Showing papers on "Interpolation published in 1975"
TL;DR: In this article, the exchange-correlation energy of a metal surface is analyzed in terms of the wavelength of the fluctuations which contribute to it, and a scheme is proposed to interpolate between the shortwavelength region properly described by the local density functional approximation and the long-wavelength regions for which an exact limiting form is established.
749 citations
TL;DR: In this article, a general theory of multistage decimators and interpolators for sampling rate reduction and sampling rate increase is presented, and a set of curves and necessary relations for optimally designing multi-stage decimator is also given.
Abstract: In this paper a general theory of multistage decimators and interpolators for sampling rate reduction and sampling rate increase is presented. A set of curves and the necessary relations for optimally designing multistage decimators is also given. It is shown that the processes of decimation and interpolation are duals and therefore the same set of design curves applies to both problems. Further, it is shown that highly efficient implementations of narrow-band finite impulse response (FIR) filters can be obtained by cascading the processes of decimation and interpolation. Examples show that the efficiencies obtained are comparable to those of recursive elliptic filter designs.
232 citations
TL;DR: In this paper, it is shown how to obtain the corresponding approximate polynomial solution for singular integral equations by means of Gaussian quadrature, and for some special cases compact formulas are given for the strength of the singularities at the endpoints of the integration interval.
Abstract: On the basis of integration of singular integral equations by means of Gaussian quadrature, it is demonstrated how to obtain the corresponding approximate polynomial solution. For some special cases compact formulas are given for the strength of the singularities at the endpoints of the integration interval.
205 citations
TL;DR: The design of optimum interpolators for band-limited sequences is considered and newly found properties of these interpolators are discussed and used to simplify the design.
Abstract: The design of optimum interpolators for band-limited sequences is considered. Newly found properties of these interpolators are discussed and used to simplify the design.
141 citations
TL;DR: In this paper, the spectral decay rate at high frequency was used as a criterion for positive definability of the correlation function. And the spectral spectral spectral properties of the spectral transform of a correlation function were investigated.
Abstract: Objective analyses using the so-called method of optimum interpolation incorporates statistical information on the variable(s) by means of the covariance or correlation functions. The concern in this contribution is with some properties of the analytic forms of the correlation functions that are used to model the statistical structure. First, some attention is directed to the question of fitting the various analytic forms (containing adjustable constants) to samples of actual correlations. All but one of the candidate forms were indistinguishable on the basis of the residuals of the statistical fitting procedure. Second, the criterion of positive-definiteness of the correlation function is extended to stipulate that the transform (or spectrum) of the function should possess some features of the spectra of actual variables—the most important one being the spectral decay rate at high wavenumber. Again, all but one of the candidate forms (the same one as above) had transforms that were acceptable. T...
78 citations
TL;DR: In this article, statistical properties of the histospline density estimate of Boneva-Kendall-Stefanov-Schoenberg (BKS) are found.
Abstract: : Statistical properties of the histospline density estimate of Boneva-Kendall-Stefanov-Schoenberg are found. This density estimate is the derivative of a cubic spline of interpolation to the sample cumulative distribution at equally spaced points. The spacing of these points is chosen in an optimal manner. The boundary values of the spline are estimated from the data. It is shown that the mean square convergence rate of this density estimate at a point achieves the best obtainable rate. (Author)
73 citations
TL;DR: In this paper, the authors proposed a method for smooth interpolation to boundary data on a triangle, and sufficient conditions are given so that the functions when pieced together form a C N−1 (Ω) function over a triangular subdivision of a polygonal region Ω and the precision sets of the interpolation functions are derived.
69 citations
TL;DR: In this article, a polynomial interpolant which interpolates a function F E CN(T), and its derivatives of order N and less, on the boundary aT of a triangle T was derived.
Abstract: Boolean sum interpolation theory is used to derive a polynomial interpolant which interpolates a function F E CN(T), and its derivatives of order N and less, on the boundary aT of a triangle T. A triangle with one curved side is also considered.
69 citations
TL;DR: In this paper, the shape of the surface representing the observed dependent variable (which may be hydraulic head, chemical concentration, or temperature) is approximated from measured samples by means of various interpolation algorithms.
