Integral Equations

where T n (x) = cos(n arccos x) are the Chebyshev polynomials and ξ = (ξ 0 , ξ 1 , ...ξ n , ...) ∈ h 2 ;that is,‖ξ‖ 2 2 = 1 ∞∑2 |ξ 0| 2 + |ξ n | 2 n 2 < ∞.n=1In this section we add the coordinate ξ 0 to vector ξ and modify the corresponding formulas. It iseasy to see that all results of previous sections are also valid in this case. Recall the definitionsof weighted Hilbert spaces associated with logarithmic integral operators:∫1L (1)2 = {f(x) : ‖f‖ 2 1 =−1|f(x)| 2 dx√ < ∞},1 − x2and∫ 1L (2)2 = {f(x) : ‖f‖ 2 2 =−1|f(x)| 2√ 1 − x 2 dx < ∞},˜W 1 2 = {f(x) : f ∈ L (1)2 , f ′ ∈ L (2)2 },Ŵ 1 2 = {f(x) : f ∈ L (2)2 , f ∗ (x) = f(x) √ 1 − x 2 , f ∗′ ∈ L (2)2 , f ∗ (−1) = f ∗ (1) = 0}.Below, we will denote the weights by p 1 (x) = √ 1 − x 2 and p −11 (x).Note that the Chebyshev polynomials specify orthogonal bases in spaces L (1)2 and L (2)2 . Inorder to obtain the matrix representation of the integral operators with a logarithmic singularityof the kernel, we will prove a similar statement for weighted space ˜W 2 1 .Lemma 6 ˜W 1 2and Φ are isomorphic.✷ Assume that ϕ ∈ Φ. We will show that ϕ ∈ ˜W 1 2 . Function ϕ(x) is continuous, and (366) isits Fourier–Chebyshev series with the coefficientsξ n = 2 π∫1−1T n (x)ϕ(x)p −11 (x)dx.∞∑The series |ξ n | 2 n 2 converges and, due to the Riesz–Fischer theorem, there exists an elementn=1f ∈ L (1)2 , such that its Fourier coefficients in the Chebyshev polynomial series are equal to nξ n ;that is, setting U n (x) = sin(n arccos x), one obtains= − 2 πnξ n = 2 π∫1−1∫1−1nT n (x)ϕ(x)p −11 (x)dx = − 2 πU n(x)(p ′ 1 (x)ϕ(x)) dxp 1 (x) = 2 π∫1−1∫1−1U ′ n(x)ϕ(x)dxU n (x)f(x) dxp 1 (x) , n ≥ 1.75