Opuscul Mtemtic Vol. 5 o. 1 005 Zygmunt Wronicz O SOME APPLICATIO OF BIORTHOGOAL SPLIE SYSTEMS TO ITEGRAL EQUATIOS Abstrct. We consider n opertor P : L p(i S n(, suc tt P f = f for f S n(, were S n( is te spce of splines of degree n wit repect to given prtition of te intervl I. Tis opertor is defined by mens of system of step functions biortogonl to B-splines. Ten we use tis opertor to pproximtion to te solution of te Fredolm integrl eqution of te second kind. Convergence rtes for te proximtion of te solution of tis eqution re given. Keywords: opertor ssocited wit step functions, B-splines, integrl eqution, pproximtion. Mtemtics Subject Clssifiction: 41A15, 45B05, 45L10, 65R0. 1. ITRODUCTIO Te purpose of te pper is to give some ppliction of biortogonl spline systems defined erlier by te utor in [15] to te Fredolm integrl eqution of te second kind. Let be given prtition of te intervl I =[, b] nd let { j } be system of normlized B-splines of degree n wit respect to. We constructed system of step functions {λ j } biortogonl to te system { j } suc tt supp λ j supp j in [15]. Ten we defined te following opertor: P : L p (I S n (,sucttp f = f for f S n (,weres n ( is te spce of splines of degree n wit repect to given prtition of te intervl I. P,f (x = j (f,λ j j (x, f L (I, (1 149
150 Zygmunt Wronicz were (f,g = f(t g(t dt nd we estimted te difference f P f wit respect to te modulus of smootness of te function f in te spce L p (I, 1 p. Consider te Fredolm integrl eqution of te second kind y(x =f(x+λ K(x, t y(t dt, ( were f C(I, K C(I nd λ R. We myfind bsic fcts on integrl equtionsndminmetods for finding te solutions of tem in [1,, 9,11].Ametod of ppliction of interpolting splines is given in [1]. Our metod of pproximtion of te solution of te eqution ( is bsed on treemetods of finding te solutions of integrl equtions: cnge te kernel K by te degenerted kernel P K, te metod of te Bubnov Glerkin nd te metod of itertion (cf. [1,, 9]. We ssume tt λ mx K(x, t dt = ϱ<1. x I Ten we pproximte te function f by te opertors of te form (1 nd te kernel K by te opertors of te form P,K (x, t = (K, λ i λ j i (x j (t, i,j were (K, λ i λ j = I K(x, t λ i (x λ j (t dx dt. Forny ε>0 we cn find n opertor P,K (see [15] nd lso [13, 14] suc tt nd P,K (x, t P (x, t <ε for (x, t I λ mx x I P,K (x, t dt < 1. Ten we solve tefollowing integrl eqution wittedegenerte kernel P,K : y(x =P,f + λ P,K (x, t y(t dt. Te solution of tis eqution isspline. We find it using te metod of itertion nd we give te recurrence formul for it in te cse of equidistnt prtitions. At teendof te pper we consider te order of pproximtion of te solution of te eqution ( in te spce W n p (I for 1 p. It seems tt te simplicity nd good properties of pproximtion of te lgoritm my ve some pplictions.
On Some Appliction of Biortogonl Spline Systems to Integrl Equtions 151. A SYSTEM OF STEP FUCTIOS BIORTHOGOAL TO B-SPLIES AD APPROXIMATIO BY SPLIES Fortesimplicity we confine to te equidistnt prtitions of te intervl I =[, b]. Let = { = t n =...= t 0 <t 1 <...<t =...= t +n = b}, (3 were t j = + j, = b, j =1,...,. Setting xk + := (mx{0,x} k, te B-spline of degree n wit respect to is defined s follows: (see [8] or [4,5,6,7] M i,n (s =M i,n (x i,...,x i+n+1 ; s =[x i,...,x i+n+1 :(x s n +], were [x i,...,x i+n+1 : f] is te (n +1 t order divided difference of f t x i,...,x i+n+1.te normlize B-spline i,n is defined s follows i,n (x = x i+n+1 x i M i,n (x. n +1 Furter we need te following properties of B-splines: supp M i,n+1 =supp i,n+1 =[x i,x i+n+1 ], i,n+1 C n 1 (I, M i,n+1 (x dx =1, I i= n i,n+1 (x =1 for x I, i,n+1 (x 0 for x I. Teorem.1 (cf. [4, 7]. Let 1 p + 1 q =1, λ i L q (I =L p(i, i = n,..., 1. Ten for ny integer j = n,..., 1 λ i ( j,n+1 =δ i,j if nd only if λ i = D n+1 f for some f suc tt f = ψ + i,n+1,were ψ + i,n+1 (x =(x t i+1 + (x t i+... (x t i+n /n! Using tis teorem we construct system of step functions {λ i } i= n biortonorml to te system of B-splines { j,n } j= n s in [15] suc ttsupp λ i [t i+k,t i+k+1 ] for i = k, k+1,..., k 1 wit n =k or n =k+1 nd for i = n,..., k 1 or i = k,..., 1suppλ i [t 0,t 1 ] or supp λ i [t,t ] respectively. Let supp λ i [t m,t m+1 ] nd λ i = n j=0 A i,j χ [τj,τ j+1(x, were τ j = t m + j for j =0,...,n, =(t m+1 t m /(n +1 nd χ [τj,τ j+1 is te crcteristic function of te intervl [τ j,τ j+1.toobtin te function λ i it suffices to solve tefollowing Crmer system of n +1 equtions wit n +1 unknowns A i,j t m+1 t m λ i (x j,n (x dx = δ i,j, j = m n,...,m.
