836 INTEGRAL EQUATIONS As was noted above, the eigenfunctions corresponding to distinct characteristic values are orthogonal. Hence, the sequence of eigenfunctions of a symmetric kernel can be made orthonormal. In what follows we assume that the sequence of eigenfunctions of a symmetric kernel is orthonormal. We also assume that the characteristic values are always numbered in the increasing order of their absolute values. Thus, if λ 1 , λ 2 , , λ n , (16.4.5.1) is the sequence of characteristic values of a symmetric kernel, and if a sequence of eigen- functions ϕ 1 , ϕ 2 , , ϕ n , (16.4.5.2) corresponds to the sequence (16.4.5.1) so that ϕ n (x)–λ n b a K(x, t)ϕ n (t) dt = 0,(16.4.5.3) then b a ϕ i (x)ϕ j (x) dx = 1 for i = j, 0 for i ≠ j, (16.4.5.4) and |λ 1 | ≤ |λ 2 | ≤ ···≤ |λ n | ≤ ··· .(16.4.5.5) If there are infinitely many characteristic values, then it follows from the fourth Fredholm theorem that their only accumulation point is the point at infinity, and hence λ n →∞as n →∞. The set of all characteristic values and the corresponding normalized eigenfunctions of a symmetric kernel is called the system of characteristic values and eigenfunctions of the kernel. The system of eigenfunctions is said to be incomplete if there exists a nonzero square integrable function that is orthogonal to all functions of the system. Otherwise, the system of eigenfunctions is said to be complete. 16.4.5-2. Bilinear series. Assume that a kernel K(x, t) admits an expansion in a uniformly convergent series with respect to the orthonormal system of its eigenfunctions: K(x, t)= ∞ k=1 a k (x)ϕ k (t)(16.4.5.6) for all x in the case of a continuous kernel or for almost all x in the case of a square integrable kernel. We have a k (x)= b a K(x, t)ϕ k (t) dt = ϕ k (x) λ k ,(16.4.5.7) and hence K(x, t)= ∞ k=1 ϕ k (x)ϕ k (t) λ k .(16.4.5.8) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 837 Conversely, if the series ∞ k=1 ϕ k (x)ϕ k (t) λ k (16.4.5.9) is uniformly convergent, then formula (16.4.5.8) holds. The following assertion holds: the bilinear series (16.4.5.9) converges in mean-square to the kernel K(x, t). If a symmetric kernel K(x, t)hasfinitely many characteristic values λ 1 , , λ n ,thenit is degenerate, because in this case, there are only n terms remaining in the sum (16.4.5.8). AkernelK(x, t)issaidtobepositive definite if for all functions ϕ(x) that are not identically zero we have b a b a K(x, t)ϕ(x)ϕ(t) dx dt > 0, and the above quadratic functional vanishes for ϕ(x)=0 only. Such a kernel has positive characteristic values only. Each symmetric positive definite continuous kernel can be decomposed in a bilinear series in eigenfunctions that is absolutely and uniformly convergent with respect to the variables x, t. The assertion remains valid if we assume that the kernel has finitely many negative characteristic values. If a kernel K(x, t) is symmetric, continuous on the square S = {a ≤ x ≤ b, a ≤ t ≤ b},and has uniformly bounded partial derivatives on this square, then this kernel can be expanded in a uniformly convergent bilinear series in eigenfunctions. 16.4.5-3. Hilbert–Schmidt theorem. If a function f (x) can be represented in the form f(x)= b a K(x, t)g(t) dt,(16.4.5.10) where the symmetric kernel K(x, t) is square integrable and g(t) is a square integrable function, then f (x) can be represented by its Fourier series with respect to the orthonormal system of eigenfunctions of the kernel K(x, t): f(x)= ∞ k=1 a k ϕ k (x), (16.4.5.11) where a k = b a f(x)ϕ k (x) dx, k = 1, 2, Moreover, if b a K 2 (x, t) dt ≤ A < ∞,(16.4.5.12) then the series (16.4.5.11) is absolutely and uniformly convergent for any function f(x)of the form (16.4.5.10). Remark. In the Hilbert–Schmidt theorem, the completeness of the system of eigenfunctions is not as- sumed. 