Submitted Version to CAMWA, September 30, 2009 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES

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1 Submitted Version to CAMWA, September 30, 2009 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES MARTIN BOHNER AND GUSEIN SH. GUSEINOV Missouri University of Science and Technology, Department of Mathematics and Statistics, Rolla, Missouri 65401, USA. Atilim University, Department of Mathematics, Incek, Ankara, Turkey. ABSTRACT. Starting with a general definition of the Laplace transform on arbitrary time scales, we specify the Laplace transform on isolated time scales, prove several properties of the Laplace transform in this case, and establish a formula for the inverse Laplace transform. The concept of convolution is considered in more detail by proving the convolution theorem and a discrete analogue of the classical theorem of Titchmarsh for the usual continuous convolution. Keywords. Isolated time scales, Laplace transform, convolution, shift, inverse transform. AMS MOS) Subject Classification. 39A10, 44A10, 44A35, 34N INTRODUCTION A time scale is an arbitrary nonempty closed subset of the real numbers. Time scales analysis unifies and extends continuous and discrete analysis, see [3, 5]. The Laplace transform on time scales was introduced by Hilger in [6], but in a form that tries to unify the continuous) Laplace transform and the discrete) Z-transform. For arbitrary time scales, the Laplace transform was introduced and investigated by Bohner and Peterson in [4] see also [3, Section 3.10]). It was further developed by the authors in [1, 2]. Let T be a time scale with the forward jump operator σ and the delta differentiation operator. Let µt) σt) t for t T the so-called graininess of the time scale). A function p : T C is called regressive if 1 + µt)pt) 0 for all t T. The set of all regressive and rd-continuous functions p : T C will be denoted by R. Suppose p R and fix s T. Then the initial value problem 1.1) y t) pt)yt), ys) 1 has a unique solution on T. This solution is called the exponential function and is denoted by e p t, s).

2 2 MARTIN BOHNER AND GUSEIN SH. GUSEINOV Assume that sup T and fix t 0 T. Below we assume that z is a complex constant which is regressive, i.e., 1 + µt)z 0 for all t T. Therefore e z, t 0 ) is well defined on T. Suppose x : [t 0, ) T C is a locally -integrable function, i.e., it is -integrable over each compact subinterval of [t 0, ) T. Then the Laplace transform of x is defined by 1.2) L{x}z) t 0 xt) t for z D{x}, e z σt), t 0 ) where D{x} consists of all complex numbers z R for which the improper integral exists. The following two concepts are introduced and investigated by the authors in [1]. For a function f : [t 0, ) T C, its shift or delay) ft, s) is defined as the solution of the problem 1.3) f t t, σs)) f s t, s), t, s T, t s t 0, ft, t 0 ) ft), t T, t t 0. For given functions f, g : [t 0, ) T C, their convolution f g is defined by 1.4) f g)t) t t 0 ft, σs))gs) s, t T, t t0. This paper is organized as follows. In Section 2, we specify and investigate the above concept of Laplace transform for time scales which have graininess that is bounded below by a strictly positive number. These are special cases of so-called isolated time scales. The concept of time scales is actually not needed there and in the remainder of this paper, since all statements and proofs are given directly without referring to this theory. Only this present Section 1 contains time scales concepts and hence shows the origin of this development. However, for a reader to follow the rest of this paper, it is not necessary to be familiar with time scales theory. In Section 3, we present the convolution theorem and a discrete analogue of a classical theorem of Titchmarsh, while Section 4 features a formula for the calculation of the inverse Laplace transform. Finally, in Section 5, we discuss several examples of time scales for which our theory applies, e.g., see [7, 8]) hz with h > 0, q N 0 with q > 1, and N p 0 with p THE LAPLACE TRANSFORM Throughout we let t n be real numbers for all n N 0 such that 2.1) lim n t n and ω n : t n+1 t n > 0 for all n N 0, while we assume in the main results of this paper that 2.2) lim n t n and ω : inf n N 0 ω n > 0, where ω n : t n+1 t n for n N 0

