Approximation of Function Belonging To Generalized Lipschitz Class Using Euler Submethod (E,p,q)_λ

Abstract

The main aim of this research is to approximate of functions belonging to generalized Lipschitz Lip (ξ(t),p), and associated functions, by Euler Submethod (E,p,q)_λ, where we assume that function f be a 2π periodic function of xand and integrable over [-π,π], in the sense of Lebesgue.
In this research there will be given sufficient conditions to be the Fourier series and its conjugate series summable using means method (E,p,q)_λ.
Thus, we investigate trigonometric polynomials associated with f∈Lip(ξ(t),p);p>1 to approximate f and f ̅ in L^pnorm to the degree of O(ξ(1/√(λ(n) )) (λ(n) )^(1/2p) ).
By proving two theorems, we use the first Fourier series theorem and compare it to the second sequential theorem conjugate the Fourier series. In both theorems (1) and (2), we rely on the many trigonometric limits resulting Using Euler Submethod(E,p,q)_λ to the sum of the sum of partial sums of Fourier series ∑_(n=0)^∞▒〖A_n (x) 〗 and its conjugats -∑_(n=1)^∞▒〖B_n (x) 〗.
Neither of the two methods can assign an approximate sum. In order to reach our desired goal, the analytical and synthetic method was adopted. We defined Euler Submethod(E,p,q)_λ and then applied to series with important applications especially in the approximation theory we can get many results, the most important of which is than the Submethod lead to the normal (classical) methods, In conclusion, we can say that our study generalizes all previously known results of this line of work.