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<ul><li><p>V.A. DITKIN and A. P. PRUDN~KOV </p><p>(MOSCOW) </p><p>(Received 25 tfay f962) </p><p>Introduction </p><p>The methods of operational calculus are very effective in the solution of many problems in applied mathematical analysis. It is usually the differentiation operator p = ri/t/dt which is considered. In this article we shall consider operators of the form </p><p>uh. = .+ ( t $ ), k = 2,:s ,._., R, </p><p>which are closely associated with the Ressel equation or its general- ization to the case of higher order equations. Therefore operators of this form can conveniently be called Ressel operators cl]. Suppose the linear set M is formed by functions f(t) defined in the region n with the ordinary operations of addition and multiplication on real or com- PI ex numbers. </p><p>As we know, the 1 inear set df is called a ring if for any of its ele- ments f, g associative multiplication f * 6 E !l, which is permutational with multiplication by numbers and distributive with respect to addition, is defined. We shall consider only commutative rings with a unit and without a divisor of zero. Any such ring h! can be extended to a field S? (the elements of the field are denoted by f/b, g f 0). Se have I/& = fl/gI when, and only when f * g1 = f, * g. Following Mikusinskii [21, the elements of the field m will be called operators. Let numbers belong to the ring hf. In this case we can consider the product of a number h by the function f f !! either as the ordinary product hf, or as the product of elements of the ring h * f. </p><p>We shall only consider rings in which both these products are the </p><p>l Zh. vych. mat. 2, No. 6. 997-1018, 1962. </p><p>1181 </p></li><li><p>same, i.e. A * f = hf. In this case h = 1 is the unit of the ring ilf. The operators of the form h/p (h and u are numbers) coincide with ordinary fractions h/p. The operators of the form j/l coincide yrith the functions of the ring II. In the field %? we can consider operators de- pending on a parameter. Such operators are called operator functions. The basic concepts of mathematical analysis, the concepts of a limit, continuous function, derivative and integral can be carried over into operator functions. To do this it is necessary for these concepts to be introduced in the initial ring bf. We shall therefore assume that the ring irf satisfies, in addition to the preceding conditions, the following additional requirements. </p><p>1. The region f, is the ha1 f-line 0 -< t c a. </p><p>2. Each function of the ring ?I is continuous in the region n. </p><p>3. If the sequences of functions j,(t) E IV and g,(t) E M respect- ively converge in the region 9 to the functions j(t) E M and g(t) E df and in any finite interval 0 < t < T the convergence is uniform, then </p><p>4. If f(t; A) depends on the parameter a < h ( F and for all a < h < p, /(t; A) f !f, af(t; h)/;)h E h! exists and the function ?j/?h is continuous with respect to the variables t and A in the region o</p></li><li><p>1186 V.A. Ditkin and A.P. Irurlnikov </p><p>We can always assume that the operator q belongs to i.e. is a function belonging to the set M. The sum and tions which are reducible in the given interval (u, p) </p><p>the initial ring, product of func- are again re- </p><p>ducible functions. Thus, reducible operator functions