LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL

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1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 55, No. 1, 214, Pges Published online: June 2, 214 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO Abstrct. We consider the Lplce trnsform s Henstock-Kurzweil integrl. We give conditions for the existence, continuity nd differentibility of the Lplce trnsform. A Riemnn-Lebesgue Lemm is given, nd it is proved tht the Lplce trnsform of convolution is the pointwise product of Lplce trnsforms. 1. Introduction Given function f :, ) R, its Lplce trnsform t x is defined by L {f}(x) = f(t)e tx dt. In this pper we consider this trnsform using the Henstock-Kurzweil integrl. This integrl generlizes the Riemnn nd Lebesgue integrls, s well s the Riemnn nd Lebesgue improper integrls. We prove some properties of the Lplce trnsform (existence, continuity nd differentibility). A Riemnn-Lebesgue Lemm is given, nd vrious conditions re imposed on the functions so tht the Lplce trnsform of convolution is the pointwise product of Lplce trnsforms. Throughout this pper we use the vrible s to indicte tht L {f}(s) is defined on rel number s, nd we use z to indicte tht L {f}(z) is defined on complex number z. 2. Preliminries Let us begin by reclling the definition of the Henstock-Kurzweil integrl. For finite intervls in R it is defined in the following wy: Definition 2.1. Let f :, b R be function. We sy tht f is Henstock- Kurzweil (shortly, HK-) integrble, if there exists A R such tht, for ech ɛ >, there is function γ ɛ :, b (, ) (nmed guge) with the property tht for ny γ ɛ -fine prtition P = {( x i 1, x i, t i )} n i=1 of, b (i.e. {x i 1, x i : i = 1,..., n} is non-overlpping prtition of, b nd for ech i, x i 1, x i t i γ ɛ (t i ), t i + γ ɛ (t i ) ), one hs Σ n i=1f(t i )(x i x i 1 ) A < ɛ. (2.1) 21 Mthemtics Subject Clssifiction. 44A1, 26A39. Key words nd phrses. Lplce trnsform, Henstock-Kurzweil integrl, bounded vrition functions. 71

2 72 SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO The number A is the integrl of f over, b nd it is denoted s A = f. In the unbounded cse, we require the following definition. Definition 2.2. Given guge function γ :, (, ) we sy tht tgged prtition P = {( x i 1, x i, t i )} n+1 i=1 of, is γ-fine, if () = x, x n+1 = t n+1 =, (b) x i 1, x i t i γ(t i ), t i + γ(t i ) for ll i = 1, 2,..., n, (c) x n, 1/γ(t n+1 ),. Definition 2.3. A function f :, R is Henstock-Kurzweil integrble on,, if there exists A R such tht, for ech ɛ >, there is guge γ ɛ :, (, ) for which (2.1) is stisfied for every tgged prtition P which is γ ɛ -fine ccording to Definition 2.2. Let f be rel function defined on n infinite intervl, ); we cn suppose tht f is defined on, ssuming tht f( ) =. Thus f is Henstock-Kurzweil integrble on, ) if f extended to, is HK-integrble. For functions defined over intervls (, nd (, ) we mke similr considertions. Let I be finite or infinite intervl. The spce of ll Henstock-Kurzweil integrble functions on I is denoted by HK(I). This spce will be considered with the Alexiewicz semi-norm, which is defined s f I = sup f, f HK(I), J I where the supremum is being tken over ll intervls J contined in I. J Definition 2.4. Let ϕ : I C be function, where I R is finite intervl. The vrition of ϕ on the intervl I is defined s { n V I ϕ = sup ϕ(x i ) ϕ(x i 1 ) } P is prtition of I. i=1 We sy tht the function ϕ is of bounded vrition on I if V I ϕ <. Now, if ϕ is function defined on n infinite intervl I, then ϕ is of bounded vrition on I, if ϕ is of bounded vrition on ech finite subintervl of I nd there exists some M > such tht V,b ϕ M for ll, b I. The vrition of ϕ on I is V I ϕ = sup{v,b ϕ, b I}. The spce of ll bounded vrition functions on I is denoted by BV(I). The following theorems re clssicl nd will be used throughout this pper. Theorem , Lemm 24 If g is HK-integrble function on, b R nd f is function of bounded vrition on, b, then fg is HK-integrble on, b nd fg inf f(t) g(t)dt t,b + g,bv,b f. Theorem 2.6 (Chrtier-Dirichlet s Test, 2). Let f nd g be functions defined on, ). Suppose tht Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

