On the best mean-square approximation of a real nonnegative finite continuous function of two variables by the modulus of a double Fourier integral. I

2009 ◽  
Vol 160 (3) ◽  
pp. 343-356 ◽  
Author(s):  
P. O. Savenko ◽  
L. P. Protsakh ◽  
M. D. Tkach
Keyword(s):  
Continuous Function ◽  
Fourier Integral ◽  
Mean Square ◽  
Function Of Two Variables

An experimental test of a model for repeated Ca2+ spikes in osteoblastic cells

1991 ◽  
Vol 69 (7) ◽  
pp. 433-441 ◽  
Author(s):  
Jack Ferrier ◽  
Angela Kesthely ◽  
Eva Lagan ◽  
Conrad Richter
Keyword(s):  
Continuous Function ◽  
Mathematical Modelling ◽  
Model Prediction ◽  
Experimental Test ◽  
Good Evidence ◽  
The Other ◽  
Osteoblastic Cells ◽  
Mean Square ◽  
Model Predictions ◽  
Electrophysiological Measurements

A model for cytosolic Ca2+ spikes is presented that incorporates continual influx of Ca2+, uptake into an intracellular compartment, and Ca2+-induced Ca2+ release from the compartment. Two versions are used. In one, release is controlled by explicit thresholds, while in the other, release is a continuous function of cytosolic and compartmental [Ca2+]. Some model predictions are as follows. Starting with low Ca2+ influx and no spikes: (1) induction of spiking when Ca2+ influx is increased. Starting with spikes: (2) increase in magnitude and decrease in frequency when influx is reduced; (3) inhibition of spiking if influx is greatly reduced; (4) decrease in the root-mean-square value when influx is increased; and (5) elimination of spiking if influx is greatly increased. Since there is good evidence that hyperpolarizing spikes reflect cytosolic Ca2+ spikes, we used electrophysiological measurements to test the model. Each model prediction was confirmed by experiments in which Ca2+ influx was manipulated. However, the original spike activity tended to return within 5–30 min, indicating a cellular resetting process.Key words: calcium, electrophysiology, mathematical modelling.

Hybrid techniques for generation of arbitrary functions

SIMULATION ◽  
1966 ◽  
Vol 7 (6) ◽  
pp. 293-308 ◽  
Author(s):  
Arthur I. Rubin
Keyword(s):  
Continuous Function ◽  
Hybrid Method ◽  
Arbitrary Function ◽  
General Purpose ◽  
Hybrid Techniques ◽  
Order Of Magnitude ◽  
True Hybrid ◽  
Discrete Values ◽  
Function Of Two Variables ◽  
Digital Processor

A discussion of the implementation of a true hybrid func tion generator module is presented. Included is a detailed description of how two of these modules, when used in conjunction with a hybrid computer, are used simultane ously to generate a continuous function of two variables, and how four modules are used to generate a continuous function of three variables. From a variety of possible con figurations, an optimum choice for a general-purpose HFG module is made, taking into account digital coefficient storage requirements and analog multiplier requirements. A detailed error analysis compares the errors produced in generating an arbitrary function using the HFG module with the errors produced by the classical hybrid (or purely digital) method of generating an arbitrary function by table look-up and interpolation for discrete values of the argu ment. The conclusion is reached that the hybrid method can be an order of magnitude more accurate than the classical discrete method for generating arbitrary functions and has further advantages in that the hybrid method does not require a 100°lo duty cycle in the digital processor for accurate function generation.

scholarly journals On degree of approximation of function by Voronoi means of its Fourier integral

2021 ◽  
Vol 19 ◽  
pp. 24
Author(s):  
L.G. Bojtsun ◽  
T.I. Rybnikova
Keyword(s):  
Continuous Function ◽  
Fourier Integral ◽  
Degree Of Approximation ◽  
Approximation Of Function

The theorem on the degree of approximation to continuous function $f(x) \in L(-\infty; \infty)$ by Voronoi means of its Fourier integral is proved.

scholarly journals A New Fuzzy System Based on Rectangular Pyramid

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Mingzuo Jiang ◽  
Xuehai Yuan ◽  
Hongxing Li ◽  
Jiaxia Wang
Keyword(s):  
Continuous Function ◽  
Membership Function ◽  
Fuzzy System ◽  
Function Approximation ◽  
Sufficient Condition ◽  
Compact Domain ◽  
Arbitrary Degree ◽  
Partial Derivatives ◽  
Function Of Two Variables

A new fuzzy system is proposed in this paper. The novelty of the proposed system is mainly in the compound of the antecedents, which is based on the proposed rectangular pyramid membership function instead of t-norm. It is proved that the system is capable of approximating any continuous function of two variables to arbitrary degree on a compact domain. Moreover, this paper provides one sufficient condition of approximating function so that the new fuzzy system can approximate any continuous function of two variables with bounded partial derivatives. Finally, simulation examples are given to show how the proposed fuzzy system can be effectively used for function approximation.

scholarly journals Erdös-Turán mean convergence theorem for Lagrange interpolation at Lobatto points

1986 ◽  
Vol 34 (3) ◽  
pp. 375-381 ◽  
Author(s):  
William E. Smith
Keyword(s):  
Continuous Function ◽  
Weight Function ◽  
Convergence Theorem ◽  
Lagrange Interpolation ◽  
Mean Square ◽  
Lagrange Polynomial ◽  
Weighted Mean ◽  
Product Integration ◽  
Lagrange Polynomials ◽  
Mean Convergence

Let {Qn} denote the orthogonal polynomials associated with the weight function p on [−1, 1] and let denote the zeros of (1−x2) Qn (x). Consider the Lagrange polynomials which interpolate a given continuous function at these points. It is shown that, as n → ∞, the Lagrange polynomial converges to the function in the W weighted mean square sense, where w (x) = ρ(x)\(1−x2), provided that W is integrable. An application to numerical product integration is noted.

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