form a sub-ring in the field of all operator functions. It is easy to extend the basic operations and concepts of mathematical analysis to reducible operator functions [21. This is done according to one general rule. A reducible operator function is said to be continuous if there exists an operator 7 (7 does not depend on A) such that q(t; A) is a continuous function of two variables in the region 0 < t < 0~; a < h < p. </p><p>A reducible operator function n(A) is said to be differentiable in the interval (a, p) if the function q~( t; A) is differentiable with re- spect to the parameter A. If, in addition, a@?, is a continuous func- tion with respect to the variables t and A, then the operator function n(A) is said to be continuously differentiable. If n(A) is a continu- ously differentiable operator function. then the operator (l/7) (%-p/ah), where q~ = 7 * a(A), is said to be the derivative of the operator function and is denoted by </p><p>4 acp a (A) Czz 9 = - -* q 0) </p><p>This definition does not depend on the choice of the operator 7. The derivative of an operator function possesses the properties of an ordinary derivative; namely, if the operator functions a(A) and b(A) have in the interval a < A < p continuous derivatives, then their sum and product possess continuous derivatives in this interval, where </p><p>Ic (A) + 6 @)I = a (A) + 6 (A) </p><p>[a (k)*b (k)] = a (A) * b(k) + a (5) + 6 (A), </p><p>Q(A) * [ 1 bcn)= a'(A)*b(~)--o(A).b'(h) b(M*b(U </p><p>assuming the existence of the left-hand side of the last equality. For reducible operator functions n(A) the integral is defined </p><p>P </p><p>\ a (A) dh. </p><p>. a </p><p>If n(h) is a reducible operator function on the interval a1 < h C ;1 and a1 ( a < ,, < p1 then, by definition, there exists an operator 7 E hl </p><p>such that 7 * n(A) = cp(t; A) E by, a1 < h < F1. Uy definition we have </p></li><li><p>1187 </p><p>The integral of an operator function possesses the properties of an ordinary integral. </p><p>The sequence of operators an is said to converge to the operator a if there exists an operator 7 such that </p><p>QWZ, = fn (t) E M, g*a = f (t) E M, </p><p>and the sequence f,(t) converges to f(t) uniformly in any interval o</p></li><li><p>n real variahl es defined in the region /?( (0 < x1 < ED, . . . , 0 < xn < 0~) and Lebesgue integrable in any finite n-dimensional parallelepipid </p><p>/?j:,. ,,..,. ,, (()</p></li><li><p>h(t)= \dU,i du,. . . ~f(uU...Un~gr(~-u~)(*-un,)... il </p><p>. . (1 - un)] dun, (i.2) then </p><p>%a </p><p>fP s . . . ff%rEz...E*)gI(Zf-El)(sz - E2). .* f% - EtJl4l4a..*4sL= </p><p>Oi a </p><p>= h (zpz2 * . * 2,). (1.3) </p><p>Thus, the Set of all functions of the form f(xln2.. .nn) forms a ring with respect to the operations of addition and contraction. To each function f(x1x2.. . xn) of this ring there corresponds a function f(t) of one variable t. The set of all such functions is denoted by f., . If the </p><p>function J(t) belongs to the set LB then, by definition of th;s set, n </p><p>4 1 </p><p>ii s . ..an.j(z~~2...z,),dj,~~2...cE~,</p></li><li><p>1191) t. ,t . ?itkin and A.?. tt.orfnikov </p><p>we obtain </p><p>Ts s . . . *~~1(512%...