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David Ferrone
David Ferrone
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Can someone provideLet $X = {x_1, ..., x_N}$ be a listfinite subset of references (both introductory$R^n$ and advancedlet $p$ and $q$ be any polynomials of degree $k$ or less. X is called $\underline{P_k-unisolvent}$ if $p(x_j) = q(x_j)$ ($j = 1, ..., N$) containing information aboutimplies that $p=q$; i.e. evaluation of a polynomial on $X$ uniquely determines that polynomial.

In one dimension, unisolvent point sets? are completely trivial. But for $n \geq 2$ even confirming whether a given set is unisolvent or not becomes complicated.

I understand that the identification of such collections ofThe Padua points for polynomialswere recently constructed - a unisolvent set in $R^n$ can become complicated$R^2$ with minimal growth of their corresponding Lebesgue constant. I am interested in the construction of similar sets in higher dimensions, and have found numerous articles on this particular subset of (at least for$R^2$.

I am also interested in $n>1$), but$L^p$-norm estimates and interpolant error bounds to Sobolev functions defined on manifolds and more exotic domains. To this end I have alsoseen seen a result used (Duchon, J. 1978) stating that the collection of all $P_k$-unisolvent point sets is open in $R^n$.

I am hoping to find other instances of the use of unisolvent sets, and information on their properties or characterization.

Can anyone provide some references (both introductory and advanced) containing information about unisolvent sets?

Can someone provide a list of references (both introductory and advanced) containing information about unisolvent point sets?

I understand that the identification of such collections of points for polynomials in $R^n$ can become complicated (at least for $n>1$), but I have also seen a result (Duchon, J. 1978) stating that the collection of all $P_k$-unisolvent point sets is open.

Let $X = {x_1, ..., x_N}$ be a finite subset of $R^n$ and let $p$ and $q$ be any polynomials of degree $k$ or less. X is called $\underline{P_k-unisolvent}$ if $p(x_j) = q(x_j)$ ($j = 1, ..., N$) implies that $p=q$; i.e. evaluation of a polynomial on $X$ uniquely determines that polynomial.

In one dimension, unisolvent sets are completely trivial. But for $n \geq 2$ even confirming whether a given set is unisolvent or not becomes complicated.

The Padua points were recently constructed - a unisolvent set in $R^2$ with minimal growth of their corresponding Lebesgue constant. I am interested in the construction of similar sets in higher dimensions, and have found numerous articles on this particular subset of $R^2$.

I am also interested in $L^p$-norm estimates and interpolant error bounds to Sobolev functions defined on manifolds and more exotic domains. To this end I have seen seen a result used (Duchon, J. 1978) stating that the collection of all $P_k$-unisolvent point sets is open in $R^n$.

I am hoping to find other instances of the use of unisolvent sets, and information on their properties or characterization.

Can anyone provide some references (both introductory and advanced) containing information about unisolvent sets?

Source Link
David Ferrone
David Ferrone
  • 11
  • 6

References regarding unisolvent sets

Can someone provide a list of references (both introductory and advanced) containing information about unisolvent point sets?

I understand that the identification of such collections of points for polynomials in $R^n$ can become complicated (at least for $n>1$), but I have also seen a result (Duchon, J. 1978) stating that the collection of all $P_k$-unisolvent point sets is open.