INTEGRATION FORMULAS INVOLVING FOURIER-FEYNMAN TRANSFORMS VIA A FUBINI THEOREM. Timothy Huffman, David Skoug, and David Storvick

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1 J. Korean Math. Soc. 38 (1), No., pp INTEGRATION FORMULAS INVOLVING FOURIER-FEYNMAN TRANSFORMS VIA A FUBINI THEOREM Timothy Huffman, David Skoug, and David Storvick Abstract. In this paper we use a general Fubini theorem established in [13] to obtain several Feynman integration formulas involving analytic Fourier-Feynman transforms. Included in these formulas is a general Parseval s relation. 1. Introduction and preliminaries Let C [, T ] denote one-parameter Wiener space, that is the space of R-valued continuous functions x(t) on [, T ] with x(). Let M denote the class of all Wiener measurable subsets of C [, T ] and let m denote Wiener measure. (C [, T ], M, m) is a complete measure space and we denote the Wiener integral of a Wiener integrable functional F by F (x)m(dx). A subset E of C [, T ] is said to be scale-invariant measurable (s.i.m.) [8,15] provided ρe M for all ρ >, and a s.i.m. set N is said to be scale-invariant null provided m(ρn) for each ρ >. A property that holds except on a scale-invariant null set is said to hold scaleinvariant almost everywhere (s-a.e.). If two functionals F and G are equal s-a.e., we write F G. For a rather detailed discussion of s.i.m. and its relation with other topics see [15]. It was also pointed out in [15, p. 17] that the concept of s.i.m., rather than Borel measurability Received July 14, Mathematics Subject Classification: 8C, 6J65. Key words and phrases: analytic Feynman integral, Wiener integral, analytic Fourier-Feynman transform, convolution product, Fubini theorem, scale-invariant measurability, Parseval s relation.

2 4 Timothy Huffman, David Skoug, and David Storvick or Wiener measurability, is precisely correct for the analytic Fourier- Feynman transform theory and the analytic Feynman integration theory. Segal [19] gives an interesting discussion of the relationship between scale change in C [, T ] and certain questions in quantum field theory. Throughout this paper we will assume that each functional F (or G) we consider satisfies the conditions: (1.1) F : C [, T ] C is defined s a.e. and is s.i.m.. (1.) F (ρx) m(dx) < for each ρ >. Let C + and C + denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part respectively. Let F satisfy conditions (1.1) and (1.) above, and for λ >, let J(λ) F (λ 1/ x)m(dx). If there exists a function J (λ) analytic in C + such that J (λ) J(λ) for all λ >, then J (λ) is defined to be the analytic Wiener integral of F over C [, T ] with parameter λ, and for λ in C + we write (1.3) anwλ F (x)m(dx) J (λ). Let q be a real parameter and let F be a functional whose analytic Wiener integral exists for all λ C +. If the following limit exists, we call it the analytic Feynman integral of F with parameter q and we write (1.4) anfq F (x)m(dx) anwλ lim F (x)m(dx) λ iq where λ iq through values in C +. Finally, for notational purposes, we let (1.5) anλ F (x)m(dx) { anwλ F (x)m(dx), λ C + anfq F (x)m(dx), λ iq C + C +. The following Fubini theorem established in [13], plays a major role in this paper.

3 Integration formulas involving Fourier-Feynman transforms 43 Theorem 1. Assume that F satisfies conditions (1.1) and (1.) above and is such that its analytic Feynman integral anf q C F (x)m(dx) exists [,T ] for all q R {}. Then for all a, b R and all (λ, β) C + C + with λ + β, ( anβ ) anλ F (ay + bz)m(dy) m(dz) an(λβ)/(λb +βa ) (1.6) anλ F (x)dx ( anβ F (ay + bz)m(dz) ) m(dy). In section below we use Theorem 1 to help us establish several Feynman integration formulas involving Fourier-Feynman transforms. Finally, in section 3 we establish additional integration formulas including a general Parseval s relation.. Fourier-Feynman transforms The concept of an L 1 analytic Fourier-Feynman transform (FFT) was introduced by Brue in [1]. In [5], Cameron and Storvick introduced an L analytic FFT. In [14], Johnson and Skoug developed an L p analytic FFT for 1 p which extended the results in [1,5] and gave various relationships between the L 1 and the L theories. In [1], Huffman, Park and Skoug defined a convolution product for functionals on Wiener space and in [11,1] obtained various results involving the FFT and the convolution product. Also see [7,16 and 18] for further work on these topics. In this paper, for simplicity, we restrict our discussion to the case p 1; however most of our results hold for all p [1, ]. Also, throughout this section, we will assume that the functionals F : C [, T ] C satisfy the hypotheses of our Fubini theorem, namely Theorem 1 above. For λ C + and y C [, T ], let (.1) (T λ (F ))(y) anwλ F (y + x)m(dx). Then for q R {} (see [1,p.663]), the L 1 analytic FFT, T q (1) (F ) of

