Integral Equations

Let us expand the functiong(s) = 1ln 2∫1−1N(q)dxπ 2 p 1 (x)in the Fourier–Chebyshev series. Again, due to the Bessel inequality, we obtain1ln 2 2∞∑|a 0m | 2 ≤ 1 ∫1∫1m=0 ln 2 2 ∣−1 −1N(q)dsπp 1 (x)2∣dsπp 1 (s) .21 Galerkin methods and basis of Chebyshev polynomialsWe describe the approximate solution of linear operator equations considering their projectionsonto finite-dimensional subspaces, which is convenient for practical calculations. Below, it isassumed that all operators are linear and bounded. Then, the general results concerning thesubstantiation and convergence the Galerkin methods will be applied to solving logarithmicintegral equations.Definition 10 Let X and Y be the Banach spaces and let A : X → Y be an injective operator.Let X n ⊂ X and Y n ⊂ Y be two sequences of subspaces with dim X n = dim Y n = n andP n : Y → Y n be the projection operators. The projection method generated by X n and P napproximates the equationAϕ = fby its projectionP n Aϕ n = P n f.This projection method is called convergent for operator A, if there exists an index N such thatfor each f ∈ Im A, the approximating equation P n Aϕ = P n f has a unique solution ϕ n ∈ X n forall n ≥ N and these solutions converge, ϕ n → ϕ, n → ∞, to the unique solution ϕ of Aϕ = f.In terms of operators, the convergence of the projection method means that for all n ≥N, the finite-dimensional operators A n := P n A : X n → Y n are invertible and the pointwiseconvergence A −1n P n Aϕ → ϕ, n → ∞, holds for all ϕ ∈ X. In the general case, one may expectthat the convergence occurs if subspaces X n form a dense set:inf ‖ψ − ϕ‖ → 0, n → ∞ (376)ψ∈X nfor all ϕ ∈ X. This property is called the approximating property (any element ϕ ∈ X can beapproximated by elements ψ ∈ X n with an arbitrary accuracy in the norm of X). Therefore,in the subsequent analysis, we will always assume that this condition is fulfilled. Since A n =P n A is a linear operator acting in finite-dimensional spaces, the projection method reduces tosolving a finite-dimensional linear system. Below, the Galerkin method will be considered asthe projection method under the assumption that the above definition is valid in terms of theorthogonal projection.Consider first the general convergence and error analysis.79✷