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An algorithm is developed for the numerical computation of minimum norm solutions of Fredholm integral equations of the first kind $Kf = g$. The algorithm is different from Tihonov’s method and avoids direct collocation. The method has the following geometrical interpretation: One is looking for a vector of minimum norm that lies in a given finite dimensional subspace and also in a given neighborhood of the set of all solutions of $Kf = g$. A convergence proof for the algorithm is presented and four examples are treated numerically.

Variational methods are applied to evaluate the least constant c in Sobolev’s inequality for the imbedding of the Sobolev space $H^1 (0,s)$, where $(0,s)$ is a real interval, into the space $C(0,s)$ of bounded continuous functions on $(0,s)$. The method yields the exact solution explicitly. It is shown that $c = \tanh ^{ - {1 / 2}} (s)( \cong 1.1459{\text{ for }}s = 1)$ and that equality holds for the hyperbolic cosine function.