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The general idea of this paper is that a limit value always can be written as the sum of an infinite series. The latter is recast in the form of a Stieltjes integral, which can be treated numerically by means of mechanical quadrature rules. This procedure is used for a unified and effective treatment of various problems as the summation of slowly convergent series and double series, as well as integrals over infinite ranges. The results of this paper are based on the methods presented in S.-Å. Gustafson [4] for the summation of certain series. By proper selection of weight functions rapid convergence and accurate results are often obtained, which is demonstrated on several examples. Realistic and strict bounds for the influence of computational errors are easily constructed as well.

In recent papers, the Prufer transformation has been used to calculate the eigenvalues of second order Sturm–Liouville boundary value problems. In the present paper, a Prüfer transformation for fourth order vibration problems is devised using four-dimensional spherical coordinates. Some properties of the resulting angular system are investigated. These results are then used to develop a new computational procedure for finding the eigenvalues of a vibrating beam. This procedure is used to compute the eigenvalues of several singular and nonsingular problems.

The starting point of this paper is a theorem, based on the theory of functions of a complex variable, which essentially gives a representation of the remainder in a quadrature rule as a contour integral. From this theorem it is shown how most of the well-known interpolatory and non-interpolatory quadrature rules may be derived. Some new quadrature rules are also given. The latter part of the paper is devoted to the evaluation of asymptotic estimates of the remainder, by consideration of particular examples. It is shown how this asymptotic analysis frequently gives excellent estimates of the remainder.

In a previous paper concerning linear multistep methods with mildly varying coefficients for the numerical solution of a single ordinary differential equation, a certain stabilizing condition was derived. Satisfaction of this condition ensured, under certain hypotheses, that the parasitic solutions of the difference equation which arise when such a method is applied to a nonlinear differential equation would not dominate the genuine solution. In this paper, it is shown that the application of a generalization of this stabilizing condition to a somewhat wider class of linear multistep methods—which are applicable to systems of differential equations—results in a class of methods with variable matrix coefficients. It is shown that methods of this class, with suitably chosen coefficients, possess a stability property similar to A-stability, and the asymptotic stability behavior of the methods when applied to a variable coefficient linear system is also investigated. Explicit and implicit examples of the new methods of order up to four are quoted, and numerical tests are reported.

This paper is primarily concerned with the problem of finding approximants for the solution of initial boundary value problems of parabolic type and obtaining a priori error estimates.Perhaps the most widely used method for the treatment of such problems is that of finite differences. Some difficulties arising in this method are stability of the scheme and the treatment of the boundary. Another approach to this problem is the so-called semidiscrete variational method. As in the finite difference approach, a difficulty in this method is the treatment of boundary values.The methods considered in this paper will be of the least-squares type. Roughly speaking, the method consists in finding that element of a suitable finite-dimensional subspace which best approximates the data of the initial-boundary value problem in a least-squares sense. A principal virtue of this approach is that the elements of the finite-dimensional subspace are not required to satisfy any boundary or initial conditions. A particular choice of the finite-dimensional subspace could be a certain class of mixed homogeneous splines.Results are obtained which show that, for a particular least-squares scheme, the error made in approximating the solution, u, of the initial boundary value problem by the least-squares approximant is the same (modulo a constant) as the error made in approximating u by any element of the finite-dimensional subspace. The proofs of the error estimates are based on the theory of interpolation spaces and the theory of approximation in Sobolev spaces.

Consider the Cauchy transform \[T\varphi (z) = \frac{1}{{\pi i}}\int_\Gamma {\frac{{\varphi (\zeta )d\zeta }}{{\zeta - z}},} \quad z \in \Gamma ,\] in which $\Gamma $ is a simple closed curve with a continuously differentiable parameterization. In order to have direct methods for the numerical solution of equations involving $T\varphi (z)$, it is desirable to have numerical integration methods for evaluating $T\varphi (z)$. In this paper, numerical methods are investigated which are based on replacing $\varphi (z)$ by a uniformly convergent sequence $\varphi _n (z)$; if these approximations are “sufficiently smooth”, then the speed of convergence of $T_n \varphi $ to $T\varphi $ is “essentially the same” as the speed of convergence of $T\varphi _n $ to $\varphi $. Specific methods are analyzed, with $\varphi _n (z)$ defined as a piecewise linear or quadratic interpolating function to $\varphi (z)$ at a given set of nodes on $\Gamma $. Numerical examples conclude the paper.