Abstract: A method to solve the inverse problem is developed. This method does not require the iterative solution of the aquifer equation, which is an essential characteristic of many current identification schemes. The shape of the surface representing the observed dependent variable (which may be hydraulic head, chemical concentration, or temperature) is approximated from measured samples by means of various interpolation algorithms. Once the various derivatives of the dependent variable are approximated, the identification problem reduces locally to algebraic equations of small dimension. It is shown that aquifer conditions of general heterogeneity and anisotropy are amenable to this method. Input may be treated as an unknown to be evaluated. The method is appraised by application to scattered solution points of a simulated solution to a nonhomogeneous aquifer equation.
66 citations
TL;DR: In this paper, the problems of interpolation by means of entire functions of exponential type which satisfy special restrictions on their growth, and the relationship of these problems to Dirichlet series expansions in an arbitrary convex polygon of functions from a Smirnov space are studied.
Abstract: In this paper, we study the problems of interpolation by means of entire functions of exponential type which satisfy special restrictions on their growth, and the relationship of these problems to Dirichlet series expansions in an arbitrary convex polygon of functions from a Smirnov space.Bibliography: 27 items.
62 citations
TL;DR: The Herbrand's Theorem for first-order formulas was introduced in this paper, where it is shown that the Herbrand Theorem is equivalent to the separation principle of the (F1, F2, F)-interpolation principle in model theory.
Abstract: DEFINITION 0.1. Let Fx, F2, F be subsets of a boolean algebra B. Then the (F1, F2, F)-Separation Principle, written Sep(F~, F2, F) for short, is the following: whenever aEF1, bEF2 and a ^ b = O , there is c~F with a<.c and cAb=O. The (F1, F2, F)-Interpolation Principle, written Int(Ft , F2, F) for short, is the following: whenever a~F1, b~F2 and a<.b there is ceF such that a<~c<~b. Clearly Int (F~, /'2, / ') is just Sep (F l, F 2 , F) where F 2 = { b:b ~F2}, so that the two notions are coextensive. Which one to use is a matter of taste, convenience and history. In practice B is often the Lindenbaum algebra of formulas reduced modulo some theory: then we replace each equivalence class [~p] by its representative formula q~, the ordering [~0]<,. [O] of the algebra becomes a valid implication ~0-,% and O denotes falsity. We also abuse notation mildly by using F~, etc., for classes of formulas. Several standard examples of separation principles occur in Recursion Theory, Descriptive Set Theory and Model Theory: however, they are usually symmetric in the sense that F1 = F2, and strict in the sense that F~ ca F2 = F. In contrast, we shall often want to dispense with these assumptions, and this is why we phrase the definition with three parameters as opposed to the usual one. The best known interpolation theorem in Model Theory is probably Craig's Theorem: this can be summarised as In t (E 2, U 2, F) where F is the set of all first order sentences in a language L and E z [respectively U 2] is the set of existential [universal] second-order sentences over L. We shall, however, be interested only in first order formulas, of low quantifier complexity, and so we begin by looking at an interpolation theorem for such formulas, Herbrand's Theorem.
TL;DR: In this paper, error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems and applied to the Galerkin method for elliptic boundary value problems.
Abstract: Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.
01 Aug 1975
TL;DR: In this article, a method of bivariate interpolation and smooth surface fitting for z values given at points irregularly distributed in the x-y plane is developed, where each polynomial is determined by the given values of z and estimated values of partial derivatives at the vertexes of the triangle.
Abstract: A method of bivariate interpolation and smooth surface fitting is developed for z values given at points irregularly distributed in the x-y plane. The interpolating function is a fifth-degree polynomial in x and y defined in each triangular cell which has projections of three data point's in the x-y plane as its vertexes. Each polynomial is determined by the given values of z and estimated values of partial derivatives at the vertexes of the triangle. Procedures for dividing the x-y plane into a number of triangles, for estimating partial derivatives at each data point, and for determining the polynomial in each triangle are described. A simple example of the application of the proposed method is shown. User information and Fortran listings are given on a computer subprogram package that implements the proposed method.
TL;DR: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions is established in this article, and precise criteria for the uniqueness of such interpolatory perfect spline are indicated.
Abstract: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions are established. In particular, the existence of ordinary perfect splines obeying boundary and interpolation conditions is demonstrated; precise criteria for the uniqueness of such interpolatory perfect splines are indicated. These are shown to solve a host of variational problems in certain Sobolev spaces.
TL;DR: An algorithm is presented for solving nonlinear ordinary differential equations that generates a broad class of implicit linear differentiation formulas that is more efficient as to the number of arithmetic operations than Brayton's algorithm, as well as more general and systematic.