3 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 3 holds. Note that, for example, the numbers t n hn, n N 0 and t n q n, n N 0, where h > 0 and q > 1, respectively, satisfy our assumption 2.2), while the numbers do not satisfy our assumption 2.2). t n n, n N 0 and t n ln n, n N Let z be a complex number such that 2.3) z 1 ω n for all n N 0. Then the solution e z t n, t m ) of the see 1.1)) problem satisfies yt n+1 ) 1 + ω n z)yt n ), yt m ) 1, m, n N 0 2.4) e z t n, t m ) and e z t n, t m ) n ω k z) if n m km 1 m ω k z) kn if n m, where the products for m n are understood, as usual, to be 1. Thus, in conformance with 1.2), we make the following definition. Definition 2.1. Assume 2.1). If x : {t n : n N 0 } C is a function, then its Laplace transform is defined by 2.5) xz) L{x}z) ω n xt n ) n 1 + ω k z) for those values z C satisfying 2.3) for which this series converges. Let us recall our assumptions 2.2) and 2.3). Define n 2.6) P n z) : 1 + ω k z), n N 0, which is a polynomial in z of degree n + 1. It is easily verified that 2.7) P n z) P n 1 z) zω n P n 1 z), n N 0 and 2.8) hold, where P 1 z) 1. 1 P n 1 z) 1 P n z) z ω n P n z), n N 0

4 4 MARTIN BOHNER AND GUSEIN SH. GUSEINOV The numbers α n ωn 1, n N 0, belong to the real axis interval [ ω 1, 0). For any δ > 0 and n N 0, we set 2.9) D δ : C \ Dδ n, where Dδ n : {z C : z α n < δ}, n N 0 so that D δ is a closed domain of the complex plane C, and the points of D δ are in distance not less than δ from the set {α n : n N 0 }. Lemma 2.2. Assume 2.2), 2.3), 2.6), and 2.9). For any z D δ, we have 2.10) P n z) δω) n+1 and P n z) δδω) n ω n for all n N 0. Moreover, 2.11) lim n P n z) for all z D δ provided δ > ω 1. Proof. For any z D δ and n N 0, we have n P n z) 1 + ω k z) n ω k z α k )) n 1 ) n ) ω n ω k z α k ω n ω n δ n+1 δδω) n ω n. Thus the proof of the second statement in 2.10) is complete. The first statement in 2.10) follows from the second statement in 2.10), and 2.11) follows from 2.10). Example 2.3. Let us show that L{1}z) 1 z and L{e α, t 0 )}z) 1 z α. We have for z D δ, with δ > ω 1, using 2.5), 2.6), 2.8), and 2.11), ω n L{1}z) P n z) 1 [ 1 z P n 1 z) 1 ] P n z) 1 [ z lim 1 1 ] 1 m P m z) z. Now we find the Laplace transform of the function e α t) e α t, t 0 ), for which we have by 2.4) and 2.6), It follows that n 1 e α t n ) 1 + ω k α) P n 1 α) for n N 0. ẽ α z) L{e α }z) ω n e α t n ) P n z) ω n P n 1 α) P n z)

5 2.12) THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 5 n 1 ω n 1 + ω k α 1 + ω n z 1 + ω k z n 1 ω n α α k. 1 + ω n z z α k Since the numbers α k, k N 0, are contained in the finite interval [ ω 1, 0), there is a sufficiently large number R 0 > 0 such that 2.13) α α k z α k 1 for all z R 0 and k N 0. 2 Therefore the series 2.12) converges for z R 0, because ω n 1 + ω n z 1 z α n 1 δ is bounded. Next, we can write, using 2.8), ω n P n 1 α) ẽ α z) ω 0 P n z) P 0 z) + ω n P n 1 α) P n1 n z) ω 0 P 0 z) + 1 [ Pn 1 α) z P n1 n 1 z) P ] n 1α) P n z) ω 0 P 0 z) + 1 [ 1 + ωn 1 α)p n 2 α) P ] n 1α) z P n1 n 1 z) P n z) ω 0 P 0 z) + 1 [ Pn 2 α) z P n 1 z) P ] n 1α) + α ω n 1 P n 2 α) P n z) z P n 1 z) n1 ω 0 P 0 z) + 1 zp 0 z) 1 z lim m 1 z + α z ẽαz), where we have used the fact that because of P m 1 α) P m z) P m 1 α) lim m P m z) and 2.13). Thus we have obtained the equality Hence P m 1 α) P m z) 0 n1 + α z ẽαz) m 1 1 α α k 1 + ω m z z α k ẽ α z) 1 z + α z ẽαz). ẽ α z) 1 z α. Theorem 2.4. Assume 2.2). If the function x : {t n : n N 0 } C satisfies the condition 2.14) xt n ) CR n for all n N 0,