In this paper the authors investigate the cases of equality in some generalizations of Schur’s inequality relating the eigenvalues and singular values of an $n \times n$ complex matrix. For example, it follows from the results in the paper that if Ais an $n \times n$ complex matrix with eigenvalues $\lambda _1 , \cdots ,\lambda _n $ (with $|\lambda _1 | \geqq \cdots \geqq |\lambda _n $) and singular values $\alpha _1 \geqq \cdots \geqq \alpha _n $ and if $h(x_1 , \cdots ,x_k )$ is a nonzero symmetric polynomial in $x_1 , \cdots ,x_k ,1 \leqq k \leqq n$, then for $q > 0$, \[h(|\lambda _1 |^q , \cdots ,|\lambda _k |^q ) \leqq h(\alpha _1^q , \cdots ,\alpha _k^q );\] if $h(x_1 , \cdots ,x_k )$ is not of the form $\Sigma _{j = 0}^r d_j (\Pi _{t = 1}^k x_t )^j ,d_j \geqq 0$, then equality holds only if there is a unitary matrix Usuch that $U^ * AU = {\operatorname{diag}} (\lambda _1 , \cdots ,\lambda _k ) + T$, where Tis an $(n - k)$-square upper triangular matrix.

This expository paper is based on the invited lecture presented by the author at the SIAM National Meeting in June 1970. Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of $\Lambda $-stability.As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations. Numerical examples are included which illustrate the theory presented in the paper.

The first part of this paper considers the effect on the solution of the difference equations from the general self-adjoint elliptic second order partial differential equation of a periodicity condition in the x-direction. There are different effects according to whether n, the number of mesh lengths in this direction, is odd or even, but for Laplace’s equation the asymptotic rates of convergence of the iteration are independent of its actual value. Hence attention can be focused on the number of mesh lengths in the other direction. Results are given for small numbers of these as well as large.The periodicity condition can either be regarded as due to a repeating pattern in a plane or due to the region's being the surface of a solid of revolution. Hence the region can be regarded either as a rectangular region with a periodicity condition, or as a region with the error function vanishing on the boundary to which any theorems can be applied which do not depend on the region being simply connected. In contrast to the results for the Dirichlet problem for Laplace’s equation, the A.D.I. fixed parameter method is asymptotically slower than point S.O.R.The second part of the paper considers the effect of a condition of periodicity in one direction on the difference equations from the biharmonic equation. Alternating direction implicit methods of both Peaceman–Rachford and Douglas–Rachford type are considered with a rectangular region as part of a repeating pattern in a plane. Also k-line block successive overrelaxation and dynamic relaxation are considered with the corresponding surface of a cylinder. The various results are compared in a table.

In this paper, we apply a 2-dimensional extension of Peano’s theorem, due to Sard, to derive error bounds for a fully symmetric cubature rule. The weight function is assumed to be unity and the region of integration is the square $[ { - 1,1} ] \times [ { - 1,1} ]$.The error bounds we establish may be used to compare the relative merits of different fully symmetric rules. By analogous techniques, the 2-dimensional results obtained in this paper may be extended to higher dimensions.

We introduce a new three-stage process for calculating the zeros of a polynomial with real coefficients. The algorithm finds either a linear or quadratic factor, working completely in real arithmetic. In the third stage the algorithm uses one of two variable-shift iterations corresponding to the linear or quadratic case. The iteration for a linear factor is a real arithmetic version of the third stage of the algorithm for complex polynomials which we studied in an earlier paper. A new variable-shift iteration is introduced in this paper which is suitable for quadratic factors. If the complex algorithm and the new real algorithm are applied to the same real polynomial, then the real algorithm is about four times as fast.We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order.The problem and algorithm may be recast into matrix form. The third stage is a quadratic form of shifted inverse powering and a quadratic form of generalized Rayleigh iteration.The results of extensive testing are summarized. For an ALGOL W program run on an.IBM 360/67 we found that for polynomials ranging in degree from 20 to 50, the time required to calculate all zeros averaged $2n^2 $ milliseconds.An ALGOL 60 implementation of the algorithm and a program which calculates a posteriors bounds on the zeros may be found in Jenkins’ 1969 Stanford dissertation [2].