~~),~~ldz,...dz,= </p><p>no 0 a,a,. . .un </p><p>= s du&,\du,. . . ~,1(uIU2...un),du,. </p><p>0 0 0 </p><p>Introducing the new variable r = uiu2.. . u,, in this integral we have </p><p>Hence, the set L, consists of all the functions f(t) defined on the </p><p>ha1 f-line 0 < t < 0 2nd satisfying condition (1.4). It follows from (1.1) that </p><p>Applying this inequality to functions of the set L, aud using (1.3). </p><p>(Lb) we find n </p><p>a,~*. ..a, </p><p>x s !.#$!* _ .~~~ua~~-;g(Ua),dun. 0 0 0 0 0 </p><p>me bundle defined for fttnctions of the set L, according to formula </p><p>(1.2) Possesses aI the properties of an ordina~~b~dle, viz. : </p></li><li><p>Opernt ionn 1 calculus of Resse 1 operators s191 </p><p>1) conmutativity, i.e. </p><p>t 1 1 </p><p>'j s dul l du,. . . \f(U~z+.. u*)g[(t - 241) (l-u*). . .(I- un)]durc = </p><p>a 0 1 t 1 </p><p>= </p><p>s I dUIL,. . . \g(uz+.. 4 f [(t - ~1) (1 - u,) . . . (1 - un)l du,; </p><p>0 0 0 </p><p>2) associativity, i.e. </p><p>3) distributivity, i. e. </p><p>f 1 . </p><p>I 5 dildT&. . . \ f (E&2 * * * En) (47 IQ - Ed (1 - s4 * * . (1 - En)1 + </p><p>l l 0 </p><p>+ h r(t - Ed (I- Ea) . . . (I- En)11 dEn = </p><p>=\ti$W.. . ~f(I~E....S$gI(~-~~)(~-~E,)...(l-h)l%+ </p><p>r(t - Ed (1 - Ea) . . . (I- Ml 4.; </p><p>ilar properties of an n-dimensional </p><p>a 0 0 </p><p>(These Properties follow from the sim bundle of functions of n-variahles. ) </p><p>4) if for all t >O </p></li><li><p>1192 v. .1 . Pitkin and A.r. Prodnikov </p><p>t </p><p>s s d%, d%, . * En) g re - El) (I - Es) * * . (1 -Ml d%,, = 0, (1.5) 43 0 0 then at least one of the functions f(t) and g(t) is equal to zero for almost all t on the ha1 f-line 0 < t < a~. </p><p>The last property follows from the theorem of Mikusinski and kyl - Nardzewski [41. In order to use this theorem it is sufficient to see that (1.5) is equivalent (see (1.3)) to </p><p>.x1 x2 </p><p>1s . ..s I (%I %z * * . %n) g [(XI - Ed (2, - Ez) . . . (xn - %n)l d%, 42 . . . d%n = 0 00 0 </p><p>for all O</p></li><li><p>Operat ionaf calculus of ijesse 1 operators 1193 </p><p>an </p><p>dx2 . . . s m (x1 x2 . . . x, ilkc . . . En) dx,. 47, </p><p>Hence, </p><p>where </p><p>G(x,, 21, . . . . xn)=r.~...ip(ll,5,,... &)d$&z...dEn. 0 0 0 </p><p>Applying the latter equality to the functions of the set noting that </p><p>we find that if </p><p>then </p><p>. . </p><p>lBn and </p></li><li><p>1194 V..t. Ditkin and A.P. Prortnikov </p><p>where </p><p>Let Ile denote the sub-set of all functions F(t) which can be put in </p><p>the form n </p><p>where f(t) E IA, and C is any constant. n </p><p>The function f( t) belongs to the set L, . For every function of the </p><p>set YB n th there exists almost everywhere the n derivative F(t). In </p><p>the set !?n let us define the operation of mu1 tip1 ication by the formula </p><p>P (1)*G (t) 2 </p><p>If </p><p>V,?. . .un__l </p><p>\ f (un) dun , </p><p>I, (I II 0 </p><p>We define addition in t!:.e set ;!,,I in the nat.urd way. The product. tle- </p><p>fined in - ( 1. fi) again be1 ongs to ;lhe set !,, , it is commutative, </p><p>associative and possesses distrihtrtivity with respect to sdditim. Tliere </p><p>fore t!le set f R forms a commutntive rin(:. Here, if one of t!le factors </p><p>in (1.6) is ~~~enn,~~~,e~ cx, then we tinve </p></li><li><p>perat ional co lcu lus of Resse 1 operntors 1195 </p><p>=u -&j d\d$[F(U,)du,=aB.;F(t)=aF(t), 2, 6 </p><p>where B = -$t Gl (see [51), i. e. the product of a number by a function </p><p>in a ring coincides with the ordinary product of a number by a function. </p><p>If both the factors in (1.6) are numbers, then the product (1.6) is the same as the ordinary product of numbers. The ring !YB has no divisor </p><p>n of zero. </p><p>We can extend </p><p>extended ring by </p><p>For the operator inverse operator </p><p>2. The operator </p><p>the ring I!!