6 6 MARTIN BOHNER AND GUSEIN SH. GUSEINOV where C and R are some positive constants, then the series in 2.5) converges uniformly with respect to z in the region D δ with δ > Rω 1 and therefore its sum xz) is an analytic holomorphic) function in D δ. Proof. By Lemma 2.2 and 2.14), for the general term of the series in 2.5), we have the estimate ω n xt n ) P n z) The series ω ncr n δδω) n ω n C δ ) n R for n N 0 and z D δ. δω ) n R δω converges if δ > Rω 1. This completes the proof. A large class of functions for which the Laplace transform exists is the class F δ of functions x : {t n : n N 0 } C satisfying the condition 2.15) δω) n xt n ) <. Theorem 2.5. Assume 2.2). For any x F δ, the series in 2.5) converges uniformly with respect to z in the region D δ, and therefore its sum xz) is an analytic function in D δ. Proof. The proof follows from the second inequality in 2.10) and from 2.15). Theorem 2.6. Assume 2.2). Let x : {t n : n N 0 } C be a function and define a new function x : {t n : n N 0 } C by x t n ) xt n + ω n ) xt n ) ω n. Suppose that x F δ. Then x F δ, too, and 2.16) L{x }z) z xz) xt 0 ). Moreover, defining x x ), we have that x F δ and that 2.17) L{x }z) z 2 xz) zxt 0 ) x t 0 ). Proof. We have δω) n x t n ) δω) n xt n+1) xt n ) ω n ω 1 δω) n [ xt n+1 ) + xt n ) ] δ δω) n 1 xt n+1 ) + ω 1 δω) n xt n ) <

7 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 7 and therefore x F δ. Next, using the definition 2.5) of the Laplace transform, we find L{x }z) ω n x t n ) P n z) xt n+1 ) P n z) xt n ) P n z) xt n+1 ) P n+1 z) xt 0) P 0 z) + z 1 + ω 0z)xt 0 ) P 0 z) xt n+1 ) xt n ) P n z) xt n ) P n z) + z [ xz) ω ] 0xt 0 ) P 0 z) xt n+1 ) P n+1 z) 1 + ω n+1z) ω n+1 xt n+1 ) P n+1 z) + z xz) xt 0 ) + z xz) so that 2.16) holds. The formula 2.17) is obtained by applying 2.16) to x. xt n ) P n z) Theorem 2.7 Initial Value and Final Value Theorem). Assume 2.2). We have: a) If x F δ for some δ > 0, then 2.18) xt 0 ) lim z {z xz)}. b) If x F δ for all δ > 0, then 2.19) lim n xt n ) lim z 0 {z xz)}. Proof. Assume x F δ for some δ > 0. It follows from 2.5) that and Hence xz) ω 0xt 0 ) 1 + ω 0 z + ω 1 xt 1 ) 1 + ω 0 z)1 + ω 1 z) + ω 2 xt 2 ) 1 + ω 0 z)1 + ω 1 z)1 + ω 2 z) ω 0 z) xz) ω 0 xt 0 ) + ω 1xt 1 ) 1 + ω 1 z + ω 2 xt 2 ) 1 + ω 1 z)1 + ω 2 z) lim xz) 0 and lim {1 + ω 0z) xz)} ω 0 xt 0 ), z z which yield 2.18). To show 2.19), assume x F δ for all δ > 0. In the proof of Theorem 2.6 we have obtained the formula xt n+1 ) xt n ) z xz) xt 0 ). P n z) Hence, using Lemma 2.2 and taking into account that it is not difficult to arrive at 2.19). lim P nz) 1 for any n N 0, z 0