In this paper we develop the Ritz–Trefftz algorithm in the context of a useful class of distributed parameter control problems. More particularly, we treat a typical linear parabolic control problem ${{\partial v}/{\partial t}} = A(x,t){{\partial ^2 v}/ {\partial x^2 + B(x,t)u}}$, with $\lim _{t \to 0^ + } v(x,t) = v_0 (x),\alpha v(0,t) + ({{\partial v} / {\partial x}})(0,t) = cf_1 (t)$, and $\beta v(1,t) + ({{\partial v}/{\partial x}})(1,t) = df_2 (t)$. The cost functional Jis given by $J[u,f_1 ,f_2 ] = \frac{1} {2}\int_0^T {\int_0^1 {[\langle {v,Q,(x,t)v} \rangle + \langle {u,R(x,t)u} \rangle ]dt\,dx + \frac{1} {2}\int_0^T {[\langle {f_1 ,r(t)f_1 } \rangle + \langle {f_2 ,s(t)f_2 } \rangle ]dt} } } $. Let $\Sigma $ be a suitable space of piecewise bicubic polynomials on a uniform mesh of norm hon the rectangle $[0,1] \times [0,T]$. It is then shown that the Ritz–Trefftz method for minimizing $J[ \cdot ]$ over $\Sigma $ leads to an approximation to $J[ \cdot ]$ of order $O(h^4 )$. It is also shown that the computed quadruple $(\bar u,\bar f_1 ,\bar f_2 ,\bar v)$ converges to the optimal quadruple $(u^ * ,f_1^ * ,f_2^ * ,v^ * )$ with an order of $O(h^2 )$. Similar statements are made for piecewise polynomial approximations of arbitrary positive order. Numerical results will appear in a forthcoming paper.

An estimate for the error of derivative interpolation is obtained. The particular interpolation to which this estimate refers has been considered in a previous paper ; the present paper is an extension of this previous work, and is of special importance for the finite element method.

Consider an operator $Q:X_1 \to X_2 $ where $X_1 $ and $X_2 $ are normed linear spaces. Let $\delta _1 , \cdots ,\delta _m $ be functionals on $X_1 $ and let $H_2 $ be a finite-dimensional affine subspace of $X_2 $. In this paper we consider the problem \[ Q(x) \in H_2 \quad {\text{subject to}}\quad \delta _i (x) = \beta _i, \quad i = 1, \cdots ,m.\]A generalized form of Newton’s method which approximates a solution of this problem by a sequence of the form \[ x_{n + 1} = x_n - \Gamma _n p_n (x_n ),\qquad n = 0,1, \cdots , \] is presented. It is shown that many important constrained problems including the ordinary differential equation boundary value problem and the Euler–Poisson equation from the calculus of variations fall very naturally within this theory.

This paper extends the direct method of cyclic reduction to linear systems which result from the discretization of separable elliptic equations with Dirichlet, Neumann, or periodic boundary conditions. For an $m \times n$ net, the operation count is proportional to $mn\log _2 n$ and $mn$ storage locations are required.

The solution of the Volterra integral equation, \[ ( * )\qquad x(t) = g_1 (t) + \sqrt {t}g_2 (t) + \int _0^t \frac {K(t,s,x(s))} {\sqrt {t - s} } ds, \quad 0 \leqq t \leqq T,\] where $g_1 (t)$, $g_2 (t)$ and $K(t,s,x)$ are smooth functions, can be represented as $x(t) = u(t) + \sqrt {t}v(t) $,$0 \leqq t \leqq T$, where $u(t)$, $v(t)$ are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate $x(t)$ via $u(t)$, $v(t)$ in a neighborhood of the origin and use (*) on the rest of the interval $0 \leqq t \leqq T$. In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson’s rule are treated in detail. The schemes are shown to be convergent of order $h^{{7 / 2}} $. Asymptotic error estimates are derived in order to examine the numerical stability of the methods.

