On the convergence of interpolating periodic spline functions of high degree

1972 ◽  
Vol 19 (2) ◽  
pp. 146-154 ◽  
Author(s):  
M. v. Golitschek
Keyword(s):  
Spline Functions ◽  
Periodic Spline ◽  
High Degree

New family of periodic spline functions

1995 ◽  
Author(s):  
Eduardo P. Serrano
Keyword(s):  
Spline Functions ◽  
Periodic Spline ◽  
New Family

Splines

1995 ◽  
Author(s):  
Marek A. Kowalski ◽  
Krzystof A. Sikorski ◽  
Frank Stenger
Keyword(s):  
Computer Graphics ◽  
Numerical Solutions ◽  
Noisy Data ◽  
Spline Functions ◽  
Complexity Bounds ◽  
B Splines ◽  
Approximation Errors ◽  
Linear Problems ◽  
High Degree ◽  
Polynomial Methods

Spline functions are important approximation tools in numerous applications for which high degree polynomial methods perform poorly, such as in computer graphics and geometric modelling, as well as for various engineering problems—especially those involving graphing of numerical solutions and noisy data. Algorithms based on spline functions enjoy minimal approximation errors in wide classes of problems and minimal complexity bounds. In this Chapter we provide a brief introduction to basic classes of polynomial splines, B-Splines, and abstract splines. Further study of spline algorithms as applied to linear problems is outlined in Chapter 7. In this section we define polynomial spline functions, exhibit their interpolatory properties, and construct algorithms to compute them. It turns out that these splines provide interpolating curves that do not exhibit the large oscillations associated with high degree interpolatory polynomials. This is why they find applications in univariate curve matching in computer graphics.

scholarly journals Inequalities of Bernstein type for splines in $L_2(\mathbb{R})$ space

2021 ◽  
Vol 16 ◽  
pp. 21
Author(s):  
V.F. Babenko ◽  
S.A. Spektor
Keyword(s):  
Sharp Inequality ◽  
Spline Functions ◽  
Bernstein Type ◽  
Minimal Defect ◽  
Periodic Spline

We obtain sharp inequality of Bernstein type in $L_2(\mathbb{R})$ space for non-periodic spline functions of degree $m$, of minimal defect, with equidistant knots.

scholarly journals Two-piece Cubic Spline Functions

2003 ◽  
Vol 2 (1) ◽  
pp. 25-33
Author(s):  
Hannah Vijayakumar
Keyword(s):  
Cubic Spline ◽  
Oscillatory Behavior ◽  
Spline Functions ◽  
Entire Interval ◽  
Natural Spline ◽  
Cubic Polynomial ◽  
Natural Cubic Spline ◽  
Cubic Spline Function ◽  
High Degree ◽  
The Given

.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

scholarly journals General Degree of Periodic Spline Functions

2013 ◽  
Vol 73 (20) ◽  
pp. 1-4
Author(s):  
Manprit Kaur ◽  
Arun Kumar
Keyword(s):  
Spline Functions ◽  
Periodic Spline

Tissue section lifting for electron microscopy

1978 ◽  
Vol 36 (2) ◽  
pp. 86-87
Author(s):  
Adrian F. van Dellen
Keyword(s):  
Infectious Disease ◽  
Electron Microscopy ◽  
Tissue Culture ◽  
Organ System ◽  
Surrounding Tissue ◽  
Paraffin Sections ◽  
Surface Areas ◽  
Specific Determination ◽  
High Degree ◽  
Accuracy And Repeatability

The morphologic pathologist may require information on the ultrastructure of a non-specific lesion seen under the light microscope before he can make a specific determination. Such lesions, when caused by infectious disease agents, may be sparsely distributed in any organ system. Tissue culture systems, too, may only have widely dispersed foci suitable for ultrastructural study. In these situations, when only a few, small foci in large tissue areas are useful for electron microscopy, it is advantageous to employ a methodology which rapidly selects a single tissue focus that is expected to yield beneficial ultrastructural data from amongst the surrounding tissue. This is in essence what "LIFTING" accomplishes. We have developed LIFTING to a high degree of accuracy and repeatability utilizing the Microlift (Fig 1), and have successfully applied it to tissue culture monolayers, histologic paraffin sections, and tissue blocks with large surface areas that had been initially fixed for either light or electron microscopy.

