Exact Asymptotic Behaviour of the Singular Values of Integral Operators With the Kernel Having Singularity on the Diagonal

Milutin Dostanić

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Milutin Dostanić (1997)

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Asymptotic behavior of singular values of certain integral operators.

Dostanić, R.Milutin, Milinković, Z.Darko (1997)

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Exact asymptotic behavior of singular values of a class of integral operators

Milutin R. Dostanić (1999)

Czechoslovak Mathematical Journal

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We find an exact asymptotic formula for the singular values of the integral operator of the form $\int_{Ω} T (x, y) k (x - y) \cdot d y L^{2} (Ω) \to L^{2} (Ω)$ ( $Ω \subset ℝ^{m}$ , a Jordan measurable set) where $k (t) = k_{0} ({(t_{1}^{2} + t_{2}^{2} + ... t_{m}^{2})}^{\frac{m}{2}})$ , $k_{0} (x) = x^{α - 1} L (\frac{1}{x})$ , $\frac{1}{2} - \frac{1}{2 m} < α < \frac{1}{2}$ and $L$ is slowly varying function with some additional properties. The formula is an explicit expression in terms of $L$ and $T$ .