Let C0[0,T] be the classical Wiener space. In [4], Cameron and Storvick introduced a Banach algebra S(L2[0,T]) of analytic Feynman integrable functions on C0[0,T]. Each function in S(L2[0,T]) is defined as a stochastic Fourier transform of a complex measure on L2[0,T]. Cameron and Storvick showed that certain functions which arise naturally in quantum mechanics are elements of the Banach algebra S(L2[0,T]). Under strengthened measurability assumptions, Cameron and Storvick showed in [3] that the analytic Feynman integral of functions F having the form

(1.1)F(x)=exp{∫0Tθ(s,x(s))ds}

gives a solution of an integral equation formally equivalent to Schrödiner equation. In (1.1), {θ(s,·),s∈[0,T]} is a family of the Fourier transforms of bounded measures on R. The functions given by equation (1.1) also are elements of the Banach algebra S(L2[0,T]), see [3, 4, 17].

A study of the analytic Fourier–Feynman transform is an interesting topic concerning with the analytic Feynman integral theory. The theory of the analytic Fourier–Feynman transform suggested by Brue [1] now plays a noteworthy role in infinite dimensional analysis.

In [9, 11], the authors used a generalized Brownian motion process (GBMP) to define a generalized analytic Feynman integral and an Lp(1≤p≤2) analytic GFFT for functions on a function space Ca,b[0,T]. The general Wiener space Ca,b[0,T] can be understood as a space of continuous sample functions of the GBMP. We refer to the references [9, 11, 19, 20] for more detailed informations about the definition of the GBMP associated with continuous functions a(·) and b(·) on the time interval [0,T], and the construction of the function space Ca,b[0,T]. Standard Brownian motion is centered and stationary in time, while in general, a GBMP is neither centered nor stationary in time.

In [9], the authors studied the Lp analytic GFFT of cylinder functions on Ca,b[0,T]. However, they provided the existences of only L1 and L2 GFFTs for cylinder functions on Ca,b[0,T] because the drift term a(t) of the GBMP makes establishing the existences of the GFFTs very difficult. The purpose of this paper is to study the cylinder functions on Ca,b[0,T] whose Lp analytic GFFT exists for all p∈[1,2]. For our purpose, we first examine certain cylinder functions which belong in a Banach algebra F(Ca,b[0,T]) of functions on the function space Ca,b[0,T]. The class F(Ca,b[0,T]) used in this paper is homeomorhic to the Banach algebra S(La,b2[0,T]) studied in [11]. We then provide an explicit formula for the GFFT of the cylinder function under our consideration.

In this section we first provide a brief background about the general Wiener space Ca,b[0,T] induced by the GBMP.

Let (Ca,b[0,T],B(Ca,b[0,T]),μ) denote the function space induced by a GBMP Y determined by continuous functions a(t) and b(t) where B(Ca,b[0,T]) is the Borel σ-algebra induced by sup-norm, see [19] and [20, Chapters 3 and 4]. We assume in this paper that a(t) is an absolutely continuous real-valued function on [0,T] with a(0)=0, a'(t)∈L2[0,T], and b(t) is an increasing, continuously differentiable real-valued function with b(0)=0 and b'(t)>0 for each t∈[0,T]. Then we can consider the coordinate process X:[0,T]×Ca,b[0,T]→R given by X(t,x)=x(t) which is the continuous realization of Y [20, Theorem 14.2]. For any t∈[0,T] and x∈Ca,b[0,T], we have X(t,x)=x(t)∼N(a(t),b(t)). We then complete this function space to obtain the measure space (Ca,b[0,T],W(Ca,b[0,T]),μ) where W(Ca,b[0,T]) is the set of all μ-Carathéodory measurable subsets of Ca,b[0,T].

A subset B of Ca,b[0,T] is said to be scale-invariant measurable (s.i.m.) provided ρB is W(Ca,b[0,T])-measurable for all ρ>0, and a s.i.m. set N is said to be a scale-invariant null set provided μ(ρN)=0 for all ρ>0. A property that holds except on a scale-invariant null set is said to hold scale-invariant almost everywhere (s-a.e.). A function F is said to be s.i.m. provided F is defined on a s.i.m. set and F(ρ·) is W(Ca,b[0,T])-measurable for every ρ>0. If two functions F and G defined on Ca,b[0,T] are equal s-a.e., then we write F≈G.

