scholarly journals Uniform Approximation of Periodical Functions by Trigonometric Sums of Special Type

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
A. S. Serdyuk ◽  
Ie. Yu. Ovsii
Keyword(s):  
Uniform Approximation ◽  
Higher Order ◽  
Order Of Approximation ◽  
Trigonometric Sums ◽  
Best Uniform Approximation ◽  
Fourier Sums ◽  
De La Vallée Poussin

The approximation characteristics of trigonometric sums Un,pψ of special type on the class Cβ,∞ψ of (ψ,β)-differentiable (in the sense of A. I. Stepanets) periodical functions are studied. Because of agreement between parameters of approximative sums and approximated classes, the solution of Kolmogorov-Nikol’skii problem is obtained in a sufficiently general case. It is shown that in a number of important cases these sums provide higher order of approximation in comparison with Fourier sums, de la Vallée Poussin sums, and others on the class Cβ,∞ψ in the uniform metric. The range of parameters in which the sums Un,pψ give the order of the best uniform approximation on the classes Cβ,∞ψ is indicated.

Role of Single Word Variables in Recall of Statistical Approximations to English

1966 ◽  
Vol 19 (2) ◽  
pp. 627-634 ◽  
Author(s):  
Mari J. K. Brown
Keyword(s):  
Free Recall ◽  
Higher Order ◽  
Order Of Approximation ◽  
Single Word ◽  
First Order ◽  
Original Order ◽  
Sequential Organization ◽  
The Individual

Free recall of lists at different orders of approximation to English was compared to the recall of the same lists when the order of the words had been scrambled to destroy their sequential organization. Recall of the organized lists showed the typical improvement with increasing order of approximation. Recall of the scrambled lists was unrelated to the original order of approximation. The results indicate that increased recall with increasing order of approximation to English is not produced by systematic differences in the characteristics of the individual words comprising the approximations. When recall of the organized lists was scored in terms of the number of longer sequences present in recall, the number of recalled sequences of any given length increased as order of approximation to English increased, with the first order list showing proportionally less organization in recall than the second and higher order lists.

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

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