Many scientists and researchers have used multilayered neural networks to approximate multivariate functions for several years [see Lin and Cao 2015, Li and Xu 2007, Suzuki 1998 & Wang and Xu 2010]. They have established both upper and lower bounds of simultaneous approximations for 1st and 2nd orders, spaces of function to approximate and approximators as well. That work solves many applicant issues in science and engineer. Our goal was to achieve that both bounds of modulus of smoothness of order for a th Lebegue integrable multivariate function that is approximated by a multi-layered feedforwrd neural network.
Given a natural number , ( ) , a function belongs to the space (, - ) under the norm defined by
? ? {(( ) ? ?| ( )| ) *| ( )| | | + (1)
Neural Network Simultaneus Approximation Hawraa Abbas Fadhil
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For (, - ) define the following from [Liflyand 2006]
? ? ( ) ? ? ( ) ( ) (2)
and
( ) ( ) ? ? ( ) ( ) (3)
In this paper, we use modulus of smoothness to measure the estimates of approximation, so we need to define the kth symmetric difference by
( ) ?( ) ( ) ( ( ) ) , (4)
and the kth modulus of smoothness by
( ) ? ? ? ( )? (5)
Now, let us state some important properties of the classical modulus of smoothness that will be minor in our proofs, such us [Dineva, A, V?rkonyi-K?czy, Tar and Piur 2015]
(1) ( ) is monotone increasing about
(2) ( ) ( ) ( )
(3) ( ) ( )
(4) ( ) ( ( ) )
In order to approximate ( ) by ( ) with p-norm, each should approximates each with p-norm.
Finally, we need to define the three hidden trigonometric layer feedforward neural network defined by [Suzuki 1998]
, - ( , -) ( , - , -) (6)
Where , -( ) , - ?{ , - ( ) , - ( )