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Minor clarification.
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edit approved Feb 24, 2014 at 15:51
Hans
edit approved Feb 24, 2014 at 15:51
Hans
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For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can takefactor out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric inunder transformation $\lambda_j+\lambda_j^{-1}$$\lambda_j\leftrightarrow\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can take out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric in $\lambda_j+\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can factor out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric under transformation $\lambda_j\leftrightarrow\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

edited body
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Lev Borisov
Lev Borisov
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For $\phi_i=0$$\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can take out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric in $\lambda_j+\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

For $\phi_i=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can take out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric in $\lambda_j+\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

For $\phi_j=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can take out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric in $\lambda_j+\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.

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Lev Borisov
Lev Borisov
  • 5.2k
  • 1
  • 22
  • 38

For $\phi_i=0$ and $t_k=k$ you have (up to a constant) the matrix $A_{jk}= \lambda_j^k - \lambda_j^{-k}$ where $\lambda_j=\exp( i w_j)$. You can take out $\lambda_j-\lambda_j^{-1}$ to get a matrix $B_{jk}$ with the entries that are symmetric in $\lambda_j+\lambda_j^{-1}$. Moreover, you have $$ B_{jk}=P_k(\lambda_j+\lambda_j^{-1}) $$ where $P_k$ is a monic polynomial of degree $k-1$. By subtracting columns (note that $P_k$ depends on $k$ only, not on $j$) you get Vandermonde matrix $(\lambda_j+\lambda_j^{-1})^{k-1}$ and you can go from there.