~ to the n </p><p>Bn=+(t$)n </p><p>field of ratios. Let us denote the </p><p>all,. We call the elements of the set al:,, operators. </p><p>l/t we introduce the notation l/t = Sn. Then for the 17; = l/B, we shall have l/B, = t. Therefore </p><p>$ j(t) = f(t -qjd%l!j de,\ d& . . . i &Ez.. . n 6 ; 0 0 </p><p>* - * %nf rtt - %l) (1 - Ed * * * (1 - %n)l d%n, </p><p>or </p><p>f (Un) dun (2-l) </p><p>If F(t) E MB and F(O) = 0, then </p><p>BnF(t)= +(I $)F(t). (2.4 </p><p>It follows from (2.1) that </p></li><li><p>lJ!lr, V.A. Ditkin and A.P. Procfnibov </p><p>* * - 5 f (ElE2. * . En) (1 - Enp d&l. 0 </p><p>In a special case </p><p>1 tm Bn = (m!) * </p><p>The equation </p><p>f(f &fhy=O (2.3) has the solution </p><p>O ( ilk kz (kl) (hqk = oFn__1 (1, 1, . . . , 1; - hf)= J&!. 0 (n fif,. . I. , </p><p>For t = 0 the function J6:,!!., o (0) = 1 (see [d ). Therefore it </p><p>follows from (2. l), (2.21, (2.3) that </p><p>Therefore </p><p>Bn - = J&t!.. 0 (n Tq, 4 +.A -& = IC,!!.. o (n fi,. (2.41 </p><p>where (see ld) </p><p>I&-f!.,o(npf) = ,Fn-,(l, 1, * * *7 1; Al). </p><p>Using the last two formulae it is easy to find </p><p>Pdting A = lo in (2.4) and using the formulne </p></li><li><p>Opt-rat ional calculus of I(e.sse 1 operator< 1197 </p><p>pt 0, o..... 0 (Xl *lifn+l)) = be$)O ,..,, o (x) + i be$b ,.... o(x), </p><p>I(n) 0. o,.... 0 (x1 +fn+l)) = ber:b - i ,..., ,, (2) beipb ,..., o (z). </p><p>where (see [cj! ) </p><p>her!,%,..., </p><p>00 (_ Qk (A-) (n+tf </p><p>O @) = k,. [r (Z/c + I)] (2k) I </p><p>03 (_ l)k (_zJ+l) WS1) </p><p>be&%,..., o (4 = 2 k_-O Ir (2k + 2)1 (2k i- 1) ! </p><p>we find </p><p>As in the ordinary operational calculus, applying well-known oper- ational methods [31, [71, iSI, to the field of operators %~f, we can </p><p>extend the table of valtles of operators consideraMy. For example, differentiating (2.4) with respect to the parameter h and then Putting A = 1 and A = -1 we obtain </p><p>where (see id) </p></li><li><p>1198 V.A. Ditkin and ;2.1. ~ro~fnik#tJ </p><p>1 </p><p>2 </p><p>3 </p><p>4 </p><p>5 </p><p>6 </p><p>7 </p><p>8 </p><p>9 </p><p>IO </p><p>41 </p><p>12 </p><p>13 </p><p>14 </p><p>Tables of formulae of the operators B, </p><p>In B _ _-.ze-lfB* </p><p>0: </p></li><li><p>- </p><p>No. </p><p>15 </p><p>17 </p><p>18 </p><p>F(Z) </p><p>43 (B, - i)- </p><p>(Table 1 conttnued) </p><p>I( t ) </p><p>i nl! t </p><p>m.nJ(n-1) m,m.....m 4 ( ) </p><p>1 PI~~~*,,, m! ?A A) </p><p>& t- n </p><p>imp-1) -%. -/x1. . , -% i ) n)/t </p><p>tiz t- ?I Imp-l) - -/,,-/* )..., -/, ( ) n Vr - </p><p>From (2.5) we find </p><p>It follows from this that </p><p>Multiplying the second of the inequalities (2.6) hy # and summing with respect to II from (1 to CD, we find </p><p>i,,TJS B,;;n+i = Bn n &-1-p * </p><p>which gives </p><p>The operational crrlciilus for the operator </p></li><li><p>1200 t.A. nitbin nnft A.P. Pforlnihov </p><p>can also be applied to the solution of differential equations. The solu- tions of differential equations of the form </p><p>where L(h) = h. q ~~~~~~ f. - - + a, is a polynomial with constant co- </p><p>efficients qi are most simply found. The substitution of the operator </p><p>by the operator ,Cn is done according to the formula </p><p>it. (t -iv] 5 (t) = BTX (t) - l&x,__, - B2,x,_2 - - - * - &&, where xk = BXx (t) [I=0 (k = 0, l, 2, . - 9 m - ) </p><p>3. 