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D G
D G
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First question: how do you choose $a_j$?. Since I expect you want an operator $P_h : X \to L^2$, you must make the choice before hand. You are unlikely to find any better estimation, since you can take analytical functions designed specifically to break any $P_h$ you construct.

Your result so far is a natural consequence of the Sobolev embedding theorem (see Brézis' book). In dimension one, $H^1 ([a,b]) \subset \mathcal C([a,b])$. In this sense you have just proved the convergence of Riemann sums in the standard way.

First question: how do you choose $a_j$?. Since I expect you want an operator $P_h : X \to L^2$, you must make the choice before hand. You are unlikely to find any better estimation, since you can take analytical functions designed specifically to break any $P_h$ you construct.

Your result so far is a natural consequence of the Sobolev embedding theorem (see Brézis' book). In dimension one, $H^1 ([a,b]) \subset \mathcal C([a,b])$. In this sense you have just proved the convergence of Riemann sums in the standard way.

You are unlikely to find any better estimation, since you can take analytical functions designed specifically to break any $P_h$ you construct.

Your result so far is a natural consequence of the Sobolev embedding theorem (see Brézis' book). In dimension one, $H^1 ([a,b]) \subset \mathcal C([a,b])$. In this sense you have just proved the convergence of Riemann sums in the standard way.

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D G
D G
  • 201
  • 2
  • 11

First question: how do you choose $a_j$?. Since I expect you want an operator $P_h : X \to L^2$, you must make the choice before hand. You are unlikely to find any better estimation, since you can take analytical functions designed specifically to break any $P_h$ you construct.

Your result so far is a natural consequence of the Sobolev embedding theorem (see Brézis' book). In dimension one, $H^1 ([a,b]) \subset \mathcal C([a,b])$. In this sense you have just proved the convergence of Riemann sums in the standard way.