8 8 MARTIN BOHNER AND GUSEIN SH. GUSEINOV 3. THE CONVOLUTION In this section we only assume 2.1). For a given function f : {t n : n N 0 } C, we consider the shifting problem see 1.3)) [ ω m ftn+1, t m+1 ) ft ] n, t m+1 ) 3.1) [ + ω n ftn, t m+1 ) ft ] n, t m ) 0, ft n, t 0 ) ft n ), n N 0. m, n N 0, n m, Theorem 3.1. Assume 2.1). For an arbitrary function f : {t n : n N 0 } C, the shifting problem 3.1) has a unique solution. Proof. Setting ft n, t m ) f n,m for brevity, we rewrite the shifting problem 3.1) in the form 3.2) ω m fn+1,m+1 f ) n,m+1 + ω n fn,m+1 f ) n,m 0, m, n N 0, n m, 3.3) f n,0 ft n ), n N 0, where f n,m defined for m, n N 0 with m n is a desired solution. Note that for m n in 3.2) there arises the term f n,n+1 in which the second index is greater than the first one, but this term arises in 3.2) in two places with the same coefficient and opposite signs and therefore this term cancels. Assume that f n,m is a solution of problem 3.2), 3.3). Putting in 3.2) m n, we get f n+1,n+1 f n,n for all n N 0. Therefore f n,n is constant for n N 0, and since f 0,0 ft 0 ) by 3.3), we obtain 3.4) fn,n ft 0 ) for all n N 0. Consequently, it is enough to show that equation 3.2) has a unique solution satisfying conditions 3.3) and 3.4). We will do this by showing that equation 3.2) can be solved recurrently under the conditions 3.3) and 3.4). For any i N 0, let us set N i [i, ) N 0. Putting m n 1 with n N 1 in 3.2), we get ω n 1 fn+1,n f n,n ) + ω n fn,n f n,n 1 ) 0, n N 1. Hence, taking into account 3.4), we get 3.5) fn+1,n 1 ω n ω n 1 and besides, by 3.3), ) ft 0 ) + 3.6) f1,0 ft 1 ). ω n ω n 1 fn,n 1, n N 1, Using the initial condition 3.6), we find f n+1,n from 3.5) recursively in a unique way for all n N 0. Next, we put m n 2 with n N 2 in 3.2) to get ω n 2 fn+1,n 1 f n,n 1 ) + ω n fn,n 1 f n,n 2 ) 0, n N 2.

9 Hence 3.7) fn+1,n 1 and besides, by 3.3), THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 9 1 ω ) n f n,n 1 + ω n fn,n 2, n N 2, ω n 2 ω n 2 3.8) f2,0 ft 2 ). In equation 3.7) the term f n,n 1 is known for all n N 1 from the first step. Therefore, using the initial value 3.8), we can find f n+1,n 1 from 3.7) recursively in a unique way for all n N 1. Repeating this procedure, we put m n i for n N i in 3.2) to get Hence ω n i fn+1,n i+1 f ) n,n i+1 + ω n fn,n i+1 f ) n,n i 0, n N i. 3.9) fn+1,n i+1 and besides, by 3.3), 3.10) fi,0 ft i ). 1 ω ) n f n,n i+1 + ω n fn,n i, n N i, ω n i ω n i In equation 3.9) the term f n,n i+1 is known for all n N i 1 from the previous step. Therefore, using the initial value 3.10), we can find f n+1,n i+1 from 3.9) recursively in a unique way for all n N i 1. Since i N can be taken arbitrarily, we see that f n,m is constructed in this way uniquely for all m, n N 0 with m n. We now introduce the following definition see 1.4)) of the convolution of two functions. Definition 3.2. Assume 2.1), let f, g : {t n : n N 0 } C be two functions, and let f be the solution of the shifting problem 3.1). Then the convolution f g of f and g is defined by f g)t 0 ) 0 and n 1 f g)t n ) ω k ftn, t k+1 )gt k ), n N 0. The following theorem is a discrete analogue of the classical theorem of Titchmarsh [9, 10] for the usual continuous convolution. Theorem 3.3. Assume 2.1) and let f, g : {t n : n N 0 } C be two functions. If f g is identically equal to zero on {t n : n N 0 }, then at least one of the functions f and g is identically equal to zero on {t n : n N 0 }.

