Integral Equations

n = 2 π∫1−1f(x)U n (x) √ 1 − x 2 dx, n = 0, 1, ... . (330)Note that f(x) may be a complex-valued function; then the Fourier coefficients a n and b nare complex numbers.Consider some examples of decomposing functions in the Fourier–Chebyshev series√1 − x2 = 2 π − 4 π∞∑n=114n 2 − 1 T 2n(x), −1 ≤ x ≤ 1;arcsin x = 4 ∞∑ 1πn=0(2n + 1) T 2n+1(x), −1 ≤ x ≤ 1;2∞∑ (−1) n−1ln (1 + x) = − ln 2 + 2T n (x), −1 < x ≤ 1.n=1nSimilar decompositions can be obtained using the Chebyshev polynomials U n of the secondkind. In the first example above, an even function f(x) = √ 1 − x 2 is decomposed; therefore itsexpansion contains only coefficients multiplying T 2n (x) with even indices, while the terms withodd indices vanish.Since |T n (x)| ≤ 1 according to (321), the first and second series converge for all x ∈ [−1, 1]uniformly in this segment. The third series represents a different type of convergence: it convergesfor all x ∈ (−1, 1] (−1 < x ≤ 1) and diverges at the point x = −1; indeed, thedecomposed function f(x) = ln(x + 1) is not defined at x = −1.In order to describe the convergence of the Fourier–Chebyshev series introduce the functionalspaces of functions square-integrable in the segment (−1, 1) with the weights 1/ √ 1 − x 2 and√1 − x2 . We shall write f ∈ L (1)2 if the integral∫1−1|f(x)| 2 dx√1 − x2(331)converges, and f ∈ L (2)2 if the integral∫1−1|f(x)| 2√ 1 − x 2 dx. (332)converges. The numbers specified by (convergent) integrals (331) and (332) are the squarednorms of the function f(x) in the spaces L (1)2 and L (2)2 ; we will denote them by ‖f‖ 2 1 and ‖f‖ 2 2,respectively. Write the symbolic definitions of these spaces:∫1L (1)2 := {f(x) : ‖f‖ 2 1 :=−1∫1L (2)2 := {f(x) : ‖f‖ 2 2 :=−161|f(x)| 2 dx√ < ∞},1 − x2|f(x)| 2√ 1 − x 2 dx < ∞}.