4 44 Timothy Huffman, David Skoug, and David Storvick F, is defined by the formula (λ C + ) (.) q (F ))(y) lim λ iq (T λ(f ))(y) for s-a.e. y C [, T ] whenever this limit exists. That is to say, anfq (.3) (F ))(y) F (y + x)m(dx) for s-a.e. y C [, T ]. We note that if T q (1) (F ) exists and if F G, then T q (1) (G) exists and T q (1) (F ) T q (1) (G). In equations (.4), (.7), (.8), (.9), (.1), (.11), (.1) and (.13) below, we establish various analytic Feynman integration formulas involving Fourier-Feynman transforms. Theorem. Let F be as in Theorem 1 above and let r > be given. Then for all q 1 and q in R {} with q 1 + q, anfrq 1 (F ))( rz)m(dz) (.4) anf(q1 q )/(q 1 +q ) anfrq1 F (x)m(dx) (F ))( ry)m(dy). Proof. Using equation (.3) and the first equality in equation (1.6) with a 1, b r, λ iq 1 and β irq, we obtain that (.5) anfrq 1 (F ))( rz)m(dz) anfrq anfq1 ( anf(rq q 1 )/(rq 1 +rq ) F ( rz + y)m(dy))m(dz) F (x)m(dx) anf(q1 q )/(q 1 +q ) F (x)m(dx).

5 Integration formulas involving Fourier-Feynman transforms 45 Also using equation (.3) and the second equality in equation (1.6) with a r, b 1, β iq and λ irq 1, we obtain that (.6) anfrq1 (F ))( ry)m(dy) anfrq1 anfq ( anf(rq1 q )/(rq +rq 1 ) F ( ry + z)m(dz))m(dy) F (x)m(dx) anf(q1 q )/(q 1 +q ) F (x)m(dx). Now equation (.4) follows from equations (.5) and (.6). Our first corollary below says that the Feynman integral with parameter q of the FFT with parameter q 1 equals the Feynman integral with parameter q 1 of the FFT with parameter q provided q 1 + q. Corollary 1 to Theorem. Let F be as in Theorem. Then for all q 1 and q in R {} with q 1 + q, (.7) anfq anfq1 1 (F ))(z)m(dz) (F ))(y)m(dy). Corollary to Theorem. Let F be as in Theorem. Then for all q in R {}, (.8) anfq anfq/ (F ))(y)m(dy) anfq F (x)m(dx) F ( x)m(dx). Proof. The first equality in equation (.8) follows by letting r 1 and q 1 q q in equation (.4). The second equality follows from the formula anfkq F (x)m(dx) anfq F (x/ k)m(dx)

6 46 Timothy Huffman, David Skoug, and David Storvick established in [13] for k >. Corollary 3 to Theorem. Let F be as in Theorem. Then for all q in R {}, (.9) anf q q/ (F ))(z)m(dz) anf q anfq/ F (x)m(dx) q (F ))(y)m(dy). Proof. Simply choose q 1 q/, q q and r 1 in equation (.4). Theorem 3. Let F be as in Theorem and let q 1, q,..., q n be elements of R {} with k q 1 q... q k for k,..., n. q j j1 Then, for s-a.e. z C [, T ], n q n 1 (... q q 1 (F )))... )))(z) (.1) where α n anfα n α n (F ))(z) q 1q... q n n j1 F (z + x)m(dx) q 1 q...q n q j 1 1 q q q n Proof. Using equation (.3), and then equation (1.6) repeatedly, we obtain that n n 1 (... (F )))... )))(z) anfq n ( anfqn 1 (... ( q 1 anfq anfq1 ( m(dy ))... )m(dy n 1 ))m(dy n ) anfα n F (z + x)m(dx) α n (F ))(z) F (z + y 1 + y + + y n )m(dy 1 ))

