([On Summability Of Double Fourier Series And It's Conjugate By Double Nörlund in The Space L_2 ([0,π]×[0,π

Abstract

Abstract: Let f be a function of two variables u,v, periodic with respect to u and with respect to v, in each case with period 2π, and summable in the square
Q:[-π,π]×[-π,π].
In this research we will proof two theorems.
The first study summability of the Double Fourier series
∑_(m=0)^∞▒∑_(n=0)^∞▒〖λ_(m,n) A_(m,n) (u,v) 〗to f at point (u,v)=(x,y) within a certain conditions, and we put the necessary lemmas for this theorem, and in the second we also study summability conjugate of the series:
∑_(m=1)^∞▒∑_(n=1)^∞▒[δ_mn cos⁡mx cos⁡ny-γ_mn sin⁡mx cos⁡ny-β_mn cos⁡mx sin⁡ny+α_mn sin⁡mx sin⁡ny ]
To [8]: f ̅(x,y)=1/(4π^2 ) ∫_0^π▒∫_0^π▒〖ψ(s,t) 1/(tan⁡〖s/2〗 tan⁡〖t/2〗 ) dsdt〗
Where
ψ(x,y)=ψ(x,y;s,t)=
1/4 {f(x+s,y+t)-f(x-s,y+t)-f(x+s,y-t)+f(x-s,y-t) }
and put all necessary conditions and lemmas for this theorem by Double Nörlundsummability, which considered bounded linear operator, to both theorems in the space L_2 ([0,π]×[0,π]).
and we can get many results, the most important of which is that the simple Fourier series or double Fourier Series, its Nörlundsummability to same function.
In conclusion, we can say that Double Nörlund method are general rarely, follow up many methods as cesaro double and Holder Double and et.
This method has wide applications in analysis mathematics and spicily in approximation theory.