Abstract
We consider a linear Abel integral operatorA α: L2(0, 1) → L2(0, 1) defined by
. We construct a scale {Xβ}β∈ℝ of Hilbert spaces of functions in (0, 1) and relate it with a Hilbert scale of Sobolev spaces. Under suitable assumptions onK, we prove that\(\left\| {A_\alpha u} \right\|_{L^2 \left( {0,1} \right)} \) gives an equivalent norm inX α. On the basis of this equivalence, we find a lower and upper estimate for the singular values ofA α and, furthermore a Hölder estimate for\(\left\| u \right\|_{L^2 \left( {0,1} \right)} \) by\(\left\| {A_\alpha u} \right\|_{L^2 \left( {0,1} \right)} \) provided that\(\left\| u \right\|_{Xq} \), withq > 0 is uniformly bounded. Finally we discuss convergence rates of regularized solutions obtained by a Tikhonov method.
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Gorenflo, R., Yamamoto, M. Operator theoretic treatment of linear Abel integral equations of first kind. Japan J. Indust. Appl. Math. 16, 137 (1999). https://doi.org/10.1007/BF03167528
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DOI: https://doi.org/10.1007/BF03167528