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if we consider a [itex]n\times n[/itex] symmetric positive definite matrix, we can prove that it hasnpositive eigenvalues andnorthogonal eigenvectors, and that such matrix can be expressed as a linear combination [itex]\sum_{i=1}^n \lambda_i e_i\otimes e_i[/itex]

Mercer's theorem extends this result to continuous symmetric positive definite functions [itex]K:[a,b]\times [a,b]\rightarrow \mathbb{R}[/itex] by stating thatK(x,y)can be expressed as [itex]\sum_{i=1}^\infty \lambda_i e_i(x)e_i(y)[/itex] wheree_iare eigenfunctions of the linear operator associated withK.

My question is: can Mercer's theorem be generalized tosquare integrablefunctionsKdefined on the whole domain [itex]\mathbb{R}^2[/itex] instead of just [itex][a,b]\times[a,b][/itex]?

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# Question on Mercer's theorem

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