Integral Equations

In terms of the summation operators, this systems can be represented asLξ + Aξ = f; ξ = (ξ 0 , ξ 1 , . . . , ξ n , . . .) ∈ h 2 ; f = (f 0 , f 1 , . . . , f n , . . .) ∈ h 4 .Here, operators L and A are defined byL = {l nj } ∞ n,j=0, l nj = 0, n ≠ j; l 00 = ln 2; l nn = 1 , n = 1, 2, . . . ;nwhereA = {a nj } ∞ n,j=0,a nj = ε nj∫ ∫Π 1⎧⎨ 1, n = j = 0,ε nj = 2, n ≤ 1, j ≤ 1,⎩ √2, for other n, j.dxdsN(q)T n (x)T j (s)π 2 p 1 (x)p 1 (s) , (373)Lemma 7 Let N(q) satisfy the continuity conditions formulated in (367). Then∞∑n=1,j=0|a nj | 2 n 4 +∞∑j=0|a 0j | 2 1ln 2 2 ≤ C 0, (374)whereC 0 = 1ln 2 2∫1∫1∣−1 −1N(q)dxπp 1 (x)2∣∫ ∫dsπp 1 (s) +Π 1p 2 1(x) |(p 1 (x)N ′ x) ′ x| 2 dxdsπ 2 p 1 (x)p 1 (s) . (375)✷ Expand a function of two variables belonging to the weighted space L 2 [Π 1 , p −11 (x)p −11 (s)] inthe double Fourier–Chebyshev series using the Fourier–Chebyshev system {T n (x)T m (s)}:where∞∑ ∞∑−p 1 (x)(p 1 (x)N x(q)) ′ ′ x = C nm T n (x)T m (s),n=0 m=0C nm = − ε2 nmπThe Bessel inequality yields= ε2 nmπ∫1−1∫ 1−1∫= nε2 1nmπ 2= n2 ε 2 nmπ 2−1T m (s) dsp 1 (s)Π 1∫1−1∫ 1T n (x)(p 1 (x)N ′ x(q)) ′ xdxT m (s) ds Tp 1 (s)n(x)p ′ 1 (x)N x(q)dx′−1∫1T m (s)p 1 (s) ds U n (x)N x(q)dx′−1∫ ∫N(q)T n (x)T m (s)dxds = n 2 ε nm a nm .p 1 (x)p 1 (s)∞∑ ∞∑∫ ∫n 4 π 2 |a nm | 2 ≤ p 2 1(x)|[p 1 (x)N x(q)] ′ ′ x| 2 dxdsn=1 m=0p 1 (x)p 1 (s) .Π 178