3 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL 73 (i) g HK(, c) for every c, nd G defined by G(x) = x g is bounded on, ); (ii) f is of bounded vrition on, ) nd lim f(x) =. x Then fg HK(, )). Theorem 2.7 (Du Bois-Reymond s Test, 2). Let f nd ϕ be functions defined on, ) nd suppose tht: (1) f HK(, c) for ll c nd F (x) = x f is bounded on, ); (2) ϕ is differentible on, ) nd ϕ is Lebesgue integrble on, ); (3) lim F (x)ϕ(x) exists. x Then fϕ HK(, )). Definition 2.8 (4). Let E, b. A function f :, b R is AC δ on E, if for every ɛ >, there exist η ɛ > nd guge δ ɛ on E such tht s f(v i ) f(u i ) < ɛ, i=1 whenever P = {( u i, v i, t i )} s i=1 is (δ ɛ, E)-fine subprtition of, b (i.e., P is δ ɛ -fine nd the tgs t i belong to E) nd s i=1 v i u i < η ɛ. We sy tht f is ACG δ on, b, if, b cn be written s countble union of sets on ech of which the function f is AC δ. If h is function in the vribles (t, s), then we use the nottion D 2 h for the prtil derivtive of h with respect to the vrible s. Theorem , Theorem 4 Let, b R. If h : R, b C is such tht (i) h(t, ) is ACG δ on, b for lmost ll t R, (ii) h(, s) is HK-integrble on R for ll s, b, then H := h(t, )dt is ACG δ on, b nd H (s) = D 2h(t, s)dt for lmost ll s (, b), if nd only if, t s for ll s, t, b. In prticulr, D 2 h(t, s)dtds = H (s ) = t s D 2 h(t, s )dt, when H 2 := D 2h(t, )dt is continuous t s. 3. Min results D 2 h(t, s)dsdt Let I be finite or infinite intervl. We use the following nottion: L(I) = {f f is Lebesgue integrble on I}, L loc = {f f is Lebesgue integrble on ech, b, )}, HK(I) = {f f is HK-integrble on I}, Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

