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Fixed definition of positive definite
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Francois Ziegler
Francois Ziegler
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Recall that a function $f: \mathbb{R} \longrightarrow \mathbb{C}$$f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}$$x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with the entries $$a_{ij} = f(x_i - x_j)$$$$a_{ij} = f(x_j - x_i)$$ is positive semi-definite.

My question is: Let $g \in L^2(\mathbb{R}^n)$. Is the function $$f(z, y) =\int_{\mathbb{R}^n} g(x - y) \overline{g(x-z)} d x$$ positive definite on $\mathbb{R}^{2n}$?

Recall that a function $f: \mathbb{R} \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}$, the matrix $(a_{ij})$ with the entries $$a_{ij} = f(x_i - x_j)$$ is positive semi-definite.

My question is: Let $g \in L^2(\mathbb{R}^n)$. Is the function $$f(z, y) =\int_{\mathbb{R}^n} g(x - y) \overline{g(x-z)} d x$$ positive definite on $\mathbb{R}^{2n}$?

Recall that a function $f: \mathbb{R}^d \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}^d$, the matrix $(a_{ij})$ with entries $$a_{ij} = f(x_j - x_i)$$ is positive semi-definite.

My question is: Let $g \in L^2(\mathbb{R}^n)$. Is the function $$f(z, y) =\int_{\mathbb{R}^n} g(x - y) \overline{g(x-z)} d x$$ positive definite on $\mathbb{R}^{2n}$?

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Matthias Ludewig
Matthias Ludewig
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Positive Definiteness of a certain function

Recall that a function $f: \mathbb{R} \longrightarrow \mathbb{C}$ is positive definite, iff for all numbers $N$ and $x_1, \dots, x_N \in \mathbb{R}$, the matrix $(a_{ij})$ with the entries $$a_{ij} = f(x_i - x_j)$$ is positive semi-definite.

My question is: Let $g \in L^2(\mathbb{R}^n)$. Is the function $$f(z, y) =\int_{\mathbb{R}^n} g(x - y) \overline{g(x-z)} d x$$ positive definite on $\mathbb{R}^{2n}$?