Order of convergence for spline interpolation in a Sobolev norm
We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{text{loc}}(mathbb{R})$.
Let us be more precise:
Let $hin mathbb{R}_{> 0}$, $boldsymbol{x} in mathbb{R}^{mathbb{Z}}$ with $boldsymbol{x}_{i} +h leq boldsymbol{x}_{i+1} forall iin mathbb{Z}$, $r,p in [1,infty], k,n in mathbb{N}_{geq 1}$, $k> n$, and $fin W^{k,p}_{text{loc}}(mathbb{R})$ with $f'in W^{k-1,p}(mathbb{R})$ be given. Let $s_{f}^{n}: mathbb{R}rightarrow mathbb{R}$ denote the spline interpolation of order $n$ of $f$, does it hold that
$$
exists mathcal{C}_{n}inmathbb{R}_{geq0}: |f - s^n_f|_{L^r((boldsymbol{x}_i,boldsymbol{x}_{i+1}))} leq mathcal{C}_n|f^{(n+1)}|_{L^p(mathbb{R})} h^{n+1+frac1r-frac1p}?
$$
We already have found a reference for a subproblem, where the previous result in a weakened form is shown for $W^{2,p}$ (see Approximation error estimates and inverse inequalities for B-splines of maximum smoothness).
However, we would be interested in a textbook or paper which presents results in the above described generality. Any hints are highly appreciated.
Thanks a lot.
analysis sobolev-spaces spline
asked Nov 23 at 6:41
Alex
243320
add a comment |
We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{text{loc}}(mathbb{R})$.
Let us be more precise:
Let $hin mathbb{R}_{> 0}$, $boldsymbol{x} in mathbb{R}^{mathbb{Z}}$ with $boldsymbol{x}_{i} +h leq boldsymbol{x}_{i+1} forall iin mathbb{Z}$, $r,p in [1,infty], k,n in mathbb{N}_{geq 1}$, $k> n$, and $fin W^{k,p}_{text{loc}}(mathbb{R})$ with $f'in W^{k-1,p}(mathbb{R})$ be given. Let $s_{f}^{n}: mathbb{R}rightarrow mathbb{R}$ denote the spline interpolation of order $n$ of $f$, does it hold that
$$
exists mathcal{C}_{n}inmathbb{R}_{geq0}: |f - s^n_f|_{L^r((boldsymbol{x}_i,boldsymbol{x}_{i+1}))} leq mathcal{C}_n|f^{(n+1)}|_{L^p(mathbb{R})} h^{n+1+frac1r-frac1p}?
$$
We already have found a reference for a subproblem, where the previous result in a weakened form is shown for $W^{2,p}$ (see Approximation error estimates and inverse inequalities for B-splines of maximum smoothness).
However, we would be interested in a textbook or paper which presents results in the above described generality. Any hints are highly appreciated.
Thanks a lot.
analysis sobolev-spaces spline
asked Nov 23 at 6:41
Alex
243320
add a comment |
We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{text{loc}}(mathbb{R})$.
Let us be more precise:
Let $hin mathbb{R}_{> 0}$, $boldsymbol{x} in mathbb{R}^{mathbb{Z}}$ with $boldsymbol{x}_{i} +h leq boldsymbol{x}_{i+1} forall iin mathbb{Z}$, $r,p in [1,infty], k,n in mathbb{N}_{geq 1}$, $k> n$, and $fin W^{k,p}_{text{loc}}(mathbb{R})$ with $f'in W^{k-1,p}(mathbb{R})$ be given. Let $s_{f}^{n}: mathbb{R}rightarrow mathbb{R}$ denote the spline interpolation of order $n$ of $f$, does it hold that
$$
exists mathcal{C}_{n}inmathbb{R}_{geq0}: |f - s^n_f|_{L^r((boldsymbol{x}_i,boldsymbol{x}_{i+1}))} leq mathcal{C}_n|f^{(n+1)}|_{L^p(mathbb{R})} h^{n+1+frac1r-frac1p}?
$$
We already have found a reference for a subproblem, where the previous result in a weakened form is shown for $W^{2,p}$ (see Approximation error estimates and inverse inequalities for B-splines of maximum smoothness).
However, we would be interested in a textbook or paper which presents results in the above described generality. Any hints are highly appreciated.
Thanks a lot.
analysis sobolev-spaces spline
asked Nov 23 at 6:41
Alex
243320
We are interested in a result stating the order of convergence of a spline interpolation for a given function in $W^{k,p}_{text{loc}}(mathbb{R})$.
Let us be more precise:
Let $hin mathbb{R}_{> 0}$, $boldsymbol{x} in mathbb{R}^{mathbb{Z}}$ with $boldsymbol{x}_{i} +h leq boldsymbol{x}_{i+1} forall iin mathbb{Z}$, $r,p in [1,infty], k,n in mathbb{N}_{geq 1}$, $k> n$, and $fin W^{k,p}_{text{loc}}(mathbb{R})$ with $f'in W^{k-1,p}(mathbb{R})$ be given. Let $s_{f}^{n}: mathbb{R}rightarrow mathbb{R}$ denote the spline interpolation of order $n$ of $f$, does it hold that
$$
exists mathcal{C}_{n}inmathbb{R}_{geq0}: |f - s^n_f|_{L^r((boldsymbol{x}_i,boldsymbol{x}_{i+1}))} leq mathcal{C}_n|f^{(n+1)}|_{L^p(mathbb{R})} h^{n+1+frac1r-frac1p}?
$$
We already have found a reference for a subproblem, where the previous result in a weakened form is shown for $W^{2,p}$ (see Approximation error estimates and inverse inequalities for B-splines of maximum smoothness).
However, we would be interested in a textbook or paper which presents results in the above described generality. Any hints are highly appreciated.
Thanks a lot.
analysis sobolev-spaces spline
analysis sobolev-spaces spline
asked Nov 23 at 6:41
Alex
243320
asked Nov 23 at 6:41
Alex
243320
edited Nov 24 at 3:55
asked Nov 23 at 6:41
Alex
243320
asked Nov 23 at 6:41
Alex
243320
asked Nov 23 at 6:41
Alex
243320
243320
add a comment |
add a comment |
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