Showing papers on "Interpolation published in 1970"

The Reconstruction of a Three-Dimensional Structure from Projections and its Application to Electron Microscopy

Abstract: A transmission electron micrograph is essentially a projection of the specimen in the direction of view. In order to reconstruct a three-dimensional image of the specimen, it is necessary to be able to combine data from a number of different views. A formal solution of this problem is given in terms of Fourier transforms. Its realization requires data reduction and interpolation. The final solution is given by a least squares approach, which also indicates how many views must be included to give a valid reconstruction of a given particle to a given degree of resolu­tion. Interpolation procedures of varying power are given, to be employed according to the economy with which the available data must be used. An alternative procedure is described for direct reconstruction without the use of Fourier transforms, but it is shown to be in general less practicable than the Fourier approach.

Solution of Vandermonde Systems of Equations

Abstract: We obtain in this paper a considerable improvement over a method developed earlier by Ballester and Pereyra for the solution of systems of linear equations with Vandermonde matrices of coefficients. This is achieved by observing that a part of the earlier algorithm is equivalent to Newton's interpolation method. This allows also to produce a progressive algorithm which is significantly more efficient than previous available methods. Algol-60 programs and numerical results are included. Confluent Vandermonde systems are also briefly discussed.

Extrapolation and interpolation of quasi-linear operators on martingales

Abstract: 5. T h e ope ra to r s S a n d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Some app l i c a t i ons of S a n d s . . . . . . . . . . . . . . . . . . . . . . . . 281 R a n d o m walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 H a a r a n d W a l s h ser ies . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Loca l conve rgence of m a r t i n g a l e t r a n s f o r m s . . . . . . . . . . . . . . . . . 285

Trigonometric Interpolation Method for One‐Dimensional Quantum‐Mechanical Problems

Abstract: A rapidly converging difference method, based on harmonic analysis, is described. It can be applied to periodic or nonperiodic bound‐state problems of the general Sturm–Liouville type. Numerical examples for the Mathieu problem and for the harmonic oscillator show considerable accuracy. Advantages and disadvantages of the method are discussed in a comparison with Harris's matrix transformation technique and with direct integration methods. The set of difference equations representing a quantum‐mechanical problem constitutes a symmetric matrix eigenvalue problem which is approximately equivalent to the algebraic problem obtained by using a finite trigonometric basis. Basis functions associated with the difference method are related to the Dirichlet kernel. In an approximation which corresponds to the difference method, these basis functions can be treated in a similar way as Dirac's δ function.

Incremental Curve Generation

Abstract: The process of converting a mathematically defined curve into unit steps along a fixed axis in digital technique is known as interpolation. The representation of the curve may be parametric or nonparametric. The parametric representation is widely used since it lends itself to realization by ordinary DDA technique. However, the nonparametric representation is shown to be fully competitive. In many cases, e.g., circle generation, it seems to be advantageous because it eliminates the risk of curve degradation.

Absorption corrections for neutron diffraction

Abstract: Absorption corrections for cylindrical and spherical crystals have been evaluated by numerical integration for values of μR in the range 0 to 1.0, where μ is the mean linear absorption coefficient and R is the radius of the crystal. For this range of μR, which is that normally required for neutron diffraction data, interpolation from existing tables is unsatisfactory. The transmission factor Ahkl is tabulated as a function of μRand sin2θ, accurate to four decimal places. The intervals chosen for the tabulation are such that linear interpolation may be used for intermediate values. Analytical expressions, which may be used for calculating the transmission factor when a lower accuracy is acceptable, are also given.

Memory Effects in Cross-Modal Matching.

Abstract: Predictions based on the model advanced by Connolly & Jones (1970) regarding the storage systems involved in cross-modality matching tasks were examined. Adult subjects made intra-modal and cross-modal matches to a range of standard stimuli under three conditions: zero delay between presentation of standard and subject producing his match, interpolation of 10-sec. unfilled interval and interpolation of 10-sec. filled interval. The results confirmed previous findings about the properties of visual and kinaesthetic storage systems and their relation to central processing capacity. The hypothesis that translation takes place before storage in cross-modality matching tasks was also confirmed.