15 Zygmunt Wronicz Exmple.1. Let = {t i } +1 i= 1, t 1 =0, t j = j for j =0,..., nd t +1 =. Ten for n =1 x i for i<x i +1, i,1 (x = i + x for i +1<x i +, i =0,...,, 0 oterwise, { 1 x for 0 <x 1, 1,1 (x = 0 oterwise, { x +1 for 1 <x,,1 (x = 0 oterwise, 1 for i<x i + 1, i =0,..., 1, λ i (x = 3 for i + 1 <x i +1, 0 oterwise, 3 for 0 <x 1, λ 1 (x = 1 for 1 <x 1, 0 oterwise. For te prtition (3 we ve ( x t0 1, (x = 1,1 { ( x t 0,1, (x = ( x ti, i, (x = 0,1 for t <x t, 0 oterwise, i =0,...,, = b, nd ( ( x λ 1, (x = 1 t0 x λ 1, λ i, (x = 1 ti λ 0, i =0,...,, { ( 1 λ x t0 0 for t <x t, λ, (x = 0 oterwise. Exmple.. Let = {t i } +1 i= 1, t = t 1 =0, t j = j for j =0,..., nd t +1 = t + =. Ten for n = (x i for i<x i +1, i, (x = (x i +3(x i 3 for i +1<x i +3, i =0,..., 3, 0 oterwise, x 3 x for 0 <x 1, ( x 1, (x = for 1 <x, 0 oterwise,
On Some Appliction of Biortogonl Spline Systems to Integrl Equtions 153 { (x 1 for 0 <x 1,, (x = 0 oterwise, 1 (x (3 4 x for <x 1, (x, (x = 1, ( x = for 1 <x, 0 oterwise, { (x 1 for 1 <x,, (x =, ( x = 0 oterwise, 7 for i +1<x i + 4 3, 10 for i + 4 3 λ i (x = i + 5 3, i = 1, 0,...,, 7 for i + 5 3 <x i +, 0 oterwise, 11 for 0 <x 1 3, 7 for 1 3 λ (x = <x 3, 1 for 3 <x 1, 0 oterwise, 1 for 1 <x 3, 7 for 3 λ (x =λ ( x = <x 1 3, 11 for 1 3 <x, 0 oterwise. nd For te prtition (3 we ve ( ( x t0 x ti, (x =,, i, (x = 0,, i = 1, 0,...,, = b, ( ( t x t x, (x = 1,,, (x =, ( x λ i, (x = 1 ti λ 0 λ, (x = 1 λ ( x t0 We my write te opertor (1 in te form, i = 1, 0,...,, (, λ, (x = 1 t x λ. P (x =P,f (x = j= n (λ j,f j,n (x (4
154 Zygmunt Wronicz nd for tenorm of te opertor P : L p (I L p (I, for 1 p we ve te estimte P Lp(I i= n were λ i,n is defined for teprtition (3 (see [15]. Furter we need te following λ i,n (t dt, Teorem. (see [15] nd lso [6, 7, 13, 14]. Tere exist constnts C k,n,p depending only on k, n nd p suc tt for 1 p, f Wp r (I, 0 k r n +1, k n f (k P (k C k,n,p r k ω(p n+1 r (f (r, for k =0,...,r, r =0,...,n+1,,f Lp(I were = b nd ω (p n+1 (f,δ = sup 0< δ n+1 i=0 ( n +1 ( 1 i i f(x + i Lp([,b n], 1 p is te (n +1 t modulus of smootness of te function f in te spce L p (I. 3. UMERICAL SOLUTIO OF THE FREDHOLM ITEGRAL EQUATIO OF THE SECOD KID Let be te prtition of te intervl I = [, b] defined by (3. Consider te following integrl eqution: y(x =P,f (x+λ P,K (x, t y(t dt, (5 were P,f is defined by (4 nd P,K (x, t = K(ξ,τ λ,i (ξ λ,j (τ dξdτ i,n (x j,n (t. i= n j= n I ow λ,i is defined s follows: Let = {x i} +n i= n, x n =... = x 1 =0, x k = k, k =0,...,, x +1 =...= x +n =. 1 λ ( x xi 0 for i =0,..., n, 1 λ,i (x = λ ( x x0 i for i = n,..., 1, 1 λ ( X x i for i = n +1,..., 1, were λ i is defined for teprtition by mens of Teorem 1(see [15].
