In the space of continuous periodic functions, we construct interpolation rational operators, use them to obtain quadrature formulas with positive coefficients which are exact on rational trigonometric functions of order 2n, and suggest an algorithm for an approximate solution of integral equations of the second kind. We estimate the accuracy of the approximate solution via the best trigonometric rational approximations to the kernel and the right-hand side of the integral equatio

Гриб, Николай Васильевич

Русак, Валентин Николаевич

Belarusian State Pedagogical University Repository

ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 2, pp. 1–8. c© Pleiades Publishing, Ltd., 2012.Original Russian Text c© V.N. Rusak, N.V. Grib, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 2, pp. 266–273.NUMERICAL METHODSRational Interpolation and Approximate Solutionof Integral EquationsV. N. Rusak and N. V. GribBelarus State University, Minsk, BelarusBelarus State Pedagogical University, Minsk, BelarusReceived March 16, 2011Abstract—In the space of continuous periodic functions, we construct interpolation rationaloperators, use them to obtain quadrature formulas with positive coefficients which are exact onrational trigonometric functions of order 2n, and suggest an algorithm for an approximate so-lution of integral equations of the second kind. We estimate the accuracy of the approximatesolution via the best trigonometric rational approximations to the kernel and the right-handside of the integral equation.DOI: 10.1134/S001226611120115Interpolation quadrature formulas designed for the integration of periodic functions well approx-imated by trigonometric polynomials were studied quite comprehensively [1, 2]. They were used forthe development of approximate methods for solving integral equations whose kernels and coeffi-cients can be approximated by polynomials (e.g., see [3, p. 511]). In the present paper, we considerthe interpolation of periodic functions by rational operators and its application to the constructionof quadrature formulas of Gauss type and to the approximate solution of integral equations of thesecond kind.Let |αk| < 1, k = 1, . . . , n, and letpin(z) =n∏k=1(z − αk)/(1− αkz)be the Blaschke product. Since |pin(z)| = 1 on the unit circle, we havepin(eiϕ) = eiΦn(ϕ), (1)where Φn(ϕ) = argpin(eiϕ) and 0 ≤ Φn(0) = arg∏nk=1(1−αk)/(1−αk) < 2pi. By using relation (1),one can readily find thatΦ′n(ϕ) =n∑k=11− |αk|2(eiϕ − αk)(e−iϕ − αk) > 0. (2)Consequently, as ϕ varies from 0 to 2pi, the argument Φn(ϕ) + ϕ/2 grows from Φn(0) toΦn(0) + (2n+ 1)pi; consequently, the function sin(Φn(ϕ) + ϕ/2) has zeros at the points {ϕk},0 ≤ ϕ0 < ϕ1 < · · · < ϕ2n < 2pi.By using relation (1) and the Euler formulas, one can readily show thatsin(Φn(ϕ) +ϕ2)=eiϕ/22iqn(ϕ)n∏k=1(1− αke−iϕ)(eiϕ − αk)− e−iϕ/2n∏k=1(1− αkeiϕ)(e−iϕ − αk)=pn+1/2(ϕ)qn(ϕ), qn(ϕ) =n∏k=1(e−iϕ − αk)(eiϕ − αk), (3)1 (Reg. No. 211, 9.3.2012)РЕПОЗИТОРИЙ БГПУ2 RUSAK, GRIBwhere pn+1/2(ϕ) is a trigonometric polynomial of half-integer order n + 1/2 with real coefficientsand with zeros at the points {ϕk}2nk=0. In this case, we have the expansion [4, p. 17]pn+1/2(ϕ) = C2n∏k=0sinϕ− ϕk2. (4)Note that cos(Φn(ϕ) + ϕ/2) is a function of the form (3) as well with some trigonometricpolynomial of half-integer order in the numerator and with the same denominator; in addition,cos(Φn(ϕk) +ϕk2)= (−1)k cos(Φn(ϕ0) +ϕ02).By Qn,1 we denote the set {pn(x)/qn(x)} of trigonometric rational functions of order ≤ n withgiven denominator qn(x); accordingly, by Qn,2 we denote the set {p2n(x)/q2n(x)} of trigonometricrational functions of order ≤ 2n with given denominator q2n(x).For any function f(ϕ) in the space C2pi of continuous 2pi-periodic functions, we construct aninterpolation rational operator Ln by settingLn(ϕ, f) =2n∑k=0f(ϕk)lk,n(ϕ), lk,n(ϕ) =sin(Φn(ϕ) + ϕ/2) cos(Φn(ϕk) + ϕk/2)sin((ϕ− ϕk)/2)(1 + 2Φ′n(ϕk)). (5)Lemma 1. The operator Ln defined in (5)1. takes each function f ∈ C2pi to a trigonometric rational function in Qn,1;2. satisfies the conditions Ln(ϕk, f) = f(ϕk), k = 0, . . . , 2n;3. is exact on the set Qn,1.Proof. By virtue of the representation (4), by dividing the half-integer trigonometric polynomialpn+1/2(ϕ) by sin((ϕ − ϕk)/2), we obtain a trigonometric polynomial pk,n(ϕ) of integer order n.By virtue of (3), the terms f(ϕk)lk,n(ϕ) differ from pk,n(ϕ)/qn(ϕ) only by constants, i.e., are rationalfunctions of order ≤ n with the same denominator qn(ϕ). Consequently,∑2nk=0 f(ϕk)lk,n(ϕ) ∈ Qn,1.It follows from (4) and (5) that lk,n(ϕj) = 0 if k 6= j. But if k = j, then, by the l’Hoˆpital rule,we havelj,n(ϕj) = limϕ→ϕjlj,n(ϕ) = limϕ→ϕjcos(Φn(ϕ) + ϕ/2)(Φ′n(ϕ) + 1/2) cos(Φn(ϕj) + ϕj/2)cos((ϕ− ϕj)/2)(1 + 2Φ′n(ϕj))/2= 1.ThenLn(ϕk, f) =2n∑k=0f(ϕj)lj,n(ϕk) =∑k 6=j0 + f(ϕk)× 1 = f(ϕk).If the trigonometric rational function rn(ϕ) belongs to Qn,1, then the difference rn(ϕ)−Ln(ϕ, rn)belongs to Qn,1 and is zero at the points ϕk, k = 0, . . . , 2n; i.e., rn(ϕ)− Ln(ϕ, rn) has 2n+ 1 zeroson [0, 2pi); therefore, rn(ϕ) − Ln(ϕ, rn) ≡ 0, since a trigonometric polynomial of order ≤ n has atmost 2n zeros in the half-strip 0 ≤ Re z < 2pi.The norm in the space C2pi is given by the relation ‖f‖C2pi = max{|f(ϕ)|, ϕ ∈ [0, 2pi]}. Notealso that the interpolation operator (5) is treated as an operator mapping C2pi into C2pi; thus(e.g., see [5, p. 201]),‖Ln(ϕ, f)‖C2pi ≤ ‖Ln‖ ‖f‖C2pi , ‖Ln‖ = max{2n∑k=0|lk,n(ϕ)|, ϕ ∈ [0, 2pi]}. (6)Using the interpolation operator (5), consider a quadrature formula for a function f ∈ C2pi,2pi∫0f(ϕ)dϕ ≈2n∑k=0Akf(ϕk), Ak =2pi∫0lk,n(ϕ) dϕ. (7)(Reg. No. 211, 9.3.2012) DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012РЕПОЗИТОРИЙ БГПУRATIONAL INTERPOLATION AND APPROXIMATE SOLUTION 3Lemma 2. The quadrature formula (7) has the following