4 74 SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO HK loc = {f f is HK-integrble on ech, b, )}, B(I) = {f f is bounded on I}, BV(I) = {f f is of bounded vrition on I}, BV (, )) = {f BV(, )) lim f(t) = }, t BV = {f f BV(b, )) for some b > nd lim f(t) = }. b t Tlvil in 11 introduced the spce HK loc BV to study the Fourier trnsform. Also Mendoz, Escmill nd Sánchez (see 5, 6, 7) hve studied this trnsform on the spces HK(, )) BV(, )) nd BV (, )). All these spces stisfied the following inclusion reltions: (1) HK(, )) BV(, )) BV (, )) HK loc BV ; (2) HK(, )) BV(, )) L(, )), (see 5, Exmple 2.1 (i)); (3) HK loc BV HK(, )) nd HK(, )) HK loc BV (see 5, Exmple 2.1 (ii)). In this pper, we nlyse the Lplce trnsform on the spces bove. It is well known tht if f L loc nd there exist constnts c R, M > nd T > such tht f(t) Me ct for ll t > T, then L {f}(s) exists for ll s > c. We give other existentil conditions. Observe tht if s R +, then e st, s function of rel vrible t, is monotone nd lim e st =. Therefore, from Chrtier-Dirichlet s Test (Theorem 2.6), it t follows tht if f HK loc nd F (x) = x f(t)dt, x <, is bounded on, ), then L {f}(s) exists for ll s >. So the next theorem follows. Theorem 3.1. If f HK(, )), then L {f}(s) exists for ll s, ). Proof. Its cler tht if f HK(, )), then L {f}() exists. Moreover, since x f(t)dt f, ) for ll x, it follows tht L {f}(s) exists for ll s, ). Exmple 3.2. For ny >, sin t t HK(, )) \ L(, )). Thus the function f :, ) R defined by { sin t f (t) = t, if t,, if t < belongs to HK(, )). Therefore L {f }(s) exists for ll s, ). The function t e zt, now when z C, is not of bounded vrition on, ), so in order to prove the existence of the Lplce trnsform we cn t use Chrtier- Dirichlet s Test. In 3 the next result is proved using Du Bois-Reymond s Test. For the ske of completeness, here we will give its proof. Theorem 3.3. If f HK loc nd F (x) = x f(t)dt, x <, is bounded on, ), then L {f}(z) exists for ll z C with Re z >. Proof. The condition Re z > implies tht lim e zt = nd so d t dt (e zt ) is Lebesgue integrble on, ). Also lim F x (x)e zx =. Therefore by Theorem 2.7, L {f}(z) = f(t)e zt dt exists. Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

5 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL 75 Corollry 3.4. If f HK(, )), then L {f}(z) exists for ll z C such tht Re z >. x Another condition for the existence of L {f}(z), tht does not require F (x) = f(t)dt to be bounded on, ), is shown in the following theorem. Theorem 3.5. If f HK loc BV then L {f}(z) exists for ll z C with Re z >. Proof. There exists b > such tht f BV(b, )) nd lim f(t) =. Since t Re z >, it follows tht x b e tz dt < 2 z for ll x b, ). This implies, from Theorem 2.6, tht f(t)e tz dt exists. Now since f HK b loc nd e tz is of bounded vrition on, b, the integrl f(t)e tz dt lso exists. Exmple 3.6. Let f : 1, ) R be function defined by f(t) = t p, where x < p < 1. Observe tht lim x t p dt = but f HK loc BV. Thus, Corollry 3.4 does not pply; however from Theorem 3.5, L {f}(z) exists for ll z C with Re z >. Remrk 3.7. If f HK(, )) BV(, )), then L {f}(z) exists for ll z C with Re z. It is cler tht this condition is stisfied when Re z >, becuse HK(, )) BV(, )) HK loc BV. Now, if Re z =, then L {f}(z) = fχ, ) (Im z), where the ltter expression is the Fourier trnsform of fχ, ) evluted t Im z. From 6, Theorem 3.1, fχ, ) (Im z) exists, since f HK(, )) BV(, )). Thus L {f}(z) lso exists when z C with Re z =. Now, we prove tht the Lplce trnsform is continuous on, ) when f HK(, )), nd is continuous on (, ) when f HK loc BV. Proposition 3.8. Let f HK(, b), b >. If F b is defined by F b (s) = f(t)e ts, then F b is continuous on, ). Proof. Tke s, ), nd note tht for s, ) with s s < δ, F b (s) F b (s ) = f(t)e ts e t(s s) 1dt f( )e ( )s,b inf t,b e t(s s) 1 + V,b e t(s s) 1 f( )e ( )s,b V,b e t(s s) 1. Since d dt (e t(s s) 1) = s s e t(s s) V,b e t(s s) 1 2 s s e δb b. Thus s s e δb, it follows tht nd hence lim s s F b (s) = F b (s ). F b (s) F b (s ) 2b s s e δb f( )e ( )s,b, Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