On Some Appliction of Biortogonl Spline Systems to Integrl Equtions 155 Let for every t IK(,t W r p (I nd for every x IK(x, W r p (I, were K(,t denotes te function K(x, t of x t fixed t. Since P,K (x, t =P,P,K(,t (x, ten by Teorem we obtin were Let K(x, t P,K (x, t Lp(I K(x, t P,K(,t (x + Lp(I P,K(,t (x P,K (x, t Lp(I ( ( C k,n,p k ω (p k n+1 k x k K, + C k,n,p k ω (p k n+1 k t k K, ( ω m (p k x k K, ( k ω (p m t k K, =sup t I =supω m (p x I for ( ω m (p k x k K(,t, ( k K(x,, tk k =0,...,r,,. ϱ = λ sup x I K(x, t dt < 1 for p =, q 1 q p ϱ = λ K(x, t p dx dt for 1 <p<, (6 ϱ = λ sup t I K(x, t dx < 1 for p =1 nd ϱ denotes te bove quntities for tekernel K (x, t. Hence tereexists 0 suc ttfor > 0 ϱ <ϱ 0 = 1+ϱ < 1. (7 Te kernel P,K is degenerted nd becuse of (7 te solution of te integrl eqution (5 is spline of degree n wit respect to te prtition.denote it by s. We ve tefollowing Teorem 3.1. Let for every t I, K(.t Wp r (I nd for every x I, K(x, Wp r (I nd ϱ stisfies (6. Ten tere exist constnts A k,n,p nd B k,n,p depending only on k, n nd p suc tt s (k Lp(I y(k
156 Zygmunt Wronicz A k,n,p r k ω(p n+1 r (y(r, + + B k,n,p P C(I y Lp (I r k ω(q n+1 r ( r t r K, were y nd s re te solutions of te equtions ( nd (5 respectively nd 1 p + 1 q =1. Proof. Let p = nd > 0.Applying te opertor P to te solution y of te eqution ( we obtin, Hence P,y (x =P,f (x+λ P,K(,t (x y(t dt. s (x P,y (x =λ P,K (x, t[s (t y(t] dt+ + λ [P,K (x, t P,K(,t (x] y(t dt nd by (7 s y C(I s P,y C(I + P,y y C(I ϱ 0 s y C(I + λ Using te properties of B-splines we obtin P,K (x, t P,K(,t (x y(t dt = = i= n P C(I y C(I sup ξ I Hence by Teorem. we obtin P,K (x, t P,K(,t (x y(t dt + P,y y C(I. [ P,K(ξ, (t K(ξ,t ] λ,i (ξ dξ i,n(x y(t dt s y C(I 1 1 ϱ 0 P,y y C(I + P,K(ξ, (t K(ξ,t dt.
On Some Appliction of Biortogonl Spline Systems to Integrl Equtions 157 + λ P C(I y C(I sup P,K (x, t P,K(,t (x dt 1 ϱ 0 x I A 0,n, 1 ϱ 0 r ω n+1 r (y (r, + + λ 1 ϱ 0 P C(I y C(I C 0,n, r ω n+1 r ( r t r K,. ow using te Mrkov inequlity nd Teorem. we obtin s y C(I s P,y C(I + P,y y C(I λ sup x I x P,K(x, t[s (t y(t] dt + [ + λ sup x I x P,K(x, t ] x P,K(,t(x y(t dt + P,y y C(I I Mλ sup P,K (x, t[s (t y(t] dt x I + + Mλ [ P,K (x, t P,K(,t (x ] y(t dt + + P,y y C(I Mλ + Mλ sup P,K (x, t dt s y C(I + x I P,K (x, t P,K(,t (x y(t dt + P,y y C(I Mϱ 0 A 0,n, r 1 ω n+1 r(y (r, + + MλB 0,n, y C(I r 1 C 1,n, r 1 ω n+1 r(y (r, ω n+1 r ( t r K, were M = M(n is constnt depending only on n tken from temrkov inequlity nd A 0,n,, B 0,n, nd C 0,n, re tken from Teorem.. Te remining inequlities we prove similrly. Te proof for p is nlogous.