10 10 MARTIN BOHNER AND GUSEIN SH. GUSEINOV Proof. Assume that f g is identically zero on {t n : n N 0 }. Then we have for any n N 0, by Definition 3.2 of the convolution and the notation ft n, t m ) f n,m, ω 0 f1,1 gt 0 ) 0, ω 0 f2,1 gt 0 ) + ω 1 f2,2 gt 1 ) 0, 3.11) ω 0 f3,1 gt 0 ) + ω 1 f3,2 gt 1 ) + ω 2 f3,3 gt 2 ) 0,. ω 0 fn,1 gt 0 ) + ω 1 fn,2 gt 1 ) ω n 1 fn,n gt n 1 ) 0, where we can take n N as large as we wish. It is sufficient to show that if f is not identically zero on {t n : n N 0 }, then g is identically zero on {t n : n N 0 }. Thus assume that f is not identically zero on {t n : n N 0 }. Let ft m ) with an m N 0 be the first of the values of ft 0 ), ft 1 ),... that is different from zero. Hence 3.12) ft 0 )... ft m 1 ) 0 and ft m ) 0. We have to show that then gt n ) 0 for all n N 0. Let us consider the possible values of m N 0 in 3.12) separately. If m 0 in 3.12), then we have ft 0 ) 0. Consider 3.11) as a homogeneous system of linear algebraic equations A 0 x 0 0 with x 0 gt 0 ), gt 1 ),..., gt n 1 )) T. The determinant of the matrix A 0 being a triangular matrix) is equal to n 1 n 1 ω k fk+1,k+1 [ft 0 )] n ω k, where we have used 3.4). Hence the determinant of A 0 is different from zero by the assumption ft 0 ) 0. Therefore A 0 is invertible and the equation A 0 x 0 0 implies x 0 0, i.e., gt 0 ) gt 1 )... gt n 1 ) 0. Since n N is arbitrary, we get that g is identically zero on {t n : n N 0 }. If m 1 in 3.12), then we have 3.13) ft 0 ) 0 and ft 1 ) 0. In this case, 3.4) implies that 3.14) fn,n 0 for all n N 0, and the system 3.11) becomes ω 0 f2,1 gt 0 ) 0, ω 0 f3,1 gt 0 ) + ω 1 f3,2 gt 1 ) 0, 3.15) ω 0 f4,1 gt 0 ) + ω 1 f4,2 gt 1 ) + ω 2 f4,3 gt 2 ) 0,. ω 0 fn,1 gt 0 ) + ω 1 fn,2 gt 1 ) ω n 2 fn,n 1 gt n 2 ) 0.