6 76 SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO Proposition 3.9. If f HK(, b), then f(t)e ts dt e s f,b, for ll u v b nd s. u Proof. Tke u v b nd s. Since e ts, s function of the vrible t, is decresing, it follows tht V u,v e ts e us e vs ; thus by Theorem 2.5, f(t)e ts dt f u,v inf u t u,v e ts + V u,v e ts f u,v e vs + (e us e vs ) f u,v e us e s f,b. Theorem 3.1. If either f HK(, )) or f HK loc BV, then L {f} is continuous on (, ). Proof. Tke s (, ) nd let ɛ > be given. Consider < δ 1 < s /2. First observe tht for ech K > nd s (s δ 1, s + δ 1 ), L {f}(s) L {f}(s ) F K (s) F K (s ) + f(t)(e ts e ts )dt. (3.1) K We clim tht there exists K > such tht K f(t)(e ts e ts )dt < ɛ 2 independently of s (s δ 1, s + δ 1 ). Assumption 1: f HK(, )). By Hke s Theorem, there exists K 1 > such tht f K1, ) < ɛ 2. (3.2) Note tht, for ll v K 1 nd s δ 1 < s s, v f(t)(e ts e ts )dt = f(t)e ts e t(s s) 1dt K 1 K 1 f( )e ( )s K1,v inf t K e t(s s) 1 1,v + V K1,ve t(s s) 1. (3.3) From Proposition 3.9, f( )e ( )s K1,v f K1,v. Also, since s s, it follows tht the function e t(s s) 1 is decresing nd not positive. Thus, the right-hnd side of the inequlity (3.3) is bounded by f K1,v 1 e K1(s s) + (e K1(s s) e v(s s) ) nd this equls f K1,v 1 e v(s s). So, tking the limit (s v ) produces f(t)(e ts e ts )dt f K 1, ), K 1 Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

7 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL 77 for ll s δ 1 < s s. Of course this inequlity is lso true when s s < s +δ 1. Assumption 2: f HK loc BV. First observe tht for ll u, v nd s (s δ 1, s + δ 1 ), (e ts e ts )dt 6. s u We set M to the right side of this inequlity. Since lim V t, ) f =, it follows tht t there exists K 2 > such tht V K2, )f < ɛ 2M. Then, from Theorem 2.5, it follows tht for every v K 2 nd s (s δ 1, s + δ 1 ), f(t)(e ts e ts )dt M inf f(t) + V K 2, vf K 2 t K 2, v M f(v) + V K2, )f. This implies, since lim t f(t) =, tht f(t)(e ts e ts )dt M V K 2, )f < M ɛ 2M = ɛ 2. K 2 Therefore, with ny of the two hypotheses, we see tht our initil ssertion is true. Finlly, from Proposition 3.8, F K is continuous, so there exists δ 2 > such tht for every s (, ) with s s < δ 2, F K (s) F K (s ) < ɛ 2. Let δ = min{δ 1, δ 2 }; then by (3.1) we hve tht for ll s (s δ, s + δ), L {f}(s) L {f}(s ) < ɛ 2 + ɛ 2 = ɛ. It cn be shown tht if f HK loc nd F (x) = x f(t)dt, x <, is bounded on, ), then lim L {f}(s) = L {f}(s ) s s + for ll s, ). We hve lredy seen the continuity of the Lplce trnsform of function, now we give nother feture of this trnsform. This property indictes tht not every rbitrry continuous function is Lplce trnsform of some function. Given b we set { } Γ(, b) = f HK(, b) lim s f(t)e ts dt = It is cler tht L(, b) Γ(, b); however Γ(, b) L(, b), see Exmple 3.12 below. Moreover, if < b, then, from Proposition 3.9, it follows tht HK(, b) = Γ(, b). When = the vercity of the equlity HK(, b) = Γ(, b) is still n open question. A clss of functions belonging to Γ(, b) is given in the next theorem: Theorem Let f be continuous function on, b such tht f is differentible except for n t most countble set A. If g(t) = f(t) + tf (t) for ll t, b \ A,. Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

