Uniform bounds for Fourier transforms of surface measures in R$^3$ with nonsmooth density

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h2) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

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Convex Models for Uncertain Imperfection Control in Multimode Dynamic Buckling

Journal of Applied Mechanics ◽

10.1115/1.2894064 ◽

1992 ◽

Vol 59 (4) ◽

pp. 937-945 ◽

Cited By ~ 24

Author(s):

H. E. Lindberg

Keyword(s):

Quality Control ◽

Fourier Coefficients ◽

Fourier Transforms ◽

Dynamic Buckling ◽

Mode Number ◽

Uniform Bounds ◽

Additional Information ◽

Probabilistic Distribution ◽

Finite Fourier Transforms

Control of uncertain imperfections by means of convex bounds on finite Fourier transforms is shown to be more direct and not as overly conservative as control based on uniform bounds, i.e., bounding maximum and minimum imperfections. With either method, conservatism in bounds on buckling response is reduced by filtering the imperfection measurements. Extraction of the needed filtered information by operating directly on the Fourier coefficients is straightforward and allows use of additional information on the variation of the coefficients with mode number. Use of this information in example multimode buckling problems gives a bound on maximum possible buckling response that is a factor of1.6 larger than the response at a reliability of 99.5 percent for hypothetical (but reasonably representative) probabilistic imperfections. The bounding response itself, of course, does not depend on any assumptions concerning the probabilistic distribution of imperfections. Two additional combined uniform and Fourier ellipsoid bound models further reduce this factor to 1.1 and 0.5, and require only a simple, unfiltered imperfection bound measurement during quality control inspection.

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Explicit uniform bounds on integrals of Bessel functions and trace theorems for Fourier transforms

Mathematische Nachrichten ◽

10.1002/mana.201700326 ◽

2018 ◽

Vol 292 (1) ◽

pp. 106-120

Author(s):

Hubert Kalf ◽

Takashi Okaji ◽

Osanobu Yamada

Keyword(s):

Bessel Functions ◽

Fourier Transforms ◽

Uniform Bounds ◽

Trace Theorems

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Computer processing of high-resolution images of periodic specimens

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100141834 ◽

1986 ◽

Vol 44 ◽

pp. 2-5

Author(s):

W. Chiu ◽

M.F. Schmid ◽

T.-W. Jeng

Keyword(s):

High Resolution ◽

Fourier Transforms ◽

Complex Structure ◽

Distribution Functions ◽

Multiple Image ◽

Cryo Electron Microscopy ◽

Structure Factors ◽

Unit Cells ◽

High Resolution Images ◽

Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

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The extended Fourier algorithm: Application in discrete optics and electron holography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100172322 ◽

1994 ◽

Vol 52 ◽

pp. 918-919

Author(s):

E. Voelkl ◽

L. F. Allard

Keyword(s):

Fourier Transform ◽

Fourier Transforms ◽

Sampling Rate ◽

Electron Holography ◽

Optical Bench ◽

Fourier Space ◽

Ccd Cameras ◽

Simulated Image ◽

Initial Sampling ◽

Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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Observations of Frozen-Hydrated Microtubules

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100181270 ◽

1990 ◽

Vol 48 (1) ◽

pp. 504-505

Author(s):

D. Chrétien ◽

D. Job ◽

R.H. Wade

Keyword(s):

Fourier Transforms ◽

Structural Complexity ◽

Image Contrast ◽

Phase Behaviour ◽

Experimental Conditions ◽

Thin Sections ◽

Surface Lattice ◽

Cilia And Flagella ◽

Outstanding Question

Microtubules are filamentary structures found in the cytoplasm of eukaryotic cells, where, together with actin and intermediate filaments, they form the components of the cytoskeleton. They have many functions and show various levels of structural complexity as witnessed by the singlet, doublet and triplet structures involved in the architecture of centrioles, basal bodies, cilia and flagella. The accepted microtubule model consists of a 25 nm diameter hollow tube with a wall made up of 13 paraxial protofilaments (pf). Each pf is a string of aligned tubulin dimers. Some results have suggested that the pfs follow a superhelix. To understand how microtubules function in the cell an accurate model of the surface lattice is one of the requirements. For example the 9x2 architecture of the axoneme will depend on the organisation of its component microtubules. We should also note that microtubules with different numbers of pfs have been observed in thin sections of cellular and of in-vitro material. An outstanding question is how does the surface lattice adjust to these different pf numbers?We have been using cryo-electron microscopy of frozen-hydrated samples to study in-vitro assembled microtubules. The experimental conditions are described in detail in this reference. The results obtained in conjunction with thin sections of similar specimens and with axoneme outer doublet fragments have already allowed us to characterise the image contrast of 13, 14 and 15 pf microtubules on the basis of the measured image widths, of the the image contrast symmetry and of the amplitude and phase behaviour along the equator in the computed Fourier transforms. The contrast variations along individual microtubule images can be interpreted in terms of the geometry of the microtubule surface lattice. We can extend these results and make some reasonable predictions about the probable surface lattices in the case of other pf numbers, see Table 1. Figure 1 shows observed images with which these predictions can be compared.

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