Let La,b2[0,T] be the space of functions on [0,T] which are Lebesgue measurable and square integrable with respect to the Lebesgue–Stieltjes measures on [0,T] induced by a(·) and b(·): i.e.,

La,b2[0,T]={v:∫0T v 2(t)db(t)<∞ and ∫0T v 2(t)d|a|(t)<∞}

where ∣a∣(·) is the total variation function of a(·). Then La,b2[0,T] is a separable Hilbert space with inner product defined by (u,v)a,b=∫0Tu(t)v(t)d[b(t)+∣a∣(t)]. For more details, see [9, 11].

Consider the function space

Ca,b′[0,T]={w∈Ca,b[0,T]:w(t)=∫0tz(s)db(s)  for some   z∈La,b2[0,T]}.

For w∈Ca,b'[0,T], let the operator D:Ca,b'[0,T]→La,b2[0,T] be defined by the formula

(2.1)Dw(t)=w'(t)b'(t).

Then Ca,b'≡Ca,b'[0,T] with inner product (w1,w2)Ca,b'=∫0TDw1(t)Dw2(t)db(t) is a separable Hilbert space.

Note that the two separable Hilbert spaces La,b2[0,T] and Ca,b'[0,T] are (topologically) homeomorphic under the linear operator given by (2.1). The inverse operator of D is given by (D-1z)(t)=∫0tz(s)db(s) for t∈[0,T]. In the case that a(t)≡0, then the operator D:C0,b'[0,T]→L0,b2[0,T] is an isometry.

In this paper, in addition to the conditions put on a(t) above, we now add the condition

(2.2)∫0T∣a'(t)∣2d∣a∣(t)<+∞

from which it follows that

∫0T|Da(t)|2d[b(t)+|a|(t)]=∫0T|a′(t)b′(t)|2d[b(t)+|a|(t)]<M‖a′‖L2[0,T]+M2∫0T|a′(t)|2d|a|(t)<+∞,

where M=supt∈[0,T](1/b'(t)). Thus, the function a:[0,T]→R satisfies the condition (2.2) if and only if a(·) is an element of Ca,b'[0,T].

Let {en}n=1∞ be a complete orthonormal set of functions in (Ca,b'[0,T],∣·∣Ca,b') such that the Den's are of bounded variation on [0,T]. For w∈Ca,b'[0,T] and x∈Ca,b[0,T], we define the Paley–Wiener–Zygmund stochastic integral (w,x)∼ as follows:

(w,x)∼=limn→∞∫0T∑j=1n(w,ej)Ca,b'Dej(t)dx(t)

if the limit exists. We will emphasize the following fundamental facts. For each w∈Ca,b'[0,T], the Paley–Wiener–Zygmund stochastic integral (w,x)∼ exists for μ-a.e. x∈Ca,b[0,T]. If Dw=z∈La,b2[0,T] is of bounded variation on [0,T], then the Paley–Wiener–Zygmund stochastic integral (w,x)∼ equals the Riemann–Stieltjes integral ∫0TDw(t)dx(t)=∫0Tz(t)dx(t). Also we note that for w,x∈Ca,b'[0,T], (w,x)∼=(w,x)Ca,b'. Furthermore for each w∈Ca,b'[0,T], the Paley–Wiener–Zygmund stochastic integral (w,x)∼ is a Gaussian random variable on Ca,b[0,T] with mean (w,a)Ca,b'=∫0TDw(t)da(t) and variance ∣w∣Ca,b'2=∫0T{Dw(t)}2db(t).

The Banach algebra F(Ca,b[0,T]) is defined as the space of all functions F on Ca,b[0,T] having the form

(3.1)F(x)=∫Ca,b'[0,T]exp{i(w,x)∼}dσ(w)

for s-a.e. x∈Ca,b[0,T], where σ is in M(Ca,b'[0,T]), the space of complex-valued Borel measures on B(Ca,b'[0,T]), the Borel σ-algebra of subsets of the Cameron–Martin space Ca,b'[0,T].

Note that every function given by (3.1) is s.i.m..

A function F on Ca,b[0,T] is called a cylinder function if

(3.2)F(x)=f((h1,x)∼,…,(hn,x)∼), x∈Ca,b[0,T]

for μ-a.e. x∈Ca,b[0,T], where f is a complex-valued Lebesgue measurable function on Rn and {h1,…,hn} is a finite set of functions in Ca,b'[0,T].