properties.1. It is exact for any trigonometric rational function rn ∈ Qn,1.2. Its coefficients Ak, k = 0, . . . , 2n, are positive; moreover ,Ak =2pi2Φ′n(ϕk) + 1, k = 0, . . . , 2n.3. The relation∑2nk=0Ak = 2pi is satisfied.Proof. If rn ∈ Qn,1, then, by Lemma 1, rn(ϕ) = Ln(ϕ, rn); by integrating this relation, we obtain2pi∫0rn(ϕ) dϕ =2pi∫0Ln(ϕ, rn) dϕ =2n∑k=0rn(ϕk)2pi∫0lk,n(ϕ) dϕ =2n∑k=0Akrn(ϕk),which completes the proof of the first property.One should evaluate the integralIk(eiϕk) =2pi∫0sin(Φn(ϕ) + ϕ/2)sin((ϕ− ϕk)/2) dϕ = eiϕk/22pi∫0ei(Φn(ϕ)+ϕ) − e−iΦn(ϕ)eiϕ − eiϕk dϕ.Using relation (1) for |z| < 1, consider the integralI˜k(z) = eiϕk/22pi∫0ei(Φn(ϕ)+ϕ) − e−iΦn(ϕ)eiϕ − z dϕ = eiϕk/22pi∫0pin(eiϕ)eiϕ − pin(eiϕ)−1eiϕ − zdeiϕieiϕ=eiϕk/2i∫|z|=1ξpin(ξ)− pin(ξ)−1(ξ − z)ξ dξ. (8)Since, by the Cauchy integral formula,∫|z|=1(ξ − z)−1pin(ξ) dξ = 2piipin(z),and the function ((ξ − z)ξpin(ξ))−1 has no singular points in the domain |ξ| > 1 and has a second-order zero at infinity, it follows from (8) that I˜k(z) = 2pieiϕk/2pin(z). Obviously,Ik(eiϕk) = limz→eiϕkI˜k(z) = 2pieiϕk/2pin(eiϕk) = 2piei(Φn(ϕk)+ϕk/2) = 2pi cos(Φn(ϕk) + ϕk/2).This relation, together with (5) and (7), implies thatAk =cos(Φn(ϕk) + ϕk/2)2Φ′n(ϕk) + 12pi∫0sin(Φn(ϕ) + ϕk/2)sin((ϕ− ϕk)/2) dϕ =2pi2Φ′n(ϕk) + 1, k = 0, . . . , 2n;the coefficients Ak are positive by virtue of inequality (2).To prove property 3, it suffices to note that rn(x) ≡ 1 ∈ Qn,1, and, by virtue of the exactness ofthe quadrature formula on Qn,1, we obtain2n∑k=0Ak =2pi∫0dϕ = 2pi.The proof of Lemma 2 is complete.DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012 (Reg. No. 211, 9.3.2012)РЕПОЗИТОРИЙ БГПУ4 RUSAK, GRIBFor a function f ∈ C2pi, we define the best uniform approximation by trigonometric rationalfunctions in Qn,1 and Qn,2 by settingRTn (f) = inf(‖f − rn‖, rn ∈ Qn,1), RT2n(f) = inf(‖f − r2n‖, r2n ∈ Qn,2).Theorem 1. The quadrature formula (7) is exact for rational functions in the set Qn,2, and itsaccuracy can be estimated as ∣∣∣∣∣2pi∫0f(ϕ) dϕ−2n∑k=0Akf(ϕk)∣∣∣∣∣ ≤ 4piRT2n(f). (9)Proof. Let us show that each trigonometric rational function r2n ∈ Qn,2 can be represented inthe formr2n(ϕ) = sin(Φn(ϕ) +ϕ2) pn−1/2(ϕ)qn(ϕ)+pn(ϕ)qn(ϕ), (10)where pn−1/2(ϕ) is a trigonometric polynomial of half-integer order and the rational functionpn(ϕ)/qn(ϕ) belongs to Qn,1.Indeed, let Ln(ϕ, r2n) be the value of the interpolation operator for the function r2n; thenLn(ϕ, r2n) = rn(ϕ) = pn(ϕ)/qn(ϕ), and the difference r2n(ϕ) − pn(ϕ)/qn(ϕ) is zero at the points{ϕk}2nk=0; therefore,r2n(ϕ)− pn(ϕ)qn(ϕ)= sin(Φn(ϕ) +ϕ2) pn−1/2(ϕ)qn(ϕ),which is equivalent to relation (10).Let us now show that2pi∫0sin(Φn(ϕ) +ϕ2) pn−1/2(ϕ)qn(ϕ)dϕ = 0. (11)Since a trigonometric polynomial of half-integer order can be represented in the formtn−1/2(ϕ) =n∑j=1(ajei(j−1/2)ϕ + bje−i(j−1/2)ϕ),we find that relation (11) follows from the relationsIj =2pi∫0sin(Φn(ϕ) +ϕ2) ei(j−1/2)ϕqn(ϕ)dϕ = 0, j = 1, . . . , n, (12)which should be proved. By using the Euler formulas again and by passing to complex variables,we obtainIj =12i2pi∫0(ei(Φn(ϕ)+ϕ/2) − e−i(Φn(ϕ)+ϕ/2))ei(j−1/2)ϕqn(ϕ)dϕ=12i2pi∫0pin(eiϕ)eiϕ/2 − pin(eiϕ)−1e−iϕ/2∏nk=1(eiϕ − αk)(e−iϕ − αk)ei(j−1/2)ϕ dϕ= −12∫|ξ|=1ξn+j−1∏nk=1(1− αkξ)2dξ +12∫|ξ|=1ξn+j−2∏nk=1(ξ − αk)2dξ = 0,(Reg. No. 211, 9.3.2012) DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012РЕПОЗИТОРИЙ БГПУRATIONAL INTERPOLATION AND APPROXIMATE SOLUTION 5because the integrand in the first integral on the right-hand side is analytic in |z| ≤ 1, and that inthe second integral is analytic in |ξ| ≥ 1 and has a second-order zero at the point at infinity.It follows from relations (10) and (11) and Lemma 2 that2pi∫0r2n(ϕ) dϕ =2pi∫0pn(ϕ)qn(ϕ)dϕ =2pi∫0rn(ϕ) dϕ =2n∑k=0Akrn(ϕk) =2n∑k=0Akr2n(ϕk);i.e., the quadrature formula (7) is exact on the set Qn,2; in other words, formula (7) is a quadratureformula of the Gauss type.Let r∗2n(ϕ) be a trigonometric rational function of the best approximation for the functionf ∈ C2pi, r∗2n(ϕ) ∈ Qn,2; i.e., the relation‖f − r∗2n‖C2pi = RT2n(f)holds. For the accuracy of the quadrature formula, we have∣∣∣∣∣2pi∫0f(ϕ) dϕ−2n∑k=0Akf(ϕk)∣∣∣∣∣ =∣∣∣∣∣2pi∫0(f(ϕ)− r∗2n(ϕ)) dϕ+2n∑k=0Ak(r∗2n(ϕk)− f(ϕk))∣∣∣∣∣≤2pi∫0|f(ϕ)− r∗2n(ϕ)| dϕ+2n∑k=0Ak|r∗2n(ϕk)− f(ϕk)| ≤ 4piRT2n(f).The proof of Theorem 1 is complete.Note that RTn (f) ≥ RT2n(f), and inequality (9) remains valid if RT2n(f) is replaced by RTn (f).Now consider the integral equationx(s)− λ2pi∫0h(s, t)x(t) dt = y(s), (13)where y(s) ∈ C2pi is a known function and the kernel h(s, t) has period 2pi with respect to eachvariable and is jointly continuous. It is known that, for small |λ|, there exists a unique solution ofEq. (13), which is a continuous periodic function (e.g., see [6, p. 109]). By replacing the integralon the left-hand side in relation (13) by a quadrature formula and by requiring the validity of thatrelation only at the node points {ϕj}2nj=0, we obtain the system of linear algebraic equationszj − λ2n∑k=0Akh(ϕj, ϕk)zk = y(ϕj), j = 0, . . . , 2n, (14)whose solution provides the approximate values of the solution of Eq. (13) at the node points.We define an approximate solution for all values of s as a rational function defined by the interpo-lation operator (5) at {zk}2nk=0,z(s) =2n∑k=0zklk,n(s). (15)It is also convenient to rewrite system (14) in the operator formK˜z0z1. . .z2n =y(ϕ0)y(ϕ1). . .y(ϕ2n)⇔z0z1. . .z2n = K˜−1y(ϕ0)y(ϕ1). . .y(ϕ2n) , (16)DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012 (Reg. No. 211, 9.3.2012)РЕПОЗИТОРИЙ БГПУ6 RUSAK, GRIBwhere K˜ is a linear operator on the space R2n+1 and K˜−1 is the corresponding inverse operator onthe same space.The integral equation (13) is also represented in the operator form Kx = y, where K is a linearoperator on the space C2pi and ‖K‖ is the norm of it treated