Error bounds for polynomial spline interpolation

Abstract: New upper and lower bounds for the L2 and L- norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. (1), and Schultz and Varga, cf. (5). 1. Introduction. In this paper, we derive new bounds for the L2 and LX norms of derivatives of the error in polynomial spline interpolation. These bounds improve and generalize the known error bounds, cf. (1) and (5), in the following important ways: (1) these bounds can be explicitly calculated and are not merely asymptotic error bounds such as those given in (1) and (5); (2) explicit lower bounds are given for the error for a class of functions; (3) the degree of regularity required of the func- tion, f, being interpolated is extended, i.e., in L1) and (5) we demand that the mth or 2mth derivative of f be in L2, if we are interpolating by splines of degree 2m - 1, while here we demand only that some pth derivative of f, where m ? p ? 2m, be in L2; and (4) bounds are given for high-order derivatives of the interpolation errors. 2. Notations. Let - o < a < b < o and for each positive integer, m, let Km(a, b) denote the collection of all real-valued functions u(x) defined on (a, b) such that u E Cm'-(a, b) and such that Dm-lu is absolutely continuous, with Dmu E L2 (a, b), where Du _ du/dx denotes the derivative of u. For each nonnegative integer, M,

Efficient Algorithms for Polynomial Interpolation and Numerical Differentiation

Abstract: Algorithms based on Newton's interpolation formula are given for: simple polynomial interpolation, polynomial interpolation with derivatives supplied at some of the data points, interpolation with piecewise polynomials having a continuous first derivative, and numerical differentiation. These algorithms have all the advantages of the corresponding algorithms based on Aitken-Neville interpolation, and are more efficient.

Besov spaces in theory of approximation

Abstract: In this paper we apply the theory of interpolation spaces to different parts of Approximation theory. We study the rate of convergence of summation processes of Fourier series and Fourier integrals. The main body of the paper is devoted to a study of the rate of convergence of solutions of difference schemes for parabolic initialvalue problems with constant coefficients and to related problems.

Accelerating convergence of trigonometric approximations

Abstract: Lanczos has recently developed a method for accelerating the convergence of trigonometric approximations for smooth, nonperiodic functions by modifying their boundary behavior. The method is reformulated here in terms of interpolation theory and is shown to be related to the theory of Lidstone interpolation. Extensions given include a new type of modifying function and the establishment of criteria for the convergence of associated interpolation series. Applications are given for the error function and its derivative.

Some Observations on Interpolation in Higher Dimensions

Abstract: This paper presents a method, based on successive displacement of the coordinates, both for finding suitable interpolation points and for constructing the interpolating polynomial for functions of more than one independent variable.

Sears function and lifting surface theory for harmonic gust fields

Abstract: Sears' function for the lift on a two-dimensional airfoil induced by a harmonic gust field in an incompressible flow is reviewed. It is observed that a much simpler function can be defined by a transformation of the gust reference point from the airfoil midchord to its leading edge. The simplicity of the modified Sears function permits accurate interpolation for the large number of reduced frequencies required in a gust frequency response analysis. Approximations to the modified Sears function are discussed for use in preliminary analysis. The form of the transformation appropriate for interpolation of gust loads obtained from lifting surface theory for finite aspect ratio wings is then discussed. Both oneand twodimensional gust fields are considered. It is shown by example calculations for compressible subsonic flow that the transformation required by lifting surface theory is analogous to that required by the two-dimensional incompressible case. It is concluded that the computational savings permitted by interpolation makes lifting surface methods economical for gust frequency response analysis. It is also concluded that the Doublet Lattice Method for subsonic flows has the versatility required for load calculations for both oneand two-dimensional gust fields.

Optimum influence radii for interpolation with the method of successive corrections1

Abstract: A method of selecting optimum influence radii for the objective analysis of a scalar field using the method of successive corrections is presented for an arbitrary weight function. The Cressman weight function is used in a computational verification of the result. A well-defined first pass optimum radius is found that increases with station separation, observational error, and wavelength of the true field for the average taken as the guess field.