11 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 11 Next, since ft 0 ) 0, we have from 3.5) that f n+1,n ω n fn,n 1, n N 1. ω n 1 Iterating this equation and taking into account 3.3), we find 3.16) fn+1,n ω n ω 0 f1,0 ω n ω 0 ft 1 ), n N 0. Considering the system 3.15) as before as a system A 1 x 1 0, the determinant of A 1 is found to be equal to n 2 n 2 ω k fk+2,k+1 [ft 1 )] n 1 ω k+1 ω 0 and hence is different from zero by 3.13). Therefore x 1 0, i.e., gt 0 ) gt 1 )... gt n 2 ) 0, and since n N is arbitrary, we get that g is identically zero on {t n : n N 0 }. If m 2 in 3.12), then we have 3.17) ft 0 ) ft 1 ) 0 and ft 2 ) 0. In this case, 3.14) and 3.16) still hold. Besides, by ft 1 ) 0, equation 3.16) yields 3.18) fn+1,n 0 for all n N 0. Therefore the system 3.11) becomes ω 0 f3,1 gt 0 ) 0, 3.19) ω 0 f4,1 gt 0 ) + ω 1 f4,2 gt 1 ) 0, ω 0 f5,1 gt 0 ) + ω 1 f5,2 gt 1 ) + ω 2 f5,3 gt 2 ) 0,. ω 0 fn,1 gt 0 ) + ω 1 fn,2 gt 1 ) ω n 3 fn,n 2 gt n 3 ) 0. Next, since f n,n 1 0 for n N 1 by 3.18), we have from 3.7) that f n+1,n 1 ω n fn,n 2, n N 2. ω n 2 Iterating the last equation, we find f n+1,n 1 ω nω n 1 ω 1 ω 0 f2,0 ω nω n 1 ω 1 ω 0 ft 2 ), n N 1. Writing the system 3.19) again as A 2 x 2 0, the determinant of A 2 is equal to n 3 n 3 ω k fk+3,k+1 [ft 2 )] n 2 ω k+2 ω k+1 ω 1 ω 0 and hence is different from zero by 3.17). Then x 2 0, i.e., gt 0 ) gt 1 )... gt n 3 ) 0, and since n N is arbitrary, we get that g is identically zero on {t n : n N 0 }.

12 12 MARTIN BOHNER AND GUSEIN SH. GUSEINOV We see that one can discuss the system A m x m 0 and argue in this way for any value of m N 0 in 3.12) in order to obtain that gt n ) 0 for all n N 0. Remark 3.4. Note that in the special case t n hn, n N 0, where h > 0 is a fixed real number, for any two functions f, g : {t n : n N 0 } C their convolution f g has the form see [2]) n 1 f g)nh) h fnh kh h)gkh) for n N 0, and therefore equations 3.11) in the proof of Theorem 3.3 take in this case the form f0)g0) 0, fh)g0) + f0)gh) 0, f2h)g0) + fh)gh) + f0)g2h) 0,. fn 1)h)g0) + fn 2)h)gh) f0)gn 1)h) 0, which is much easier for illustrating the reasoning made in the proof of Theorem 3.3. Theorem 3.5 Convolution Theorem). Assume 2.1) and let f, g : {t n : n N 0 } C be two functions such that L{f}z), L{g}z), and L{f g}z) exist for a given z C satisfying 2.3). Then, at the point z, 3.20) L{f g}z) L{f}z) L{g}z). Proof. For brevity let us set Then 2.4) gives 3.21) 3.22) 3.23) e n,m z) : e z t n, t m ) and fn,m : ft n, t m ). e n,n z) 1 for all n N 0 e n+1,m z) 1 + ω n z)e n,m z) for all n, m N 0 with n m e n,m+1 z) e n,mz) 1 + ω m z for all n, m N 0 with n m + 1, and the shifting problem 3.1) can be rewritten as 3.2), 3.3). Using definition 2.5) of the Laplace transform and Definition 3.2 for the convolution, we have ω n f g)t n ) ω n n 1 L{f g}z) ω k fn,k+1 gt k ) e n+1,0 z) e n+1,0 z) Substituting here n1 ω k gt k ) nk+1 n1 ω n fn,k+1 e n+1,0 z). e n+1,0 z) e n+1,k+1 z)e k+1,0 z),

