2,ldots N}{mathop{max }},{{left| sumlimits_{n=1}^{N}{{{(-1)}^{{{delta }_{k,n}}}}{{a}_{n}}} right|}^{2}}$, where ${{delta }_{k,n}}$ represents a Kronecker delta.
In the particular case of a uniform distribution, i. e., ${{a}_{n}}=frac{1}{sqrt{N}}text{ }forall n$, we have ${{S}_{2}}=frac{{{(N-2)}^{2}}}{N}$ and ${{S}_{1}}-{{S}_{2}}=N-frac{{{(N-2)}^{2}}}{N}=frac{4(N-1)}{N}$.
I wanted to find general upper bounds, as tight as possible, for the following three quantities: ${{S}_{2}}$, ${{S}_{1}}-{{S}_{2}}$, and $left| {{S}_{1}}-{{S}_{2}} right|$. The only restriction on ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ is that $sumlimits_{n=1}^{N}{{{left| {{a}_{n}} right|}^{2}}}=1$. Thank you!
calculus sequences-and-series algebra-precalculus upper-lower-bounds
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
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add a comment |
$begingroup$
Let us suppose that ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ are $N$ complex numbers such that $sumlimits_{n=1}^{N}{{{left|a_nright|}^{2}}}=1$, and define finite sums ${{S}_{1}}={{left| sumlimits_{n=1}^{N}{{{a}_{n}}} right|}^{2}}$ and ${{S}_{2}}=underset{k=1,2,ldots N}{mathop{max }},{{left| sumlimits_{n=1}^{N}{{{(-1)}^{{{delta }_{k,n}}}}{{a}_{n}}} right|}^{2}}$, where ${{delta }_{k,n}}$ represents a Kronecker delta.
In the particular case of a uniform distribution, i. e., ${{a}_{n}}=frac{1}{sqrt{N}}text{ }forall n$, we have ${{S}_{2}}=frac{{{(N-2)}^{2}}}{N}$ and ${{S}_{1}}-{{S}_{2}}=N-frac{{{(N-2)}^{2}}}{N}=frac{4(N-1)}{N}$.
I wanted to find general upper bounds, as tight as possible, for the following three quantities: ${{S}_{2}}$, ${{S}_{1}}-{{S}_{2}}$, and $left| {{S}_{1}}-{{S}_{2}} right|$. The only restriction on ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ is that $sumlimits_{n=1}^{N}{{{left| {{a}_{n}} right|}^{2}}}=1$. Thank you!
calculus sequences-and-series algebra-precalculus upper-lower-bounds
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
$endgroup$
add a comment |
$begingroup$
Let us suppose that ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ are $N$ complex numbers such that $sumlimits_{n=1}^{N}{{{left|a_nright|}^{2}}}=1$, and define finite sums ${{S}_{1}}={{left| sumlimits_{n=1}^{N}{{{a}_{n}}} right|}^{2}}$ and ${{S}_{2}}=underset{k=1,2,ldots N}{mathop{max }},{{left| sumlimits_{n=1}^{N}{{{(-1)}^{{{delta }_{k,n}}}}{{a}_{n}}} right|}^{2}}$, where ${{delta }_{k,n}}$ represents a Kronecker delta.
In the particular case of a uniform distribution, i. e., ${{a}_{n}}=frac{1}{sqrt{N}}text{ }forall n$, we have ${{S}_{2}}=frac{{{(N-2)}^{2}}}{N}$ and ${{S}_{1}}-{{S}_{2}}=N-frac{{{(N-2)}^{2}}}{N}=frac{4(N-1)}{N}$.
I wanted to find general upper bounds, as tight as possible, for the following three quantities: ${{S}_{2}}$, ${{S}_{1}}-{{S}_{2}}$, and $left| {{S}_{1}}-{{S}_{2}} right|$. The only restriction on ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ is that $sumlimits_{n=1}^{N}{{{left| {{a}_{n}} right|}^{2}}}=1$. Thank you!
calculus sequences-and-series algebra-precalculus upper-lower-bounds
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
$endgroup$
Let us suppose that ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ are $N$ complex numbers such that $sumlimits_{n=1}^{N}{{{left|a_nright|}^{2}}}=1$, and define finite sums ${{S}_{1}}={{left| sumlimits_{n=1}^{N}{{{a}_{n}}} right|}^{2}}$ and ${{S}_{2}}=underset{k=1,2,ldots N}{mathop{max }},{{left| sumlimits_{n=1}^{N}{{{(-1)}^{{{delta }_{k,n}}}}{{a}_{n}}} right|}^{2}}$, where ${{delta }_{k,n}}$ represents a Kronecker delta.
In the particular case of a uniform distribution, i. e., ${{a}_{n}}=frac{1}{sqrt{N}}text{ }forall n$, we have ${{S}_{2}}=frac{{{(N-2)}^{2}}}{N}$ and ${{S}_{1}}-{{S}_{2}}=N-frac{{{(N-2)}^{2}}}{N}=frac{4(N-1)}{N}$.
I wanted to find general upper bounds, as tight as possible, for the following three quantities: ${{S}_{2}}$, ${{S}_{1}}-{{S}_{2}}$, and $left| {{S}_{1}}-{{S}_{2}} right|$. The only restriction on ${{a}_{n}}in mathbb{C},text{ }n=1,2,ldots N,$ is that $sumlimits_{n=1}^{N}{{{left| {{a}_{n}} right|}^{2}}}=1$. Thank you!
calculus sequences-and-series algebra-precalculus upper-lower-bounds
calculus sequences-and-series algebra-precalculus upper-lower-bounds
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
asked Dec 12 '18 at 10:59
HS TQHS TQ
426
426
add a comment |
add a comment |
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