Abstract: An algorithm is presented for solving nonlinear ordinary differential equations that generates a broad class of implicit linear differentiation formulas. Specific interest is concerned in one type of formula which for constant timestep reduces to the well-known Gear formulas. Although these formulas are mathematically fully equivalent to the BDF formulas as presented by Brayton et al. [1], their construction is quite different. Instead of using previously calculated function values, we employ predictions extrapolated from these values to set up and evaluate the differentiation formula. A recursive relation for these predictions is derived in order to simplify their calculation in the next timestep. As predictions are needed for order and error control, our algorithm appears to be more efficient as to the number of arithmetic operations than Brayton's algorithm, as well as more general and systematic. Moreover, a change of the order can be accomplished without extra work. Due to the available predictions, an interpolation to determine function values at intermediate time instants, as for instance required in plotting procedures, can be performed in a fast way.
Patent•
12 Aug 1975TL;DR: In this article, a numerical control system with both linear and circular interpolating capabilities is described, where data from a permanent storage medium such as tape are input to a preprocessor.
Abstract: A numerical control system having both linear and circular interpolating capabilities is described. Data from a permanent storage medium such as tape are input to a preprocessor. The preprocesor processes the data into a form acceptable to the system logic and makes all parameters for both linear and circular interpolation available to the system logic. Because linear interpolation is used, straight lines as well as arbitrary curves which are simulated by a series of short straight line segments can be drawn or cut. Additionally, because circular interpolation is used, arbitrary curves can be simulated by a series of arcs, each of the arcs being a portion of a different circle and the various circles having varying radii in accordance with the dictates of the arbitrary curve being cut or drawn. The ability to use both circular and linear interpolation substantially reduces the data processing because the interpolation technique requiring the least definition can be conveniently employed. Also, because of the particular interpolation techniques employed, a minimum of multiplication and division is required so that the system logic is substantially reduced. Additionally, because of the particular linear and circular interpolation techniques and because of the data processing technique employed, the execution time of real time tasks is reduced. This results in a substantial reduction in demands on the computing capabilities of the system, thereby significantly increasing the overall capabilities of the system.
01 Jan 1975
TL;DR: In this paper, a general theory of multistage decimators and interpolators for sampling rate reduction and sampling rate increase is presented, and a set of curves and necessary relations for optimally de-signing multi-age DECIMators is also given.
Abstract: In this paper a general theory of multistage decimators and interpolators for sampling rate reduction and sampling rate increase is presented. A set of curves and the necessary relations for optimally de- signing multistage decimators is also given. It is shown that the pro- cesses of decimation and interpolation are duals and therefore the same set of design curves applies to both problems. Further, it is shown that highly efficient implementations of narrow-band finite impulse response (FIR) fiiters can be obtained by cascading the processes of decimation and interpolation. Examples show that the efficiencies obtained are comparable to those of recursive elliptic filter designs.
TL;DR: In this paper, a wave-form generator capable of producing a desired wave-shape by sampling one period of the waveshape at a coarse interval and calculating amplitudes with a fine interval between the basic amplitudes is described.
Abstract: The invention is directed to a waveshape generator capable of producing a desired waveshape by previously storing basic amplitudes obtained by sampling one period of the waveshape at a coarse interval and calculating amplitudes with a fine interval between the basic amplitudes. While basic amplitudes A and B are sequentially produced at a coarse interval in response to an integer portion of the input data, a function X (c) is produced in response to a fraction portion of the input data. Waveshape amplitudes are interpolated between the basic amplitudes by carrying out calculation of A + (B - A) × X(c) in response to these values A, B and X(c). A special form of function X(c) is also used for applying interpolation by a partial waveshape of a trigonometric function wave. An example of a musical tone waveshape generator is also described in which different waveshapes are produced depending upon different tone ranges by moving the position of a radix point for each of the different tone ranges.
Patent•
31 Oct 1975
TL;DR: In this paper, the magnitudes of the separations or differences in value between the sampled data points are computed in accordance with the magnitude of the difference in values between the data points.
Abstract: A display apparatus includes a computational circuit that is connected to the output of a memory to introduce between a stored digital representation of successive sampled data points, by an interpolation method, additional virtual or interpolated data points. The interpolated data points are computed in accordance with the magnitudes of the separations or differences in value between the sampled data points. In a CRT display or hard copy recorder, this allows a higher cathode ray sweep speed, and hence, a higher density of raster lines, and consequently, an improvement in resolution, smoothness and continuity of the display, for a given size memory.