13 we get 3.24) Let us set THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 13 L{f g}z) 3.25) Ψ m nk+1 L{g}z) nm ω k gt k ) e k+1,0 z) nk+1 nk+1 ω n fn,k+1 e n+1,k+1 z) ω n fn,k+1 e n+1,k+1 z). ω n fn,m e n+1,m z), m N 0. We will show that Ψ m is independent of m N 0, which then implies ω n fn,k+1 e n+1,k+1 z) ω n fn,0 e n+1,0 z) ω n ft n ) e n+1,0 z) L{f}z), and thus yields 3.20) by using 3.24). So, it remains to show that the quantity Ψ m defined by 3.25) does not depend on m N 0. We have, putting e n,m : e n,m z) and using 3.2) and 3.21), 3.22), and 3.23), ω n fn,m+1 ω n fn,m + ω m fn,m+1 ω m fn+1,m+1 Ψ m+1 e n+1,m+1 nm+1 nm+1 nm+1 ω n fn,m e n+1,m+1 ω m nm+1 nm+1 e n+1,m+1 [ fn+1,m+1 e n+1,m+1 f n,m+1 e n,m+1 + f n,m+1 e n,m+1 ω n fn,m e n+1,m 1 + ω m z) + ω m fm+1,m+1 e m+1,m+1 ω m nm+1 f n,m+1 e n+1,m+1 ω n z 1 + ω m z)ψ m ω m fm,m e m+1,m 1 + ω m z) + ω m fm+1,m+1 ω m zψ m ω m z)ψ m ω m fm,m + ω m fm+1,m+1 ω m zψ m ω m z)ψ m ω m zψ m+1, where we have used the fact that f n,n ft 0 ) for all n N 0. Consequently 1 + ω m z)ψ m ω m z)ψ m, and hence Ψ m+1 Ψ m as 1 + ω m z 0 under condition 2.3). 4. THE INVERSE LAPLACE TRANSFORM f n,m+1 e n+1,m+1 In this section we establish an inversion formula for the Laplace transform. Theorem 4.1 Uniqueness Theorem). Assume 2.2) and let x : {t n : n N 0 } C be a function in the space F δ, i.e., x satisfies 2.15). Further, let xz) be the Laplace transform of x defined by 2.5) for z D δ. If xz) 0 for z D δ, then xt n ) 0 for all n N 0. ]

14 14 MARTIN BOHNER AND GUSEIN SH. GUSEINOV Proof. By the assumption, we have 4.1) ω 0 xt 0 ) 1 + ω 0 z + ω 1 xt 1 ) 1 + ω 0 z)1 + ω 1 z) + ω 2 xt 2 ) 1 + ω 0 z)1 + ω 1 z)1 + ω 2 z) for z D δ. Multiplying 4.1) by 1 + ω 0 z and then passing to the limit as z we can take a term-by-term limit due to the uniform convergence proved in Theorem 2.5), we get xt 0 ) 0. Now we multiply the remaining equation use xt 0 ) 0 in 4.1)) ω 1 xt 1 ) 1 + ω 0 z)1 + ω 1 z) + ω 2 xt 2 ) 1 + ω 0 z)1 + ω 1 z)1 + ω 2 z) by 1 + ω 0 z)1 + ω 1 z) and pass then to the limit as z to obtain xt 1 ) 0. Repeating this procedure, we find that xt 0 ) xt 1 ) xt 2 ) Theorem 4.1 implies that the inverse Laplace transform exists. The following theorem gives an integral formula for the inverse Laplace transform. Theorem 4.2 Inverse Laplace Transform). Assume 2.2), let x F δ, and let xz) be its Laplace transform defined by 2.5). Then 4.2) xt n ) 1 2πi n 1 xz) 1 + ω k z) for n N 0, where is any positively oriented closed curve in the region D δ that encloses all the points α k ω 1 k for k N 0. Proof. Integrating the equality 4.3) xz) ω 0xt 0 ) 1 + ω 0 z + ω 1 xt 1 ) 1 + ω 0 z)1 + ω 1 z) + ω 2 xt 2 ) 1 + ω 0 z)1 + ω 1 z)1 + ω 2 z) +... over the curve with respect to z and noting that we can integrate term-by-term by the uniform convergence of the series proved in Theorem 2.5, we get xz) ω 0 xt 0 ) 1 + ω 0 z + ω 1xt 1 ) 1 + ω 0 z)1 + ω 1 z) Next, and n ω 0 z 1 ω 0 2πi z α 0 ω 0 0 for all n N \ {1} 1 + ω k z) because if P z) is any polynomial of degree greater than or equal to two and if is any closed contour that encloses all the roots of the polynomial P z), then P z) 0.