838 INTEGRAL EQUATIONS 16.4.5-4. Bilinear series of iterated kernels. By the definition of the iterated kernels, we have K m (x, t)= b a K(x, z)K m–1 (z, t) dz, m = 2, 3, (16.4.5.13) The Fourier coefficients a k (t)ofthekernelK m (x, t), regarded as a function of the variable x, with respect to the orthonormal system of eigenfunctions of the kernel K(x, t) are equal to a k (t)= b a K m (x, t)ϕ k (x) dx = ϕ k (t) λ m k .(16.4.5.14) On applying the Hilbert–Schmidt theorem to (16.4.5.13), we obtain K m (x, t)= ∞ k=1 ϕ k (x)ϕ k (t) λ m k , m = 2, 3, (16.4.5.15) In formula (16.4.5.15), the sum of the series is understood as the limit in mean-square. If, in addition to the above assumptions, inequality (16.4.5.12) is satisfied, then the series in (16.4.5.15) is uniformly convergent. 16.4.5-5. Solution of the nonhomogeneous equation. Let us represent an integral equation y(x)–λ b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b,(16.4.5.16) where the parameter λ is not a characteristic value, in the form y(x)–f(x)=λ b a K(x, t)y(t) dt (16.4.5.17) and apply the Hilbert–Schmidt theorem to the function y(x)–f (x): y(x)–f(x)= ∞ k=1 A k ϕ k (x), A k = b a [y(x)–f(x)]ϕ k (x) dx = b a y(x)ϕ k (x) dx – b a f(x)ϕ k (x) dx = y k – f k . Taking into account the expansion (16.4.5.8), we obtain λ b a K(x, t)y(t) dt = λ ∞ k=1 y k λ k ϕ k (x), and thus λ y k λ k = y k – f k , y k = λ k f k λ k – λ , A k = λf k λ k – λ .(16.4.5.18) 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 839 Hence, y(x)=f(x)+λ ∞ k=1 f k λ k – λ ϕ k (x). (16.4.5.19) However, if λ is a characteristic value, i.e., λ = λ p = λ p+1 = ··· = λ q ,(16.4.5.20) then for k ≠ p, p +1, , q, the terms (16.4.5.19) preserve their form. For k = p, p +1, , q, formula (16.4.5.18) implies the relation f k = A k (λ – λ k )/λ, and by (16.4.5.20) we obtain f p = f p+1 = ··· = f q = 0. The last relation means that b a f(x)ϕ k (x) dx = 0 for k = p, p + 1, , q, i.e., the right-hand side of the equation must be orthogonal to the eigenfunctions that correspond to the characteristic value λ. In this case, the solutions of equations (16.4.5.16) have the form y(x)=f(x)+λ ∞ k=1 f k λ k – λ ϕ k (x)+ q k=p C k ϕ k (x), (16.4.5.21) where the terms in the first of the sums (16.4.5.21) with indices k = p, p + 1, , q must be omitted (for these indices, f k and λ – λ k vanish in this sum simultaneously). The coefficients C k in the second sum are arbitrary constants. 16.4.5-6. Fredholm alternative for symmetric equations. The above results can be unified in the following alternative form. A symmetric integral equation y(x)–λ b a K(x, t)y(t) dt = f(x), a ≤ x ≤ b,(16.4.5.22) for a given λ, either has a unique square integrable solution for an arbitrarily given function f(x) L 2 (a, b), in particular, y = 0 for f = 0, or the corresponding homogeneous equation has fi nitely many linearly independent solutions Y 1 (x), , Y r (x), r > 0. For the second case, the nonhomogeneous equation has a solution if and only if the right-hand side f(x) is orthogonal to all the functions Y 1 (x), , Y r (x) on the interval [a, b]. Here the solution is defined only up to an arbitrary additive linear combination A 1 Y 1 (x)+···+ A r Y r (x). 