Error bounds for Gauss-Hermite quadrature, for analytic functions
$begingroup$
I am working with the class of analytic functions and I want to derive some estimates, using error bounds for Gauss-Hermite quadrature, for analytic functions (based on Bernstein ellipses).
The issue is that I only know the result for the class of functions with finite smoothness, which says that: if $f$ is such that $i)$ $f,f^{prime},dots,f^{(k-1)}$ are absolutely continuous in $(-infty,+infty)$ and $ii)$ for $j=0,1,dots,k$, for some $k ge 2$ such that
$$underset{x rightarrow infty}{lim} e^{-x^2/2} f^{(i)}(x)=0,quad U=sqrt{int_{-infty}^{+infty} e^{-x^2} [f^{(k+1)}(x)]^2 dx} < infty.$$
Then for each $N ge k/2+1$,
begin{align}
|I[f]-Q_N^{text{GH}}[f]|le frac{1.632 sqrt{pi (N-1) }U}{(k-1) sqrt{(2N-3) dots (2N-k-2)}}
end{align}
where $I[f]= int_{-infty}^{+infty} f(x) dx$ and $Q_N^{text{GH}}[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
reference-request numerical-methods analyticity analytic-functions gaussian-integral
asked Jul 22 '18 at 13:35
user144209user144209
62214
$endgroup$
add a comment |
$begingroup$
I am working with the class of analytic functions and I want to derive some estimates, using error bounds for Gauss-Hermite quadrature, for analytic functions (based on Bernstein ellipses).
The issue is that I only know the result for the class of functions with finite smoothness, which says that: if $f$ is such that $i)$ $f,f^{prime},dots,f^{(k-1)}$ are absolutely continuous in $(-infty,+infty)$ and $ii)$ for $j=0,1,dots,k$, for some $k ge 2$ such that
$$underset{x rightarrow infty}{lim} e^{-x^2/2} f^{(i)}(x)=0,quad U=sqrt{int_{-infty}^{+infty} e^{-x^2} [f^{(k+1)}(x)]^2 dx} < infty.$$
Then for each $N ge k/2+1$,
begin{align}
|I[f]-Q_N^{text{GH}}[f]|le frac{1.632 sqrt{pi (N-1) }U}{(k-1) sqrt{(2N-3) dots (2N-k-2)}}
end{align}
where $I[f]= int_{-infty}^{+infty} f(x) dx$ and $Q_N^{text{GH}}[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
reference-request numerical-methods analyticity analytic-functions gaussian-integral
asked Jul 22 '18 at 13:35
user144209user144209
62214
$endgroup$
add a comment |
$begingroup$
I am working with the class of analytic functions and I want to derive some estimates, using error bounds for Gauss-Hermite quadrature, for analytic functions (based on Bernstein ellipses).
The issue is that I only know the result for the class of functions with finite smoothness, which says that: if $f$ is such that $i)$ $f,f^{prime},dots,f^{(k-1)}$ are absolutely continuous in $(-infty,+infty)$ and $ii)$ for $j=0,1,dots,k$, for some $k ge 2$ such that
$$underset{x rightarrow infty}{lim} e^{-x^2/2} f^{(i)}(x)=0,quad U=sqrt{int_{-infty}^{+infty} e^{-x^2} [f^{(k+1)}(x)]^2 dx} < infty.$$
Then for each $N ge k/2+1$,
begin{align}
|I[f]-Q_N^{text{GH}}[f]|le frac{1.632 sqrt{pi (N-1) }U}{(k-1) sqrt{(2N-3) dots (2N-k-2)}}
end{align}
where $I[f]= int_{-infty}^{+infty} f(x) dx$ and $Q_N^{text{GH}}[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
reference-request numerical-methods analyticity analytic-functions gaussian-integral
asked Jul 22 '18 at 13:35
user144209user144209
62214
$endgroup$
I am working with the class of analytic functions and I want to derive some estimates, using error bounds for Gauss-Hermite quadrature, for analytic functions (based on Bernstein ellipses).
The issue is that I only know the result for the class of functions with finite smoothness, which says that: if $f$ is such that $i)$ $f,f^{prime},dots,f^{(k-1)}$ are absolutely continuous in $(-infty,+infty)$ and $ii)$ for $j=0,1,dots,k$, for some $k ge 2$ such that
$$underset{x rightarrow infty}{lim} e^{-x^2/2} f^{(i)}(x)=0,quad U=sqrt{int_{-infty}^{+infty} e^{-x^2} [f^{(k+1)}(x)]^2 dx} < infty.$$
Then for each $N ge k/2+1$,
begin{align}
|I[f]-Q_N^{text{GH}}[f]|le frac{1.632 sqrt{pi (N-1) }U}{(k-1) sqrt{(2N-3) dots (2N-k-2)}}
end{align}
where $I[f]= int_{-infty}^{+infty} f(x) dx$ and $Q_N^{text{GH}}[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
reference-request numerical-methods analyticity analytic-functions gaussian-integral
reference-request numerical-methods analyticity analytic-functions gaussian-integral
asked Jul 22 '18 at 13:35
user144209user144209
62214
asked Jul 22 '18 at 13:35
user144209user144209
62214
edited Dec 2 '18 at 8:33
user144209
asked Jul 22 '18 at 13:35
user144209user144209
62214
asked Jul 22 '18 at 13:35
user144209user144209
62214
asked Jul 22 '18 at 13:35
user144209user144209
62214
62214
add a comment |
add a comment |
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