15 Therefore we find THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 15 xt 0 ) 1 xz). 2πi Now multiplying 4.3) by 1 + ω 0 z and then integrating over with respect to z, we obtain 1 + ω 0 z) xz) ω 0 xt 0 ) Next, and Therefore we find 0, k1 + ω 1 xt 1 ) 1 + ω 1 z 1 ω ω 1 z + ω 2 xt 2 ) 1 + ω 1 z)1 + ω 2 z) πi z α 1 ω 1 0 for all n N \ {1}. n 1 + ω k z) xt 1 ) 1 xz)1 + ω 0 z). 2πi Repeating this procedure, we can obtain formula 4.2) for an arbitrary n N EXAMPLES Example 5.1. Let t n hn, n N 0, where h > 0 is a fixed real number. In this case ω n t n+1 t n n + 1)h nh h for all n N 0. Note that 2.2) holds with ω h. For a function x : {hn : n N 0 } R, its Laplace transform 2.5) becomes xz) L{x}z) h xnh) 1 + hz) n+1. The inversion formula 4.2) takes the form xnh) 1 xz)1 + hz) n, n N 0, 2πi where is a positively oriented curve that encloses the point 1/h. Example 5.2. Let t n q n, n N 0, where q > 1 is a fixed real number. Then we have ω n t n+1 t n q n+1 q n q 1)q n q q n for all n N 0, where q q 1.

16 16 MARTIN BOHNER AND GUSEIN SH. GUSEINOV Note that 2.2) holds with ω q. For a function x : {q n : n N 0 } R, its Laplace transform 2.5) becomes xz) L{x}z) q q n xq n ). n 1 + q q k z) The inversion formula 4.2) takes the form xq n ) 1 n 1 xz) 1 + q q k z), n N 0, 2πi where is a positively oriented curve that encloses all the points q q k ) 1 with k N 0. Example 5.3. Let t n n p, n N 0, where p is a positive real number. Then we have ω n t n+1 t n n + 1) p n p for all n N 0. Next, applying the mean value theorem to the function fx) x p on the interval [n, n + 1], we obtain n + 1) p n p fn + 1) fn) f c) pc p 1, where n < c < n + 1. Therefore, taking into account that the function x p 1 is nondecreasing on [0, ) if p 1 and decreasing on 0, ) if p < 1, we get and pn p 1 ω n pn + 1) p 1 if p 1 pn + 1) p 1 < ω n < pn p 1 if p < 1. Hence, we see that condition 2.1) holds for all p > 0, while condition 2.2) holds if and only if p 1. REFERENCES [1] M. Bohner and G. Sh. Guseinov. The convolution on time scales. Abstr. Appl. Anal., 2007:24, Art. ID 54989, [2] M. Bohner and G. Sh. Guseinov. The h-laplace and q-laplace transforms. J. Math. Anal. Appl., 3651):75 92, [3] M. Bohner and A. Peterson. Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, [4] M. Bohner and A. Peterson. Laplace transform and Z-transform: Unification and extension. Methods Appl. Anal., 91): , Preprint in Ulmer Seminare 6. [5] M. Bohner and A. Peterson. Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, [6] S. Hilger. Special functions, Laplace and Fourier transform on measure chains. Dynam. Systems Appl., 83-4): , Special Issue on Discrete and Continuous Hamiltonian Systems, edited by R. P. Agarwal and M. Bohner.

17 THE LAPLACE TRANSFORM ON ISOLATED TIME SCALES 17 [7] V. Kac and P. Cheung. Quantum Calculus. Universitext. Springer-Verlag, New York, [8] W. G. Kelley and A. C. Peterson. Difference Equations: An Introduction with Applications. Academic Press, San Diego, second edition, [9] E. C. Titchmarsh. The zeros of certain integral functions. Proc. London Math. Soc., 25: , [10] E. C. Titchmarsh. Introduction to the Theory of Fourier Integrals. Chelsea Publishing Co., New York, third edition, 1986.