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Error bounds for Gauss-Hermite quadrature, for analytic functions
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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
$$undersetx rightarrow inftylim e^-x^2/2 f^(i)(x)=0,quad U=sqrtint_-infty^+infty e^-x^2 [f^(k+1)(x)]^2 dx < infty.$$
Then for each $N ge k/2+1$,
beginalign
|I[f]-Q_N^textGH[f]|le frac1.632 sqrtpi (N-1) U(k-1) sqrt(2N-3) dots (2N-k-2)
endalign
where $I[f]= int_-infty^+infty f(x) dx$ and $Q_N^textGH[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
calculus integration analysis reference-request numerical-methods
asked Jul 22 at 13:35
user144209
5213
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up vote
0
down vote
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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
$$undersetx rightarrow inftylim e^-x^2/2 f^(i)(x)=0,quad U=sqrtint_-infty^+infty e^-x^2 [f^(k+1)(x)]^2 dx < infty.$$
Then for each $N ge k/2+1$,
beginalign
|I[f]-Q_N^textGH[f]|le frac1.632 sqrtpi (N-1) U(k-1) sqrt(2N-3) dots (2N-k-2)
endalign
where $I[f]= int_-infty^+infty f(x) dx$ and $Q_N^textGH[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
calculus integration analysis reference-request numerical-methods
asked Jul 22 at 13:35
user144209
5213
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
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
$$undersetx rightarrow inftylim e^-x^2/2 f^(i)(x)=0,quad U=sqrtint_-infty^+infty e^-x^2 [f^(k+1)(x)]^2 dx < infty.$$
Then for each $N ge k/2+1$,
beginalign
|I[f]-Q_N^textGH[f]|le frac1.632 sqrtpi (N-1) U(k-1) sqrt(2N-3) dots (2N-k-2)
endalign
where $I[f]= int_-infty^+infty f(x) dx$ and $Q_N^textGH[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
calculus integration analysis reference-request numerical-methods
asked Jul 22 at 13:35
user144209
5213
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
$$undersetx rightarrow inftylim e^-x^2/2 f^(i)(x)=0,quad U=sqrtint_-infty^+infty e^-x^2 [f^(k+1)(x)]^2 dx < infty.$$
Then for each $N ge k/2+1$,
beginalign
|I[f]-Q_N^textGH[f]|le frac1.632 sqrtpi (N-1) U(k-1) sqrt(2N-3) dots (2N-k-2)
endalign
where $I[f]= int_-infty^+infty f(x) dx$ and $Q_N^textGH[f]$ is the Gauss-Hermite quadrature.
Is there any reference or a way to derive similar result for the class of analytic functions. Thanks.
calculus integration analysis reference-request numerical-methods
asked Jul 22 at 13:35
user144209
5213
edited Jul 23 at 7:08
asked Jul 22 at 13:35
user144209
5213
asked Jul 22 at 13:35
user144209
5213
asked Jul 22 at 13:35
user144209
5213
5213
add a comment |Â
add a comment |Â
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