# modulus of continuity

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## modulus of continuity

[′mäj·ə·ləs əv ‚känt·ən′ü·əd·ē]
(mathematics)
For a real valued continuous function ƒ, this is the function whose value at a real number r is the maximum of the modulus of ƒ(x) - ƒ(y) where the modulus of x-y is less than r; this function is useful in approximation theory.
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A function [omega] of a modulus of continuity type on the interval [0,2[pi]] satisfies the condition [mathematical expression not reproducible]
In particular, we estimated the deviation [mathematical expression not reproducible] by the function of modulus of continuity type in the case when conjugate function [f.
Now we need the following property of complete modulus of continuity which proved in [9]
9 If X is bounded and bounds for the density fx and its modulus of continuity are known explicitly, the last result is strong enough to allow, in principle, perfect simulation using von Neumann's rejection method; see Devroye (2001) for the case of infinitely divisible perpetuities with approximation of densities by Fourier inversion, Devroye, Fill, and Neininger (2000) for the case of the Quicksort limit distribution and Devroye and Neininger (2002) for more general fixed-point equations of type (2).
In order to make the bounds of Section 4 explicit in applications, we need to bound the absolute value and modulus of continuity of the density of the fixed-point.
Again the inf in the tail term ensures that if f is a polynomial of degree [less than or equal to] r - 1, then the modulus of continuity vanishes identically, as is expected from an rth order modulus.
For the case where Q is of faster than polynomial growth, the modulus of continuity becomes more complicated, as again there are endpoint effects, close to [+ or -] C[q.
Section 2 deals with preliminary notions such as the modulus of continuity and the best approximation, but in that use an abstract setting the cosine operator function.
MODULUS OF CONTINUITY, BEST APPROXIMATIONS, ROGOSINSKI- AND BLACKMAN-TYPE OPERATORS
By using well-known properties of the modulus of continuity and applying the Cauchy-Schwartz inequality, we obtain the formula below:
From definition modulus of continuity, we can write [omega](f; [delta]) < M[[delta].
MODULUS OF CONTINUITY, THE BEST APPROXIMATIONS, AND BLACKMAN- AND ROGOSINSKI-TYPE OPERATORS

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