Nonlinear Interpolation of Multivariable Functions by the Monte Carlo Method

Abstract: The nonlinear interpolation of functions of very many variables is discussed. Deterministic termwise assessment of a prohibitively large number of terms naturally leads to a choice of random sampling from these numerous terms. After introduction of an appropriate higher order interpolation formula, a working algorithm is established by the Monte Carlo method. Numerical examples are also given.

Trigonometric Interpolation and Curve-Fitting

Abstract: Some algorithms are introduced, whereby a function defined on an arbitrarily spaced set of abscissas may be interpolated or approximated by trigonometric or hyperbolic polynomials. The interpolation may be ordinary or osculatory. Least squares approximation is included; the approximant may be a pure sine series or a cosine series or a balanced trigonometric or hyperbolic polynomial. An application to a periodicity-search is described. An extensive set of algorithms is available for functional approximation and interpolation in terms of polynomials. The present article develops some corresponding algorithms for nonpolynomial approximants. The classes of approximant (interpolant) considered are sine polynomial, cosine polynomial, balanced trigonometric poly- nomial and their analogs in terms of hyperbolic functions. The classes of approxi- mation considered are interpolation on ordinates, osculatory and hyperosculatory interpolation, weighted least-squares approximation, weighted least-squares ap- proximation subject to some ordinate and derivative constraints. Trigonometric Analogs of Lagrange and Hermite Interpolation. Lagrangian and Hermitian interpolation in terms of sine polynomials were dealt with in (1), and the adaptation to cosine polynomials is straightforward. For Lagrangian interpolation in terms of balanced trigonometric polynomials there is a classical algorithm (2, p. 38), but we wish to develop an alternative which has some advantages with respect to economy and ease of generalization. Let there be N points (xi, fi), i = 1, * N; let all the abscissas xi be distinct and strictly within an interval I. It is required to construct a function 2) a

Interpolation functors and banach couples

Abstract: The theory of interpolation spaces originally arose from an attempt to generalize the classical interpolation theorems of M Riesz and Marcinkiewicz to a more abstract setting However it should more correctly be described as a theory of "families" of abstract spaces : Given a number of (usually two) spaces contained in a common "large" space, we try to find as many "families" of new such spaces as possible The primary goal is to gain a deeper insight into such classical cases as Lp spaces, Lipa spaces etc Thus ultimately the whole theory should be judged from the point of view to what extent it provides tools useful in other domains of analysis

Nonlinear Magnetic Field Analysis of DC Machines, Part III: Equipotential Plots Drawn by Computer

Abstract: A computer program for establishing the equipotential plots in the cross section of a dc machine is described. The necessary steps are as follows: 1) scanning of the gridlines in the cross section to find the points of definite magnetic potentials from the discrete vector potential; 2) finding the location of the point of the mesh section containing the given vector potential by interpolation; 3) expressing the positions of the points in rectangular coordinates and finding and printing out all points of the same vector potentials by a CalComp computer. This procedure is carried out for several values of the vector potentials to obtain a comprehensive plot of flux lines.

Computation of the Divergence Coefficient for Seismic Phases

Abstract: : By using a cubic spline interpolation method a representation of seismological travel time tables is achieved which is highly continuous. Divergence coefficients for seismic phases computed from this representation are free from non-essential discontinuities and are thus more meaningful than those obtained using other methods of interpolation. Results are compared with those obtained by others. For six phases results are given in graphs and tables. (Author)

Interpolation means for displacement measurement

Abstract: The invention contemplates employment of plural preset comparators, such as flip-flop set to trigger at different reference-voltage levels, to accurately identify and quantize discrete voltage levels in the electric-signal wave form by which displacement increments are tracked, or picked off, for measurement. The fast change of state in each of the elements of such a device makes possible the prompt, accurate and unambiguous identification of each particular displacement increment. The described device is inherently applicable to digital encoding for remote transmission, and several embodiments are described.