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The paper studies a finite element algorithm giving approximations to the minimum-norm solution of ill-posed problems of the form A f = g Af = g , where A is a bounded linear operator from one Hubert space to another. It is shown that the algorithm is norm convergent in the general case and an error bound is derived for the case where g is in the range of A A ∗ A{A^\ast } . As an example, the method has been applied to the problem of evaluating the second derivative f of a function g numerically.

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.

A critical system consisting of a regular infinite array of cylindrical channels of any cross section in a homogeneous multiplying medium is divided into equivalent cells of finite height. For such...

Friedrichs’ inequality for the Sobolev space $H^1 (0,s),s > 0$, is $\|f\|_1 \leq c(f(0)^2 + f(s)^2 + \|f'\|^2 )^{{1 / 2}} ,f \in H^1 (0,s)$, where $f'$ is the first derivative of f and $\| \cdot \|_1 $ and $\| \cdot \| $ are the norms of $H^1 (0,s)$ and $L_2 (0,s)$ respectively. The usual proofs for the existence of such a constant c are based on nonconstructive functional analytic ideas. It is shown that the least constants c can be evaluated by variational techniques and are given by the unique solution of a simple transcendental equation. In the special case $s=1$ one has, for example, $c = 1.07869 \cdots $.

Let f be an element of C [o, 1] and {g 1 ,..., g n } a Markoff system in C [o, 1]. We consider the last set as a subspace of L [o, 1]. Due to a known theorem of D. Jackson there exists for every f in C[o, 1] a unique best L 1 -approximation in the sub-space sp{g 1 ,..., g n } spanned by {g 1 ,..., g n } in L 1 [o,1]. 1967, K. H. USOW [4] established an algorithm for the computation of such best approximations. However, Usow succeeded to prove the convergence of his algorithm only for conditions on f and on {g 1 ,..., g n } which are fairly restrictiv and difficult to verify.