8 78 SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO then lim s g(t)e ts dt =. Proof. Let G(t) = tf(t), then G (t) = g(t) for ll t, b \ A. From 2, Theorem 4.7, g HK(, b) nd by 2, Theorem 1.12, g(t)e ts HK(, b). Integrting by prts we obtin g(t)e ts dt = e ts G(t) b b + G(t)se ts dt = e bs bf(b) + f(t)tse ts dt. Let h :, ) R defined by h(x) = xe x, then h is mesurble function stisfying 1 r 1 lim h(x)dx = lim 1 1e r r r r r re r = ; so, since f L, b, by the generlized Riemmn-Lebesgue Lemm (1, Theorem 4.4.1), lim s h(ts)f(t)dt =. Thus e bs bf(b) + f(t)tse ts dt when s. Exmple Define g :, 1 R by { ( π ) cos + π ( π ) g(t) = t t sin, if t (, 1, t, if t =. Then g HK(, 1), but g L(, 1). Note tht g(t) = cos ( ) ( π t + t d dt cos π ) t. Thus by Theorem 3.11, nd hence g Γ(, 1). lim s 1 g(t)e ts dt =, Theorem 3.13 (Riemnn-Lebesgue Lemm). Let f :, ) R be function such tht f Γ(, b) for some b >. If either (1) f BV (b, )) or (2) f HK(b, )), then lim L {f}(s) =. s Proof. Let s >. If f BV (b, )), then f( )e ( )s HK(b, )) nd f(t)e ts dt 2 s f(b) + V b, )f. b Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

9 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL 79 Therefore the conclusion of the theorem follows since f Γ(, b). On the other hnd, if f HK(b, )), then by Proposition 3.9, f(t)e ts dt f b, )e bs, nd gin the conclusion of the theorem is stisfied. b The following result is shown in 8, Theorem 3.3. Lemm Let, b R. If g :, ) R nd h :, ), b C re functions such tht (i) g BV (, )), h is mesurble, bounded nd (ii) there exists M > such tht, for ech y, b, h(x, y)dx M, then for ll v, g(x)h(x, y) dxdy = g(x)h(x, y) dydx. Theorem If f BV (, )) nd s (, ), then L {f} is differentible t s, nd (L {f}) (s ) = tf(t)e ts dt. (3.4) Proof. There exist, b, M > with < s < b such tht, for ech s, b, e ts dt < M (3.5) nd te ts dt < M (3.6) for ll v. In order to show (3.4) we use Theorem 2.9. The function f(t)e t( ) is differentible on, b for ll t, ), so f(t)e t( ) is ACG δ on, b for ll t, ). By (3.5) nd Theorem 2.6, f( )e ( )s is HK-integrble on, ) for ll s, b. Then if is continuous t s, nd t s (L {f}) (s ) = Γ := tf(t)e ts dtds = tf(t)e ts dt, tf(t)e t( ) dt t s tf(t)e ts dsdt Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

10 8 SALVADOR SÁNCHEZ-PERALES AND JESÚS F. TENORIO for ll s, t, b. The first ssertion follows using (3.6) nd similr rgument s in the proof of Theorem 3.1 (Assumption 2). The second clim is true due to (3.6) nd Lemm If f nd g re functions defined on the intervl, ), then their convolution is the function f g defined by f g(y) = y f(y x)g(x)dx. It is cler tht if f HK(, )) nd g BV(, )), then f g exists on, ) by the Multiplier Theorem (see 2), nd f g(y) = g f(y) for ll y, ). In 9, the equlity L {f g}(s) = L {f}(s)l {g}(s) is estblished, under certin conditions. However the uthors use 11, Lemm 25() nd it hs n omission unless f hs compct support. We provide other conditions nd different proof. Theorem If f HK(, )) B(, )) nd g L(, )) BV(, )), then for ll z C with Re z >. L {f g}(z) = L {f}(z)l {g}(z) Proof. Tke z C with Re z >. From Corollry 3.4, it follows tht L {f}(z) nd L {g}(z) exist, nd L {f}(z)l {g}(z) = g(x) x f(y x)e yz dydx. Let D = {(x, y) x, x y} nd consider { f(y x)e yz, if (x, y) D, h(x, y) =, if (x, y) (, ), )) \ D. For ech, x, g(x) h(x, y)dy g(x) f( )e ( )z. Thus, since g L(, )), the Dominted Convergence Theorem nd Hke s Theorem imply the second equlity of the following: L {f}(z)l {g}(z) = = lim g(x)h(x, y) dydx g(x)h(x, y) dydx. (3.7) On the other hnd, since h(x, y)dx f for ll v, it follows, from Lemm 3.14, tht the right side of (3.7) is equl to lim g(x)h(x, y) dxdy. Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)