0n the solutions of the differential equation </p><p>It was shown in t51 t!:at the operational calculus for the operator </p><p>B = $- t -$ is generated hy t!ie equation to w!?ic! </p><p>the Cessel equation </p><p>ym+;yt -(I - -$)y= 0 (3.1) for = 0 is reduced by the su!xtitution d = 2vc. </p><p>Let us consider the 1 inear ordinary differential equation of third order with variable coefficients </p><p>y + f y _i_ 1 - 3n12 +;3rnn - 3n? y, $ </p><p>i * +l + n) (2m -$ n) (n </p><p>- 2n)] y = 0 (3.2) </p><p>In the theory of third order equations this equation is tl:e analogue of the Bessel equation (3. I). </p><p>If m = n = 0 in equation (3.2) then we shall have </p><p>(3.3) </p></li><li><p>Pr~t z = 3+%. Then the preceding equation takes the form </p><p>(3.4) </p><p>Ey analogy with the way it was done in I:; for the operator </p><p>i% =- $ t -$ , starting from a new definition of a bundle and the pro- perties of solutions of equation (3.4) we can construct an onerational </p><p>calculus for the operator 1 - $ t -$- t -$ (see :!I!, !I01 ) . </p><p>:Yithout loss of generality, in equation (3.4) we C3f nut h = J anCJ </p><p>consider the equation </p><p>t2yS" $ 31y + y -+ y = 0. (3.5) </p><p>Let us use the operational method. J,et </p><p>Then </p><p>(3.7) </p><p>M </p><p>p e-Pftm $$ dt = (- -&j* [pncp (p)], m>n, </p><p>0 </p><p>cc </p><p>s . e+tm $gdl = (- -$j~n[p~c@(p)] -+- (- l)7n-q-(n,~1,,l p-m-'y(O) + (I (34 </p><p>f : (n -- 2) ! </p><p>(n _ m _ zj! p=m-%J (Of _i . * 1 + u2 ! p-m-) (O)], lfl </p></li><li><p>1202 V./l. Ditbin and A.P. Prodnlkou </p><p>equation (3.10) becomes </p><p>23qJ+z(z+I)cp-cp=o. (3.11) </p><p>Equation (3.11) is a special case of an equation depending on the para- meter v: </p><p>z3qY + [ 2 (2 + 1) - -5 2 ( 2) +~-[I- (G)]v= 0, (3.12) </p><p>when this parameter is equal to zero. If the parameter v is not a whole number, then the functions </p><p>where </p><p>&r (z, Y) = zJv (2 vi,, (3.13) G2 (2, v> = ZY (2 v-3, (3.14) </p><p>(p3 (2, Y) := zn, (2 Vi), (3.15) </p><p>Jv (z) = mto G; ($;:I , (Bessel functions) </p><p>Y (2) = COS (nv) J, (2) - J_, (2) </p><p>(sin JCv) , </p><p>I-I (2) = cos(Jg) ; (- l)m (2/2)2rn r (m + 1 + v/2) r (m + 1 - v/2) (Poisson function) m=o </p><p>form a fundamental system of solutions of equation (3.12). For v = 0 this no longer is the case, since as v - 0 the Poisson function n,,(z) degenerates to the Bessel function Iv(z) and the three linearly inde- pendent solutions (3.13), (3.14), (3.15) of equation (3.12) degenerate into two linearly independent solutions of equation (3.11) </p><p>~1 (z) = Lb.& (z, v) = hi & (2, 4 = zJ, (2 vi,, (3.16) 92 (2) = !iIy p2 (2, Y) = ZY, (2 Vi). (3.17) </p><p>To find the third linearly independent solution of equation (3.11) let us introduce the expression </p><p>To apply form in the </p><p>lim ~, (, ) - (p3 (, ) v-.0 </p><p>V </p><p>1IlGpitals rule for the evaluation of the indeterminate last expression we find </p></li><li><p>opernt ionn 1 cu lcu lus of Resse 1 operrrtnrs 1203 </p><p>9((z) = $nr (2). (Jsing the two last equalities we find easily </p><p>lim ~~ (, )...</p></li></ul>