This paper presents error estimates for finite difference approximate solutions of initial-boundary problems for parabolic partial differential equations. Methods having matrices of essentially positive type are considered using many techniques from Bramble, Hubbard and Thomeé [2] where elliptic partial differential equations were examined.It will be shown that if the approximation has “accuracy” $\nu $, the right-hand side of the differential equation fhas “smoothness” $\lambda $ and the initial and boundary data $\varphi $ and $\Phi $ have “smoothness” $\mu $, then letting$| \cdot |_A^{(r)} $ denote a Hölder-type norm of order rin the set Awe have for $\mu \ne \nu $ that the maximum error is bounded by \[ C\{ h^{\min (\lambda ,v)} | f |_\mathcal{L}^{(\lambda )} + h^{\min (\mu ,\nu )} | {\varphi ,\Phi } |_{\mathcal{R} \times \mathcal{L}}^{(\mu )} \} \] and for $\mu = \nu $ and $f \equiv 0$, this error is dominated by \[ Ch^\nu \left( {\ln \frac{1}{h}} \right)| {(\varphi ,\Phi )} |_{\mathcal{R} \times \partial \mathcal{L}}^{(\nu )} .\]

This paper examines the order of convergence of Runge–Kutta methods for the ordinary differential equation $y'(t) = f(t,y(t))$ in Banach space, where the function fis badly behaved off the solution trajectory, and the solution yhas derivatives only of low order. Under the assumption that fis everywhere Lipschitz continuous in its second argument, we get orders of convergence for the case when fhas no derivatives at all, and for when it has only directional derivatives along the solution trajectory $y(t)$.Utilization of the authors’general theory of convergence, stability and consistency enables us to obtain results which apply to actual computations in the presence of rounding error.Some numerical experiments which support the conjecture that these convergence results are the best possible are briefly described.

This paper fills a gap in the theory of multipoint iteration function exemplified by the question: How does the secant method converge to a multiple zero? To answer this question a theory for the asymptotic behavior of a certain type of homogeneous, nonlinear difference, equation is developed, and it is shown that two broad classes of iterative methods can fit this theory. The results of numerical investigations based on the theory suggest that Muller’s method applied to a zero of multiplicity greater than 2 will inevitably produce complex iterates.

In this paper, we will present and analyze an algorithm for finding ${\bf x}$ and $\lambda$ such that \[ A{\bf x} = \lambda B{\bf x},\] where Aand Bare $n \times n$ matrices. The algorithm does not require matrix inversion, and may be used when either or both matrices are singular. Our method is a generalization of Rutishauser’s $LR$-method [20] for the standard eigenvalue problem $A{\bf x} = \lambda {\bf x}$ and closely resembles the $QZ$-algorithm given by Moler and Stewart [13] for the generalized problem given above. Unlike the $QZ$-algorithm, which uses orthogonal transformations, our method, the $LZ$-algorithm, uses elementary transformations. When either Aor Bis complex, our method should be more efficient.

This paper concerns a class of update methods, or, as they are also called, quasi-Newton methods, variable metric methods, or modification methods. A general theory of rank-one and symmetric rank-two update formulas is presented which covers many of the special methods proposed in the literature. Recently, Broyden, Dennis and Moré found a local convergence theorem for a class of these methods. A new and unified proof of this theorem is given here which uses a geometrically more intuitive and also more general convergence condition than the original theorem. The proof utilizes elliptic-norm estimates to derive a majorizing system of difference inequalities. Then more careful estimates involving a generalized Frobenius norm show that under rather general conditions the methods under consideration are superlinearly convergent.

Let Xbe a normed linear space and let f, $\delta _1 $, $ \cdots $, $\delta _m $, be functionals defined on X. In this paper we extend the theory of Newton’s method developed by the author in [8] too the problem \[{\text{Minimize}}\quad f(x)\] subject to \[ \delta _i (x) = 0,\quad i = 1, \cdots ,m.\] A generalized form of Newton’s method which approximates a solution of this problem by a sequence of the form \[ x_{n + 1} = x_n - \Gamma _n P_n (x_n ),\quad n = 0,1,2, \cdots ,\] is presented.