11 LAPLACE TRANSFORM USING THE HENSTOCK-KURZWEIL INTEGRAL 81 Therefore lim g f(y)e yz dy = lim y = lim = L {f}(z)l {g}(z). g(x)f(y x)e yz dxdy g(x)h(x, y) dxdy Agin using Hke s Theorem, we obtin tht g f( )e ( )z HK(, )) nd L {f g}(z) = L {f}(z)l {g}(z). References 1 G. Bchmn, L. Nrici, E. Beckenstein, Fourier nd Wvelet Anlysis, Springer-Verlg, MR R. G. Brtle, A Modern Theory of Integrtion, Grd. Studies in Mth., Vol. 32, Amer. Mth. Soc., Providence, 21. MR M.S. Chudhry nd Snket A. Tikre, On Guge Lplce Trnsform, Int. J. Mth. Anl. 5, No. 35, 211, MR R. A. Gordon, The Integrls of Lebesgue, Denjoy, Perron, nd Henstock, Grd. Studies in Mth., Vol. 4, Amer. Mth. Soc., Providence, MR F. J. Mendoz Torres, J. A. Escmill Reyn nd S. Sánchez Perles, Inclusion Reltions for the Spces L(R), HK(R) BV(R) nd L 2 (R), Russ. J. Mth. Phys. 16, No. 2, 29, MR F. J. Mendoz Torres, J. A. Escmill Reyn nd S. Sánchez Perles, Some results bout the Henstock-Kurzweil Fourier trnsform, Mth. Bohem. 134 (29), MR F. J. Mendoz Torres, On pointwise inversion of the Fourier trnsform of BV functions, Ann. Funct. Anl. 1, No. 2, 21, MR S. Sánchez-Perles, F. J. Mendoz Torres, J. A. Escmill Reyn, Henstock-Kurzweil Integrl Trnsforms, Int. J. Mth. Mth. Sci. vol. 212, 11 pges. MR Snket A. Tikre nd M.S. Chudhry, On some results of HK-convolution, Interntionl Electronic Journl of Pure nd Applied Mthemtics 2, No. 2, 21, E. Tlvil, Necessry nd sufficient conditions for differentiting under the integrl sign, Amer. Mth. Monthly, 18 (21) MR E. Tlvil, Henstock-Kurzweil Fourier trnsforms, Illinois J. Mth. 46 (22), MR Slvdor Sánchez-Perles Instituto de Físic y Mtemátics, Universidd Tecnológic de l Mixtec, Km. 2.5, Crreter Actlim, 69 Hujupn de León, Oxc, Mexico es21254@yhoo.com.mx Jesús F. Tenorio Instituto de Físic y Mtemátics, Universidd Tecnológic de l Mixtec, Km. 2.5, Crreter Actlim, 69 Hujupn de León, Oxc, Mexico jtenorio@mixteco.utm.mx Received: Jnury 23, 213 Accepted: August 19, 213 Rev. Un. Mt. Argentin, Vol. 55, No. 1 (214)