7 Integration formulas involving Fourier-Feynman transforms 47 for s-a.e. z C [, T ]. Choosing q j q for j 1,,..., n, we obtain the following corollary to Theorem 3. Corollary 1 to Theorem 3. Let F be as in Theorem 3 and let q be an element of R {}. Then for s-a.e. z C [, T ], (.11) q q (F )))(z) q/ (F ))(z) anf q (.1) q q q (F ))))(z) q/3 (F ))(z) anf q and in general, F (z + x)m(dx), F (z + 3x)m(dx), (.13) (... (F ))... )))(z) q/n anfq (F ))(z) F (z + nx)m(dx). Corollary to Theorem 3. Let F be as in Theorem 3 and let q 1 and q be elements of R {} with q 1 + q. Then for s-a.e. z C [, T ], (.14) q q 1 (F )))(z) q 1 q q 1 +q (F ))(z) q 1 q (F )))(z). Remark 1. We note that the hypotheses (and hence the conclusions) of Theorems 1-3 and their corollaries above are indeed satisfied by many large classes of functionals. These classes of functionals include: (a) The Banach algebra S defined by Cameron and Storvick in [6]; also see [9,1,18]. (b) Various spaces of functionals of the form T F (x) exp{ f(t, x(t))dt}

8 48 Timothy Huffman, David Skoug, and David Storvick for appropriate f : [, T ] R C; see for example [5,11 and 14]. (c) Various spaces of functionals of the form T T F (x) f( α 1 (t)dx(t),..., α n (t)dx(t)) for appropriate f as discussed in [1,16]. (d) Various spaces of functionals of the form T T F (x) exp{ f(s, t, x(s), x(t))dsdt} for appropriate f : [, T ] R C; see for example [1]. Remark. In a unifying paper [17], Lee obtains some similar results for several different integral transforms including the Fourier-Feynman transform. However, the results in this paper hold for much more general functionals F. For example, in our notation, Lee requires the functional F (x + λy) to be an entire function of λ over C for each x and y in C [, T ] whereas we don t even require F to be a continuous function. The classes of functionals studied by Yeh in [] and Yoo in [1] for the Fourier-Wiener transform are similar to those used by Lee in [17]. 3. Further applications First we state the definition of the convolution product of two functionals F and G on C [, T ] as given by Huffman, Park and Skoug in [1, p. 663]. This definition is different than the definition given by Yeh in [] and used by Yoo in [1]. In [] and [1], Yeh and Yoo study the relationship between their convolution product and Fourier-Wiener transforms. For λ C +, the convolution product (if it exists) of F and G is defined by the formula (see equation (1.5) above) (3.1) (F G) λ (y) anλ F ( y + x )G( y x )m(dx) for s-a.e. y C [, T ]. When λ iq, we usually denote (F G) λ by (F G) q.

9 Integration formulas involving Fourier-Feynman transforms 49 In our first theorem of this section, we show that the Fourier-Feynman transform of the convolution product is a product of their transforms. Theorem 4. Let F be as in Theorem 1 and assume that G : C [, T ] C satisfies the same conditions as F ; i.e., G is s.i.m., G(ρx) m(dx) < for all ρ > and the analytic Feynman integral of G exists for all q R {}. Furthermore assume that T q (1) ((F G) q ) exists for all q R {}. Then for all q R {}, (3.) q ((F G) q ))(z) q (F ))(z/ ) q (G))(z/ ) for s-a.e. z in C [, T ]. Proof. Because of the assumptions on F and G, all three of the transforms in equation (3.) exist; thus we only need to establish the equality. For λ >, using (.1) and (3.1), we see that (T λ ((F G) λ ))(z) (F G) λ (z + λ 1/ y)m(dy) F ( z +λ 1/ ( y + x ))G( z +λ 1/ ( y x ))m(dx)m(dy) for s-a.e. z C [, T ]. But w 1 (y + x)/ and w (y x)/ are independent standard Wiener processes, and hence (T λ ((F G) λ ))(z) ( F (z/ + w 1 / λ)g(z/ + w / λ) m(dw ))m(dw 1 ) F (z/ + w 1 / λ)m(dw 1 ) G(z/ + w / λ)m(dw ) (T λ (F ))(z/ )(T λ (G))(z/ ) for s-a.e. z C [, T ]. Now by analytic extensions through C + we obtain that (3.3) (T λ ((F G) λ ))(z) (T λ (F ))(z/ λ)(t λ (G))(z/ )