H. L. Stone has proposed an iterative method, here called a factorization method, to solve elliptic difference equations. Because of the complexity of the algorithm defining the iteration matrix, Stone described the procedure as strongly implicit. Analysis of its convergence properties is difficult. One reason is that the iteration matrix is nonsymmetric. This motivates the construction of symmetric factorization methods. In this paper the class of strongly implicit symmetric factorization methods is determined. From this, it is possible to determine the class of such methods that are second order. This is a property of the Stone method that seems fundamental to its success. Since the iteration matrix of any of the resulting methods is defined by an algorithm with undesirable numerical properties, it may be inferred that no numerically satisfactory second order strongly implicit symmetric factorization method exists.

Practical interpolation schemes for functions of more than one variable need not be finite dimensional as in the case of univariate functions. The so-called “blending function methods” are one example of such transfinite schemes. Another is the method of “pseudo-harmonic interpolation” described in this paper. For any bounded, convex planar domain $\mathcal{D}$, we show how to construct a bivariate function Uwhich interpolates to arbitrary Dirichlet boundary conditions on the perimeter of $\mathcal{D}$. The interpolant $U(Q)$ satisfies the familar “maximum principle” and, in the case in which the domain $\mathcal{D}$ is circular, is actually a harmonic function in $\mathcal{D}$. Higher order schemes are described in which the function $U(Q)$ interpolates to specified normal derivatives on the boundary of $\mathcal{D}$ as well as to the Dirichlet boundary conditions. The interpolation schemes are also viewed as idempotent linear operators (i.e., projectors) on families of continuous functions on $\overline{\mathcal{D}} $, and the algebraic and interpolatory properties of such projectors are examined. Numerical; techniques for evaluating the interpolant Uare described together with examples, illustrations and applications.

This paper is concerned with best two-sided scaling of a general square matrix, and in particular with a certain characterization of that best scaling: namely that the first and last singular vectors (on left and right) of the scaled matrix have components of equal modulus. Necessity, sufficiency, and its relation with other characterizations are discussed. Then the problem of best scaling for rectangular matrices is introduced and a conjecture made regarding a possible best scaling. The conjecture is verified for some special cases.

Certain methods for the approximate solution of $x - Kx = 0$, where Kis a compact operator, lead to a family of operator equations $x - K_n x = 0$, $n \geqq 1$, where $K_n $,$n \geqq 1$, are collectively compact and converge pointwise to K. In this paper a new and simplified proof of the existence and uniqueness of a solution of $x - K_n x = 0$ in a neighborhood of an isolated solution of $x - Kx = 0$ is given.

A theorem of Stroud and Mysovskikh ascertains that we can construct a two-dimensional integration formula of degree $N = 2m - 1$ with the $m^2 $ distinct, finite common zeros of two orthogonal polynomials $P1(x,y)$ and $P2(x,y)$ of degree mas points.This paper gives the pairs of orthogonal polynomials of second degree, dependent on a characteristic number introduced by Mysovskikh, whose zeros give rise to a third degree integration formula with four real points and positive weights.

This paper presents a new algorithm for solving systems of nonlinear equations, which combines elements of the method of local variations with a sequential secant method. The new algorithm converges superlinearly and its convergence does not depend upon a good initial guess at a solution.

In this paper, we show that the $H^{m,\infty } $-splines obtained by taking the limits of the $H^{m,p} $-splines, as $p = p_n \to \infty $, are unique for certain data. In particular, when Hermite data is specified, the $H^{m,p} $-splines converge to a unique $H^{m,\infty } $-spline in $H^{m,2} $. This method enables us to calculate the “natural” Hermite $H^{2,\infty } $-splines explicitly.

This paper considers two classes of iterative procedures for the computation of a real solution $\xi $ of the nonlinear equation $f(x) = 0$. In the first class of methods we choose a sequence $\{ F_i \} $ of approximations to fand coincidentally define a sequence $\{ x_i \} $ of approximations to $\xi $, by choosing $x_{i + 1} $ as a zero of $F_i $. In the second class we suppose fhas an inverse function gon some interval K, choose a sequence $\{ G_i \} $ of approximations to gand coincidentally define a sequence $\{ x_i \} $ of approximations to $\xi $ by choosing $x_{i + 1} = G_i (0)$. Several convergence theorems for both of these classes of algorithms are presented along with several special cases.