158 Zygmunt Wronicz Te solution of te eqution (5 we my obtin using te metod of itertion. We proceed s follows: Let nd we put P,f (x = d i,j = j= n c j j (x, P,K (x, t = i (x j (x dx s,m+1 (x =P,f (x+λ were s,m is spline of te m t step ofitertion. Putting s,m (x = k= n i= n j= n i,j i (x j (t, P,K (x, t s,m (t dt, (8 b k,m k (x in (8 nd compring te coefficients b k t k we obtin b k,m+1 = c k + λ i= n k,i j= n b j,m d i,j, m =1,,..., k = n,..., 1. (9 Remrk 3.1. We cn lso use tis metod for te numericl solution of te Volter integrl eqution of te second kind x y(x =f(x+λ K 0 (x, t y(t dt. Let D = {(x, t: x b, t x. Putting { K 0 (x, t for (x, t D, K(x, t = 0 for (x, t / D, we obtin te Fredolm integrl eqution (. owsupp P,K D = {(x, t: x b, t min [x +(n +1,b]}, were =(b /.Iftefirst condition from (6 is stisfied, ten we my solve tis eqution s bove. Unfortuntely te function s,m+1 from te recurrence reltions (8 is not spline of degree n wit respect to te prtition. Hence we cnnot pply (9. If ϱ 1, ten we re looking te solution of te eqution (5 in te following form y(x = λ k φ k (x, (10 k=0
On Some Appliction of Biortogonl Spline Systems to Integrl Equtions 159 were φ 0 (x =P,K (x, φ k+1 (x = x n P,K (x, t φ k (t dt, were x n =min[x +(n +1,b], k =0, 1,... As in[3, 11] we my prove tt φ k (x < λk M k P,f C(I [b + k(n +1 ], k! k =1,,..., were M = sup P,K (x, t. Hence for >(b Mλe te series (x,t D (10 is convergent uniformly to te solution of te eqution (5 on teintervl [, b]. REFERECES [1] Atkinson K.: Te numericl solution of integrl equtions of te second kind. Cmbridge, Cmbridge University Press 1997. [] Atkinson K., Hn W.: Teoreticl numericl nlysis. ew York, Springer-Verlg 001. [3] Berezin I. S., Zidkov. P.: umericl metods, vol. II, Moskv 196 (Russin. [4] de Boor C.: On locl liner functionls wic vnis t ll B-splines but one. In: Teory ofapproximtion wit Applictions, Lw A., Sney A.(Eds, ew York, Acdemic Press 1976, 10 145. [5] de Boor C.: Splines s liner combintions of B-splines. In: Approximtion TeoryII,LorenzG.G.,Cui C. K., Scumker L. L. (Eds, ew York, Acdemic Press 1976, 1 47. [6] Ciesielski Z.: Constructive function teory nd spline systems. Studi Mt. 53 (1975, 78 30. [7] Ciesielski Z.: Lectures on Spline Teory. Gdńsk University, 1979 (Polis. [8] Curry H.B.,Scoenberg I.J.:IV: Te fundmentl spline functions nd teir limits. J. d Anlyse Mt. 17 (1966, 71 107. [9] Krsnov M. L., Kiselev A. I., Mkrenko G. I.: Problems in integrl equtions. Wrszw, PW 197 (Polis. [10] Miclin S.G., Smolicki C. L.: Metods of pproximtion of te solution of differentil nd integrl equtions. Wrszw 197 (Polis. [11] Petrovskii I. G.: Lectures on te teory of integrl equtions. Moscow 1984 (Russin. [1] Subbotin Yu.., Steckin S. B.: Splines in te umericl Anlysis. Moscow, uk 1976 (Russin.
160 Zygmunt Wronicz [13] Wronicz Z.: Approximtion by complex splines. Zeszyty uk. Uniw. Jgiellońskiego, Prce Mt. 0 (1979, 67 88. [14] Wronicz Z.: Systems conjugte to biortogonl spline systems. Bull.Polis Acd. Sci. Mt. 36 (1988, 73 78. [15] Wronicz Z.: On some complex spline opertors.opuscul Mtemtic 3 (003, 99 115. Zygmunt Wronicz wronicz@uci.g.edu.pl AGH University of Science ndtecnology Fculty ofapplied Mtemtics l. Mickiewicz 30, 30-059 Crcow, Polnd Received: My 10, 004.