Electron Microscopy in Molecular Biology

1967 ◽  
Vol 25 ◽  
pp. 2-3
Author(s):  
Cecil E. Hall
Keyword(s):  
Electron Microscopy ◽  
Molecular Biology ◽  
Noise Level ◽  
Molecular Structures ◽  
Material Structure ◽  
The Arts ◽  
Shadow Casting ◽  
Electron Image ◽  
High Degree ◽  
Very High

The visualization of organic macromolecules such as proteins, nucleic acids, viruses and virus components has reached its high degree of effectiveness owing to refinements and reliability of instruments and to the invention of methods for enhancing the structure of these materials within the electron image. The latter techniques have been most important because what can be seen depends upon the molecular and atomic character of the object as modified which is rarely evident in the pristine material. Structure may thus be displayed by the arts of positive and negative staining, shadow casting, replication and other techniques. Enhancement of contrast, which delineates bounds of isolated macromolecules has been effected progressively over the years as illustrated in Figs. 1, 2, 3 and 4 by these methods. We now look to the future wondering what other visions are waiting to be seen. The instrument designers will need to exact from the arts of fabrication the performance that theory has prescribed as well as methods for phase and interference contrast with explorations of the potentialities of very high and very low voltages. Chemistry must play an increasingly important part in future progress by providing specific stain molecules of high visibility, substrates of vanishing “noise” level and means for preservation of molecular structures that usually exist in a solvated condition.

A High Angle, Double Tilting Cold Stage for the AEI EM7

1974 ◽  
Vol 32 ◽  
pp. 450-451
Author(s):  
P.R. Swann ◽  
A.E. Lloyd
Keyword(s):  
Habit Plane ◽  
Temperature Control ◽  
Tilt Angle ◽  
Cooling System ◽  
Thermal Drift ◽  
High Angle ◽  
Cold Stage ◽  
High Degree ◽  
Better Than

Figure 1 shows the design of a specimen stage used for the in situ observation of phase transformations in the temperature range between ambient and −160°C. The design has the following features a high degree of specimen stability during tilting linear tilt actuation about two orthogonal axes for accurate control of tilt angle read-out high angle tilt range for stereo work and habit plane determination simple, robust construction temperature control of better than ±0.5°C minimum thermal drift and transmission of vibration from the cooling system.

New Aspects in the Design of Electron Optical Magnification Systems

1972 ◽  
Vol 30 ◽  
pp. 606-607
Author(s):  
Willem H.J. Andersen
Keyword(s):  
Electron Microscope ◽  
Imaging System ◽  
Chromatic Aberration ◽  
Research Tool ◽  
Energy Losses ◽  
Objective Lens ◽  
Theoretical Limit ◽  
Optical Magnification ◽  
Chromatic Aberrations ◽  
High Degree

Electron microscope design, and particularly the design of the imaging system, has reached a high degree of perfection. Present objective lenses perform up to their theoretical limit, while the whole imaging system, consisting of three or four lenses, provides very wide ranges of magnification and diffraction camera length with virtually no distortion of the image. Evolution of the electron microscope in to a routine research tool in which objects of steadily increasing thickness are investigated, has made it necessary for the designer to pay special attention to the chromatic aberrations of the magnification system (as distinct from the chromatic aberration of the objective lens). These chromatic aberrations cause edge un-sharpness of the image due to electrons which have suffered energy losses in the object.There exist two kinds of chromatic aberration of the magnification system; the chromatic change of magnification, characterized by the coefficient Cm, and the chromatic change of rotation given by Cp.

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