Justification of the method proposed in the first part of the paper is completed.

This paper contains $L^\infty $-error bounds for piecewise Lagrange polynomial approximation over (i) a Cartesian product of polyhedral domains, (ii) a rectangular parallelepiped, and (iii) the Cartesian product of a domain of the type (i) and a domain of the type (ii). Bounds are given for the error as well as all its possible partial derivatives, in terms of characteristic mesh lengths of the partitions, for functions with specified smoothness properties.

In this paper, a multiple exchange algorithm for calculation of best restricted approximations is presented and its convergence is proved. The algorithm presented is an extension of the single exchange algorithm for calculation of best restricted approximations which was recently given by Taylor and Winter [1]. Based on computational experience with a class of approximation problems encountered in linear phase digital filter design, a comparison of single exchange and multiple exchange convergence rates and C.P.U. times per iteration is given.

Two error bounding methods for two-point boundary value problems are presented. Using the simple factorization method and interval analysis we find the numerical solution of two-point boundary value problems of linear type with complete error bounds. The numerical solutionis given as an interval which always contains the exact solution.In the second part of this paper we present a corrector method applicable to linear and nonlinear problems—the $(2 \times 2)$ interval discretization method—which is a slight modification of Hansen’s method [3] and enables us to improve previously given results.

The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems. The purpose of this paper is to extend this concept to rectangular linear systems \[ ( * )\qquad Ax = b, \] in two directions. Firstly, this is accomplished by replacing $A^{ - 1} $ by $A^\dag $, the Moore–Penrose inverse of A, and, secondly, by considering matrices that leave a cone invariant.Let $A \in R^{m \times n} $. The splitting $A = M - N$ is called proper if $R(A) = R(M)$ and $N(A) = N(M)$. Such is the case when Aand Mare nonsingular. Consider the iteration \[ ( * * )\qquad x^{i + 1} = M^ \dag Nx^i + M^ \dag b. \] It is shown that for a proper splitting, $\rho (M^ \dag N)$ (the spectral radius of $M^ \dag N$) is less than l, if and only if the iteration (**) converges to $A^ \dag b$, the best least squares approximate solution of the system (*). This approach has the advantage of avoiding the normal system $A^T Ax = A^T b$ in solving (*). Necessary and sufficient conditions for $\rho (M^ \dag N) < 1$ are given, extending results of Varga, Ortega and Rheinboldt, Mangasarian, and Vandergraft. Results of Collatz and Schroder and of Mangasarian on monotone iterations in the nonsingular case are also extended.

The purpose of this paper is to present a method for estimating the error constants required for the application of a cubature error bound due to Chawla [2].

Some established results concerning best $L_p $-approximations are introduced, and it is shown that a minimax approximation can generally be found as the limit of a sequence of such approximations. The values of prequired, however, are too large to be practicable, and the main part of this paper is devoted to the development of an extrapolation scheme which enables minimax approximations to be predicted from best $L_p $-approximations for comparatively small values of p. Numerical evidence is presented to show that the scheme works well in practice.

In this paper we obtain error bounds for the quadrature error in a fully symmetric integration rule. The function to be integrated is assumed to be analytic in the disc $| z | \leqq r$,$r > 1$. The techniques employed are generalizations of methods suggested by Lanczos [14] and Stenger [17] for treating the error in Gaussian integration.

The problem of the existence and construction of a table of interpolating rational functions is considered. Some general properties of this table and relations between its elements are proved. Sufficient conditions for the existence of continued fractions whose convergents form the elements of the table are given.The results in this paper are mainly obtained by using methods which are similar to certain methods used in the theory of Padé approximation.

In this paper some of the numerical problems associated with computing the generalized inverse of a matrix are discussed and illustrated by a detailed analysis of an iteration of Ben-Israel and Cohen.

Optimal approximation in Hilbert spaces can be found if the space has a reproducing kernel function and if this kernel can be obtained in closed form. This paper contains the reproducing kernel function for the general Sard spaces of the type B. A numerical illustration of optimal approximation for a “nonstandard” Hilbert space is given.