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YCor
YCor
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Korovkin setsubset of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be anyfor every sequence $(T_n)$ of positive linear operators on $C(K)$ satisfying, the condition $\|T_n(f)\to f\|_\infty\to 0$$\|T_n(f)- f\|_\infty\to 0$ for all $f\in A$ ensures that $\|T_n(g)\to g\|_\infty\to 0$$\|T_n(g)- g\|_\infty\to 0$ for anyall $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

Korovkin set of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $\|T_n(f)\to f\|_\infty\to 0$ for all $f\in A$ ensures $\|T_n(g)\to g\|_\infty\to 0$ for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

Korovkin subset of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\infty\to 0$ for all $f\in A$ ensures that $\|T_n(g)- g\|_\infty\to 0$ for all $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

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Tanmoy Paul
Tanmoy Paul
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Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $T_n(f)\to f$ uniformly$\|T_n(f)\to f\|_\infty\to 0$ for all $f\in A$ ensures $T_n(g)\to g$ uniformly$\|T_n(g)\to g\|_\infty\to 0$ for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $T_n(f)\to f$ uniformly for all $f\in A$ ensures $T_n(g)\to g$ uniformly for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $\|T_n(f)\to f\|_\infty\to 0$ for all $f\in A$ ensures $\|T_n(g)\to g\|_\infty\to 0$ for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.

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Tanmoy Paul
Tanmoy Paul
  • 521
  • 3
  • 13

Korovkin set of $C(\mathbb{T})$

Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if (T_n) be any sequence of positive linear operators on $C(K)$ satisfying $T_n(f)\to f$ uniformly for all $f\in A$ ensures $T_n(g)\to g$ uniformly for any $g\in C(K)$.

The famous Korovkin Theorem says $\{1,x,x^2\}$ is a Korovkin set for $C_{\mathbb{R}}[0,1]$.

My question is is it true that $\{1,z,z^2\}$ is Korovkin set for $C_{\mathbb{C}}(\mathbb{T})$?. Here $\mathbb{T}$ is the unit circle in the complex plane.