10 43 Timothy Huffman, David Skoug, and David Storvick holds throughout C +. Finally, equation (3.) follows from equation (3.3) by letting λ iq, since all three of the transforms in (3.) exist. Corollary 1 to Theorem 4. Let F be as in Theorem 4. Then for all q R {}, q ((F 1) q ))(z) q (F ))(z/ ). Furthermore, if T q (1) ((F F ) q ) exists, then q ((F F ) q ))(z) [ q (F ))(z/ )]. In our next theorem, we establish a general Parseval s relation; for related results in the Fourier-Wiener theory see Cameron [], Cameron and Martin [3,4], and Il Yoo [1]. Theorem 5. Let F and G be as in Theorem 4. Furthermore, assume that F and G are continuous on C [, T ]. Then for all q R {}, (3.4) anf q anfq q (F ))(z/ )(T q (1) (G))(z/ )m(dz) F (x/ )G( x/ )m(dx). Proof. Because of our assumptions on F and G, the analytic Feynman integrals on both sides of equation (3.4) certainly exist. Also recall that in the definition of the analytic Feynman integral (1.4), we assumed that λ could approach iq in an arbitrary fashion through values in C +. Thus, using (3.), Theorem 1, (3.1), and the continuity of F and

11 G, we obtain that Integration formulas involving Fourier-Feynman transforms 431 anf q anf q as desired. q (F ))(z/ )(T q (1) (G))(z/ )m(dz) lim p + lim p + lim p + lim p + (T q (1) ((F G) q ))(z)m(dz) anwp+qi anwp+qi (T p qi ((F G) p qi (z)m(dz) ( ) anwp qi (F G) p qi (z + y)m(dy) m(dz) anw(p +q )/p (F G) p qi (w)m(dw) p (F G) p qi ( p + q w)m(dw) ( anwp qi p lim F ( p + p + q w + x ) ) p G( p + q w x )m(dx) m(dw) ( ) anfq anfq F (x/ )G( x/ )m(dx) F (x/ )G( x/ )m(dx) m(dw) Our first corollary gives an alternative form of Parseval s relation. Corollary 1 to Theorem 5. Let F and G be as in Theorem 5. Then for all q in R {}, (3.5) anf q (1) q/ (F ))(z)(t q/ (G))(z)m(dz) anfq F (x)g( x)m(dx).

12 43 Timothy Huffman, David Skoug, and David Storvick Corollary to Theorem 5. Let F be as in Theorem 5 and assume that T q (1) ((F F )q) exists for all q R {}. Then (3.6) anf q [ (F ))(z/ anfq )] m(dz) F (x/ )F ( x/ )m(dx). Theorem 6. Let F be as in Theorem 1. Furthermore, assume that F is continuous on C [, T ]. Then for all q in R {}, (3.7) for s-a.e. y C [, T ]. q (T q (1) (F )))(y) F (y) Proof. Proceeding as in the proof of Theorem 5 and using equations (.), (.1), (1.6) and the continuity of F, we obtain that for s-a.e. y C [, T ], q (T q (1) (F )))(y) lim p +(T p+qi(t p qi (F )))(y) ( ) anwp qi F (y + z + x)m(dx) m(dz) anwp+qi lim p + lim p + lim p + anw(p +q )/(p) F (y). F (y + w)m(dw) p F (y + p + q w)m(dw) F (y)m(dw) Corollary 1 to Theorem 6. Let E be the class of all continuous functionals F : C [, T ] C satisfying the hypotheses of Theorem 1. Let T (1) denote the identity map; i.e., T (1) (F ) F. Then { T (1) q } q R

