Symmetric kernel

Builds on Kernel of an integral equation

Definition

Let $a < b$ . A symmetric kernel on $[a, b]$ is a real-valued kernel $K \colon [a, b] \times [a, b] \to \R$ such that $K(x, s) = K(s, x) \quad \text{for all } x, s \in [a, b].$

Remarks

Example: $K(x, s) = \min(x, s)$ on $[0, 1]$ . The check: the minimum of two numbers does not depend on their order.

Non-example: $K(x, s) = x - s$ . Here $K(1, 0) = 1$ and $K(0, 1) = -1$ .

A symmetric kernel is a Hermitian kernel that takes real values. For square-integrable kernels the condition is often required only for almost all $(x, s)$ . Some authors call a complex kernel with $K(x, s) = K(s, x)$ symmetric as well; that condition is different from the Hermitian one.

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Kernel of an integral equation concept
Symmetric kernel definition

Tag	`XY7H05ANPD`
URI	`m2://XY7H05ANPD`
Kind	definition
Writing	W0 stub
MSC	45C05, 45P05
Defines	Symmetric kernel
Uses	Kernel of an integral equation

Symmetric kernel

Definition

Remarks

Reading path

Further reading

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