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References in periodicals archive ?
Let f [member of] [L.sup.p], 0 [less than or equal to] [beta] < 1 - 1 p and let a function of modulus of continuity type [omega] satisfy the conditions: for r [member of] N
An estimation of the rate of convergence can be obtained using the modulus of continuity for two dimensional real valued functions.
By using Lemma 3 case (ii) and the well-known properties of the modulus of continuity, we have
That is, the first -order modulus of smoothness of f is the same as modulus of continuity of f.
In this section, using weighted modulus of continuity, we study the rate of Abel convergence.
Unlike Part 1 where the "modulus of continuity method" will be used only in one step of the proof, Part 2 is completly based on the use of the "modulus of continuity method".
Because of linearity of [F.sub.n,m,a,b] properties of modulus of continuity, use Cauchy-Schwartz inequality and (4) , (11) we have
Here, the kth modulus of continuity ([3], p.76) is defined for f [member of] C(R) and [delta] [greater than ro equal to] 0 by
For this approximation we can give a rate of convergence, depending on the modulus of continuity of the density of the fixed-point, which is defined by
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.
In this paper, firstly we study some approximation properties of the sequence of linear positive operators given by (5) in the space of continuous functions on compact set [[??].sub.AB] = [0, A] x [0, B] and find the order of this approximation using full and partial modulus of continuity. We finally investigate the convergence of the sequence of linear positive operators [L.sub.n,m], defined on a weighted space of functions of two variables, and find the rate of this convergence by means of weighted modulus of continuity.