I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$ ). Further $A$ is a symmetric positive definite matrix, $J$ is the all ones matrix and $F$ indicates the Frobenius norm (the zero or one norm will work as well).

Any hope?

Thanks a lot!

Fabio

Note: an equivalent problem would be if $t_{b}(A)$ is the hadamard $p$-power and we substitute the integral with the sum $\sum_{p=1}^{\infty}$. I also tried to solve the integral in this last formulation but without success.

I am trying to solve the following integral $$ \int_{-1}^{1}\;db\;||[t_{b}(A),J]||_{F}^{2} $$ where $t_{b}$ is the entrywise threshold of the matrix A ($0$ if $a_{ij}<b$, $a_{ij}$ if $a_{ij}>b$, with $|a_{i,j}|\in [0,1]$ ). Further $A$ is a symmetric positive definite matrix, $J$ is the all ones matrix and $F$ indicates the Frobenius norm (the zero or one norm will work as well).

Any hope?

Thanks a lot!

Fabio

Note: an equivalent problem would be if $t_{b}(A)$ is the hadamard $p$-power of $A$ and we substitute the integral with the sum $\sum_{p=1}^{\infty}$. I also tried to solve the integral in this last formulation but without success.