13 Integration formulas involving Fourier-Feynman transforms 433 forms an abelian group acting on E with q ) 1 T (1) q. Remark 3. Looking at equation (3.) above, together with its proof, it is quite tempting to conjecture that for s-a.e. z in C [, T ], (3.8) 1 ((F G) q ))(z) q 1 q (F ))(z/ ) q 1 q (G))(z/ ) q 1 +q q 1 +q provided that q 1 + q. But in general, equation (3.8) holds if and only if q 1 q q, in which case equation (3.8) reduces to equation (3.). The proof given above to establish equation (3.) fails to work for equation (3.8) since for λ > and β >, (T λ ((F G) β ))(z) ( F ( z + m(dx))m(dy), y λ + x β )G( z + y λ x β ) y λ + while w 1 if and only if λ β. x β and w y λ x β are independent processes In particular (see section 3 of [1] for the appropriate definitions), for F and G in the Banach algebra S with corresponding finite Borel measures f and g in M(L [, T ]) and using equations (3.) and (3.5) of [1], it is easy to see that (3.9) q 1 q (F ))(z/ ) q 1 +q L [,T ] exp{ i T v(t)dz(t) i(q 1 + q ) 4q 1 q T v (t)dt}df(v), (3.1) q 1 q (G))(z/ ) q 1 +q L [,T ] exp{ i T w(t)dz(t) i(q 1 + q ) 4q 1 q T w (t)dt}dg(w),

14 434 Timothy Huffman, David Skoug, and David Storvick and that (3.11) q 1 ((F G) q ))(z) exp{ i T [v(t) + w(t)]dz(t)} L [,T ] exp{ i(q 1 + q ) 4q 1 q exp{ i(q 1 q ) q 1 q T T [v (t) + w (t)]dt} v(t)w(t)dt}df(v)dg(w). Now a careful examination of equations (3.9), (3.1) and (3.11) shows that equation (3.8) holds if and only if q 1 q. References [1] M. D. Brue, A functional transform for Feynman integrals similar to the Fourier transform, Thesis, University of Minnesota (197). [] R. H. Cameron, Some examples of Fourier-Wiener Transforms of analytic functionals, Duke Math. J. 1 (1945), [3] R. H. Cameron and W. T. Martin, Fourier-Wiener transforms of analytic functionals, Duke Math. J. 1 (1945), [4], Fourier-Wiener transforms of functionals belonging to L over the space C, Duke Math. J. 14 (1947), [5] R. H. Cameron and D. A. Storvick, An L analytic Fourier-Feynman transform, Michigan Math. J. 3 (1976), 1 3. [6], Some Banach algebras of analytic Feynman integrable functionals, in Analytic Functions (Kozubnik, 1979), Lecture Notes in Math., Springer-Verlag 798 (198), [7] S. J. Chang and D. L. Skoug, The effect of drift on the Fourier-Feynman transform, the convolution product and the first variation, Psnomei. Math. J. 1 (), [8] K. S. Chang, Scale-invariant measurability in Yeh-Wiener space, J. Korean Math. Soc. 19 (198), [9] K. S. Chang, G. W. Johnson, and D. L. Skoug, Functions in the Banach algebra S(v), J. Korean Math. Soc. 4 (1987), [1] T. Huffman, C. Park, and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), [11], Convolutions and Fourier-Feynman transforms, Rocky Mountain J. Math. 7 (1997), [1], Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), [13] T. Huffman, D. Skoug, and D. Storvick, A Fubini theorem for analytic Feynman integrals with applications, J. Korean Math. Soc. 38 (1), 49 4.

15 Integration formulas involving Fourier-Feynman transforms 435 [14] G. W. Johnson and D. L. Skoug, An L p analytic Fourier-Feynman transform, Michigan Math. J. 6 (1979), [15], Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), [16] J. Kim, J. Ko, C. Park, and D. Skoug, Relationships among transforms, convolutions and first variations, Int. J. Math. Math. Sci. (1999), [17] Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (198), [18] C. Park, D. Skoug, and D. Storvick, Relationships among the first variation, the convolution product, and the Fourier-Feynman transform, Rocky Mountain J. Math. 8 (1998), [19] I. Segal, Transformations in Wiener space and squares of quantum fields, Adv. Math. 4 (197), [] J. Yeh, Convolution in Fourier-Wiener transform, Pacific J. Math. 15 (1965), [1] I. Yoo, Convolution in Fourier-Wiener transform on abstract Wiener space, Rocky Mountain J. Math. 5 (1995), Timothy Huffman Department of Mathematics Northwestern College Orange City, IA 5141, USA timh@nwciowa.edu David Skoug Department of Mathematics and Statistics University of Nebraska Lincoln, NE , USA dskoug@math.unl.edu David Storvick School of Mathematics University of Minnesota Minneapolis, MN 55455, USA storvick@math.umn.edu