16.4.5-7. Resolvent of a symmetric kernel. The solution of a Fredholm equation of the second kind (16.4.5.22) can be written in the form y(x)=f(x)+λ b a R(x, t; λ)f(t) dt,(16.4.5.23) 840 INTEGRAL EQUATIONS where the resolvent R(x, t; λ) is given by the series R(x, t; λ)= ∞ k=1 ϕ k (x)ϕ k (t) λ k – λ .(16.4.5.24) Here the collections ϕ k (x)andλ k form the system of eigenfunctions and characteristic values of (16.4.5.22). It follows from formula (16.4.5.24) that the resolvent of a symmetric kernel has only simple poles. 16.4.5-8. Extremal properties of characteristic values and eigenfunctions. Let us introduce the notation (u, w)= b a u(x)w(x) dx, u 2 =(u, u), (Ku, u)= b a b a K(x, t)u(x)u(t) dx dt, where (u, w)istheinner product of functions u(x)andw(x), u is the norm of a func- tion u(x), and (Ku, u)isthequadratic form generated by the kernel K(x, t). Letλ 1 be the characteristic value of the symmetric kernel K(x, t) with minimum absolute value and let y 1 (x) be the eigenfunction corresponding to this value. Then 1 |λ 1 | =max y 0 |(Ky, y)| y 2 ;(16.4.5.25) in particular, the maximum is attained, and y = y 1 is a maximum point. Let λ 1 , , λ n be the first n characteristic values of a symmetric kernel K(x, t)(in the ascending order of their absolute values) and let y 1 (x), , y n (x) be orthonormal eigenfunctions corresponding to λ 1 , , λ n , respectively. Then the formula 1 |λ n+1 | =max |(Ky, y)| y 2 (16.4.5.26) is valid for the characteristic value λ n+1 following λ n . The maximum is taken over the set of functions y which are orthogonal to all y 1 , , y n and are not identically zero, that is, y ≠ 0, (y, y j )=0, j = 1, , n;(16.4.5.27) in particular, the maximum in (16.4.5.26) is attained, and y = y n+1 is a maximum point, where y n+1 is any eigenfunction corresponding to the characteristic value λ n+1 which is orthogonal to y 1 , , y n . Remark. For a positive definite kernel K(x, t), the symbol of modulus on the right-hand sides of (16.4.5.26) and (16.4.5.27) can be omitted. 16.4.5-9. Skew-symmetric integral equations. By a skew-symmetric integral equation we mean an equation whose kernel is skew- symmetric, i.e., an equation of the form y(x)–λ b a K(x, t)y(t) dt = f(x)(16.4.5.28) whose kernel K(x, t) has the property K(t, x)=–K(x, t). Equation (16.4.5.28) with the skew-symmetric kernel has at least one characteristic value, and all its characteristic values are purely imaginary. 16.4. LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 841 16.4.6. Methods of Integral Transforms 16.4.6-1. Equation with difference kernel on the entire axis. Consider an integral equation of convolution type of the second kind with one kernel y(x)+ 1 √ 2π ∞ –∞ K(x – t)y(t) dt = f(x), –∞ < x < ∞.(16.4.6.1) Let us apply the (alternative) Fourier transform to equation (16.4.6.1). In this case, taking into account the convolution theorem (see Paragraph 11.4.1-3), we obtain Y(u)[1 + K(u)] = F(u). (16.4.6.2) Thus, on applying the Fourier transform we reduce the solution of the original integral equation (16.4.6.1) to the solution of the algebraic equation (16.4.6.2) for the transform of the unknown function. The solution of equation (16.4.6.2) has the form Y(u)= F(u) 1 + K(u) .(16.4.6.3) Formula (16.4.6.3) gives the transform of the solution of the original integral equation in terms of the transforms of the known functions, namely, the kernel and the right-hand side of the equation. The solution itself can be obtained by applying the Fourier inversion formula: y(x)= 1 √ 2π ∞ –∞ Y(u)e –iux du = 1 √ 2π ∞ –∞ F(u) 1 + K(u) e –iux du.