as an operator mapping C2pi into C2pi.We assume that the kernel h(s, t) can be approximated by rational functions in the set Qn,1 withrespect to each variable; i.e., there exist functionsh1(s, t) =1qn(t)(a′0(s)2+n∑k=1a′k(s) cos kt+ b′k(s) sin kt),h2(s, t) =1qn(s)(a′′0(t)2+n∑k=1a′′k(t) cos ks+ b′′k(t) sin ks), (17)with continuous 2pi-periodic coefficients and these functions satisfy the inequalities|h(s, t)− h1(s, t)| ≤ RT∞,n(h), |h(s, t)− h2(s, t)| ≤ RTn,∞(h), 0 ≤ s, t ≤ 2pi. (18)Theorem 2. If x(s) is the exact solution of Eq. (13) whose kernels satisfy condition (18) andz(s) ∈ Qn,1 is an approximate solution found from Eqs. (14) and (15), then the following estimateholds :‖x(s)− z(s)‖C2pi ≤ (‖Ln‖ ‖K˜−1‖ ‖K‖+ 1)(2pi|λ|RTn,∞(h)‖x‖+RTn (y))+ 4pi‖Ln‖ |λ| ‖K˜−1‖RT∞,n(h)(RTn (y) + ‖x‖(1 + 2pi|λ|RTn,∞(h))). (19)Proof. First, we approximate the exact solution by the rational functionx(s) = λ2pi∫0h2(s, t)x(t) dt+ y(s), (20)where y(s) ∈ Qn,1 is the rational function of the best approximation to y(s); therefore,‖y(s) − y(s)‖ = RTn (y). By subtracting relation (20) from (13) and by taking into account condi-tions (18), we obtain‖x(s)− x(s)‖C2pi =∥∥∥∥∥λ2pi∫0(h(s, t)− h2(s, t))x(t) dt+ y(s)− y(s)∥∥∥∥∥ ≤ 2pi|λ|RTn,∞(h)‖x‖+RTn (y).(21)Now we define a rational function x˜(s) from the nodal values {ϕj}2nj=0 by setting [with regardof (16)]x˜(ϕ0)x˜(ϕ1). . .x˜(ϕ2n) = K˜−1u0u1. . .u2n , uj = x(ϕj)− λ2pi∫0h(ϕj, t)x(t) dt, j = 0, . . . , 2n. (22)By virtue of relations (14) and (16), the values of the rational function x(s) can be determinedin the formx(ϕ0)x(ϕ1). . .x(ϕ2n) = K˜−1v0v1. . .v2n , vj = x(ϕj)− λ2n∑k=0Akh(ϕj, ϕk)x(ϕk), j = 0, . . . , 2n. (23)(Reg. No. 211, 9.3.2012) DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012РЕПОЗИТОРИЙ БГПУRATIONAL INTERPOLATION AND APPROXIMATE SOLUTION 7By subtracting relations (23) from (22) and by estimating the values of the operator R2n+1 via itsnorm, we obtain the inequalitymax0≤j≤2n|x(ϕj)− x˜(ϕj)| ≤ ‖K˜−1‖ max0≤j≤2n∣∣∣∣∣λ2pi∫0h(ϕj, t)x(t) dt− λ2n∑k=0Akh(ϕj, ϕk)x(ϕk)∣∣∣∣∣. (24)By using the kernel h1(s, t), which, by virtue of the representation (17), is a rational function oft in the set Qn,1, and by using the fact that, for any s, the product h1(s, t)x(t) is a rational functionin Qn,2, from Theorem 1, we obtain the relation2pi∫0h1(ϕj, t)x(t) dt =2n∑k=0Akh1(ϕj, ϕk)x(ϕk). (25)It follows from relations (24), (25), (18), Lemma 2, and inequality (21) thatmax0≤j≤2n|x(ϕj)− x˜(ϕj)|≤ ‖K˜−1‖ max0≤j≤2n∣∣∣∣∣λ2pi∫0(h(ϕj, t)− h1(ϕj, t))x(t) dt+ λ2n∑k=0Ak(h1(ϕj, ϕk)− h(ϕj, ϕk))x(ϕk)∣∣∣∣∣≤ 4pi‖K˜−1‖|λ|‖x‖RT∞,n(h) ≤ 4pi‖K˜−1‖|λ|RT∞,n(h)(‖x‖(1 + 2pi|λ|RTn,∞(h)) +RTn (y)). (26)Then, by taking into account (6), for the uniform norm, we obtain‖x(ϕ)− x˜(ϕ)‖C2pi ≤ 4pi‖Ln‖‖K˜−1‖|λ|RT∞,n(h)(‖x‖(1 + 2pi|λ|RTn,∞(h)) +RTn (y)). (27)If {zj}2nj=0 is a solution of system (14), then, by virtue of Eqs. (13) and (16), we havez0z1. . .z2n = K˜−1y(ϕ0)y(ϕ1). . .y(ϕ2n) , y(ϕj) = x(ϕj)− λ2pi∫0h(ϕj, t)x(t) dt, j = 0, . . . , 2n, (28)where x(t) is an exact solution of the integral equation.It follows from relations (22), (28), (13), and (21) thatmax0≤j≤2n|x˜(ϕj)− zj| ≤ ‖K˜−1‖ max0≤j≤2n∣∣∣∣∣x(ϕj)− x(ϕj) + λ2pi∫0h(ϕj, t)(x(t)− x(t)) dt∣∣∣∣∣≤ ‖K˜−1‖‖K‖‖x(s)− x(s)‖ ≤ ‖K˜−1‖‖K‖(2pi|λ|RTn,∞(h)‖x‖+RTn (y));therefore, by virtue of (6), for the uniform norm ‖x˜(s)− z(s)‖ [see (15)], we obtain the estimate‖x˜(s)− z(s)‖ ≤ ‖Ln‖‖K˜−1‖‖K‖(2pi|λ|RTn,∞(h)‖x‖+RTn (y)). (29)By combining the estimates (21), 927), and (29), as a result, we get‖x(s)− z(s)‖ ≤ ‖x(s)− x(s)‖+ ‖x(s)− x˜(s)‖+ ‖x˜(s)− z(s)‖≤ (2pi|λ|RTn,∞(h)‖x‖+RTn (y))(1 + ‖Ln‖‖K˜−1‖‖K‖)+ 4pi‖Ln‖‖K˜−1‖|λ|RT∞,n(h)(‖x‖(1 + 2pi|λ|RTn,∞(h)) +RTn (y)),which coincides with the estimate (19).DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012 (Reg. No. 211, 9.3.2012)РЕПОЗИТОРИЙ БГПУ8 RUSAK, GRIBRemark. All quantities occurring on the right-hand side in the estimate (19) are bounded;this fails only for the norm ‖Ln‖ of the interpolation operator, which, in a number of importantcases, has logarithmic growth with respect to n. This right-hand side depends essentially on theparameters {αk}nk=1, which are chosen depending on properties of the kernel h(s, t) and the right-hand side y(s) of the integral equation. If, for integer values of n and for a positive α, there existsets {αk}nk=0 such thatRT∞,n(h) = O(1nα), RTn,∞(h) = O(1nα), RTn (y) = O(1nα),then it readily follows from Theorem 2 that the accuracy ‖x(s)−z(s)‖C2pi satisfies the order estimate‖x(s)− z(s)‖C2pi = O(‖Ln‖nα).In conclusion, we note that the rational approximation was used in [7] for the estimate of theaccuracy of an approximate solution of a characteristic singular integral equation.REFERENCES1. Nikol’skii, S.M., Kvadraturnye formuly (Quadrature Formulas), Moscow, 1974.2. Turetskii, A.Kh., On Quadrature Formulas with an Even Number of Nodes, Which Are Exact forTrigonometric Polynomials, Dokl. Akad. Navuk BSSR, 1960, vol. 4, no. 9, p. 365.3. Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz v normirovannykh prostranstvakh (FunctionalAnalysis in Normed Spaces), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1959.4. Turetskii, A.Kh., Teoriya interpolirovaniya v zadachakh (Theory of Interpolation in Problems), Minsk,1968.5. Korovkin, P.P., Lineinye operatory i teoriya priblizhenii (Linear Operators and Approximation Theory),Moscow, 1959.6. Kantorovich, L.V. and Krylov, V.I., Priblizhennye metody vysshego analiza (Approximated Methods ofHigher Analysis), Leningrad; Moscow, 1941.7. Mitenkov, V.I. and Rusak, V.N., An Error Estimate for an Approximation of a Characteristic SingularIntegral Equation, Differ. Uravn., 2001, vol. 37, no. 3, pp. 410–414.(Reg. No. 211, 9.3.2012) DIFFERENTIAL EQUATIONS Vol. 48 No. 2 2012РЕПОЗИТОРИЙ БГПУ

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