TL;DR: A PDP-11-to-Raman spectrometer interface for data acquisition and plotting is described and the superiority of an exponential-quadratic baseline function over common interpolation procedures such as the cubic spline, the fitting of simple shape-functions to the background, or polynomial expansions is shown.
Abstract: A PDP-11-to-Raman spectrometer interface for data acquisition and plotting is described. The background correction problem, which occurs in the transition from data acquisition to the data analysis of chemical interest, has been studied for the nontrivial case of considerable curvature near the Rayleigh wing. The conflict between a baseline function with sufficient freedom to compensate properly for the background and sufficient constraint to achieve uniqueness and reproducibility has been demonstrated. A comparative study has shown the superiority of an exponential-quadratic baseline function over common interpolation procedures such as the cubic spline, the fitting of simple shape-functions to the background, or polynomial expansions.
TL;DR: In this paper, three two-dimensional plasma simulation models have been compared using an electrostatic two-stream instability as a test case, and the major result is that the evolution of electric field energy did not depend strongly on the choice of model.
TL;DR: In this paper two interpolation methods, namely linear interpolation and interpolation via generalized Lagrange polynomials are considered and it is shown that these techniques can be modified to accomodate the causality constraint.
Abstract: Given a finite set {(xi, yi)} of ordered pairs from X × Y where X, Y are Hilbert spaces over the same field, there are numerous techniques for constructing a function, ƒ, on X to Y such that ƒ(xi) = yi. However, when X, Y have a causality structure and ƒ must be causal then the data interpolation problem is much more complicated. In this paper two interpolation methods, namely linear interpolation and interpolation via generalized Lagrange polynomials are considered. It is shown that these techniques can be modified to accomodate the causality constraint. The development is indicative of the modifications that must be made in any existing data interpolation algorithm if causal interpolation is required.
TL;DR: In this paper, it was shown that only a single parameter is necessary or desirable to describe these regions, which can be determined from ordinary and extraordinary ray measurements, at any latitude.
Abstract: Increasing accuracy in the conversion of h′(f) to N(h) profiles is obtained by increasing the order of the polynomial used to interpolate between measured points. Linear and parabolic lamination techniques correspond to first- and second-order interpolation. Fourth-order interpolation (as in the 5-term overlapping polynomial method) is about optimum. In comparing different methods, it is essential that fixed boundary conditions be employed; when this is done an adjacent polynomial technique is much less accurate than overlapping polynomials. All methods (including least-squares procedures) are equally sensitive to errors in the virtual height data. Possible procedures for reducing the errors caused by underlying and valley ionization are critically reviewed. It is concluded that, in general, only a single parameter is necessary or desirable to describe these regions. This parameter can be determined from ordinary and extraordinary ray measurements, at any latitude.
01 Jan 1975
TL;DR: In this paper, the authors give the definition of regular splines, and prove that the problem of interpolation is solvable if the interpolation points are close enough together, i.e., the solution converges to the interpolated function with the fourth order of the maximal distance between adjacent points.
Abstract: We give the definition of regular splines, slightly improving the axioms stated earlier by R. Schaback. Considering splines that are twice continuously differentiable, we can prove that the problem of interpolation is solvable if the interpolation points are close enough together. The solution converges to the interpolated function with the fourth order of the maximal distance between adjacent points.The regular splines are then used to define an implicit scheme for the integration of initial value problems of ordinary differential equations, following Loscalzo–Talbot and Runge. Again fourth order convergence is established. The method may be particularly useful in treating solutions with movable singularities.
Patent•
08 May 1975
TL;DR: In this article, the flanks of gear teeth are associated with a system of co-ordinates and a grinding program is employed that specifies involute modifying grinding feed in terms of the coordinates.
Abstract: For grinding the flanks of gear teeth to modify an involute tooth profile determined by the generating method of grinding, the tooth flank area is associated with a system of co-ordinates and a grinding programme is employed that specifies involute modifying grinding feed in terms of the co-ordinates. By comparison of actual and programmed feed positions at instantaneous points in the co-ordinate system, feed corrections are applied at those points to obtain the programmed profile. By interpolation between successive comparison points, incremental feed corrections are obtained to give a smoother change of profile.