(16.4.6.4) In fact, formula (16.4.6.4) solves the problem; however, sometimes it is not convenient because it requires the calculation of the transform F(u) for each right-hand side f (x). In many cases, the representation of the solution of the nonhomogeneous integral equa- tion via the resolvent of the original equation is more convenient. To obtain the desired representation, we note that formula (16.4.6.3) can be transformed to the expression Y(u)=[1 – R(u)]F(u), R(u)= K(u) 1 + K(u) .(16.4.6.5) On the basis of (16.4.6.5), by applying the Fourier inversion formula and the convolution theorem (for transforms) we obtain y(x)=f (x)– 1 √ 2π ∞ –∞ R(x – t)f(t) dt,(16.4.6.6) where the resolvent R(x – t) of the integral equation (16.4.6.1) is given by the relation R(x)= 1 √ 2π ∞ –∞ K(u) 1 + K(u) e –iux du.(16.4.6.7) Thus, to determine the solution of the original integral equation (16.4.6.1), it suffices to find the function R(x) by formula (16.4.6.7). To calculate direct and inverse Fourier transforms, one can use the corresponding tables from Sections T3.3 and T3.4, and the books by Bateman and Erd ´ elyi (1954) and by Ditkin and Prudnikov (1965). 842 INTEGRAL EQUATIONS Example. Let us solve the integral equation y(x)–λ ∞ –∞ exp α|x – t| y(t) dt = f(x), –∞ < x < ∞,(16.4.6.8) which is a special case of equation (16.4.6.1) with kernel K(x – t) given by the expression K(x)=– √ 2πλe –α|x| , α > 0.(16.4.6.9) Let us find the function R(x). To this end, we calculate the integral K(u)=– ∞ –∞ λe –α|x| e iux dx =– 2αλ u 2 + α 2 . (16.4.6.10) In this case, formula (16.4.6.5) implies R(u)= K(u) 1 + K(u) =– 2αλ u 2 + α 2 – 2αλ , (16.4.6.11) and hence R(x)= 1 √ 2π ∞ –∞ R(u)e –iux du =– 2 π ∞ –∞ αλ u 2 + α 2 – 2αλ e –iux du. (16.4.6.12) Assume that λ < 1 2 α. In this case the integral (16.4.6.12) makes sense and can be calculated by means of the theory of residues on applying the Jordan lemma (see Subsection 11.1.2). After some algebraic manipulations, we obtain R(x)=– √ 2π αλ √ α 2 – 2αλ exp –|x| √ α 2 – 2αλ (16.4.6.13) and finally, in accordance with (16.4.6.6), we obtain y(x)=f (x)+ αλ √ α 2 – 2αλ ∞ –∞ exp –|x – t| √ α 2 – 2αλ f(t) dt,–∞ < x < ∞. (16.4.6.14) 16.4.6-2. Equation with the kernel K(x, t)=t –1 Q(x/t)onthesemiaxis. Here we consider the following equation on the semiaxis: y(x)– ∞ 0 1 t Q x t y(t) dt = f (x). (16.4.6.15) To solve this equation we apply the Mellin transform which is defined as follows (see also Section 11.3): ˆ f(s)=M{f(x), s} ≡ ∞ 0 f(x)x s–1 dx,(16.4.6.16) where s = σ +iτ is a complex variable (σ 1 < σ < σ 2 )and ˆ f(s) is the transform of the function f(x). In what follows, we briefl y denote the Mellin transform by M{f(x)} ≡ M{f (x), s}. For known ˆ f(s), the original function can be found by means of the Mellin inversion formula f(x)=M –1 { ˆ f(s)} ≡ 1 2πi c+i∞ c–i∞ ˆ f(s)x –s ds, σ 1 < c < σ 2 ,(16.4.6.17) where the integration path is parallel to the imaginary axis of the complex plane s and the integral is understood in the sense of the Cauchy principal value. . (16.4.6.2) has the form Y(u)= F(u) 1 + K(u) .(16.4.6.3) Formula (16.4.6.3) gives the transform of the solution of the original integral equation in terms of the transforms of the known functions,. λ .(16.4.5.24) Here the collections ϕ k (x )and k form the system of eigenfunctions and characteristic values of (16.4.5.22). It follows from formula (16.4.5.24) that the resolvent of a symmetric kernel has only. transform we reduce the solution of the original integral equation (16.4.6.1) to the solution of the algebraic equation (16.4.6.2) for the transform of the unknown function. The solution of equation