Example 3.1. Let F1:Ca,b[0,T]→C be given by

(3.3)F1(x)=f((w1,x)∼,…,(wn,x)∼),

where {w1,…,wn} is a lineally independent set of functions in Ca,b'[0,T]. The GFFT of functions given by the right-hand side of (3.3) are studied in [9]. Let 0=t0<t1<⋯<tn≤T be a subdivision of [0,T].

• (i)

  For each l∈{1,…,n}, let wl(t)=∫0tχ[0,tl](s)db(s) on [0,T]. Then we can rewrite equation (3.3) as

  (3.4)F2(x)=f(x(t1),…,x(tn)).

• (ii)

  For each l∈{1,…,n}, let wl(t)=∫0tχ[tl-1,tl](s)db(s) on [0,T]. Then we can rewrite equation (3.3) as

  (3.5)F3(x)=f(x(t1),x(t2)-x(t1),…,x(tn)-x(tn-1)).

Letting a(t)=0 and b(t)=t on [0,T], the general Wiener space Ca,b[0,T] reduces to the classical Wiener space C0[0,T]. In [2, 5, 6, 14], the authors studied certain classes of functions of the forms (3.4) and (3.5) on C0[0,T] and they used those classes to complete their researches concerning the analytic Feynman integral and the analytic Fourier–Feynman transform on C0[0,T].

Let S:Ca,b'[0,T]→Ca,b'[0,T] be the linear operator given by

(3.6)Sw(t)=∫0tw(s)db(s).

Then the adjoint operator S* of S is given by

S*w(t)=w(T)b(t)-∫0tw(s)db(s)=∫0t[w(T)-w(s)]db(s).

It is easily shown that S* is injective. For a more detailed study of the operator S and S*, see [10].

Example 3.2.

Let F4:Ca,b[0,T]→C be given by

(3.7)F4(x)=f(∫0T z1 (t)x(t)db(t),…,∫0T zn (t)x(t)db(t)),

where {z1,…,zn} is a lineally independent subset of La,b2[0,T]. Then

{w1,…,wn}={∫0⋅ z 1(s)db(s),…,∫0⋅ z n(s)db(s)}

is a lineally independent subset of Ca,b'[0,T], see [10]. Since S* is linear and injective, {S*w1,…,S*wn} also is an independent subset of Ca,b'[0,T]. Furthermore, by an integration by parts formula, it follows that

(3.8)(S*wl,x)∼=∫0Tx(t)Dwl(t)db(t)=∫0Tx(t)zl(t)db(t)

for each l∈{1,…,n}. Hence

F4(x)=f((S*w1,x)∼,…,(S*wn,x)∼)

is a cylinder function on Ca,b[0,T].

Let 0=t0<t1<⋯<tn≤T be a subdivision of [0,T] and for each l∈{1,…,n}, let zl(s)=χ[0,tl](s) on [0,T]. Then we can rewrite equation (3.7) as

F5(x)=f(∫0t1x(s)db(s),∫0t2x(s)db(s),…,∫0tnx(s)db(s)).

In view of the fact that L1(Rn)∖L∞(Rn)≠∅, one can see that every cylinder function on Ca,b[0,T] is not necessarily in the Banach algebra F(Ca,b[0,T]). Thus the rest of this section, we consider a class of cylinder functions on Ca,b[0,T] and provide necessary and sufficient conditions for the cylinder functions given by (3.2) to be in the Banach algebra F(Ca,b[0,T]).

Let M(Rn) denote the space of complex-valued Borel measures on B(Rn), the Borel σ-algebra of Rn. Let ν be in M(Rn). Then the Fourier transform ν of ν given by the formula

(3.9)ν^(u→)=∫ℝn exp{i∑ l=1nul vl}dσ(v→),

is a complex-valued function on Rn.

Next theorem provide necessary and sufficient conditions for the cylinder functions on Ca,b[0,T] to be in F(Ca,b[0,T]). This result subsumes similar known results given in [5, 6, 7, 13].

Theorem 3.3. Let {w1,…,wn} be a linearly independent subset of Ca,b'[0,T]. Let F:Ca,b[0,T]→C be a cylinder function on Ca,b[0,T] given by the right-hand side of (3.3). Then F is in F(Ca,b[0,T]) if and only if there exists a measure σ∈M(Rn) such that σ=f almost everywhere on Rn.

We will provide a more basic theorem ensuring that various functions are in F(Ca,b[0,T]).

Theorem 3.4. Let (Q,Σ,γ) be a σ-finite measure space and let φl:Q→Ca,b'[0,T] be Σ–B(Ca,b'[0,T]) measurable for each l∈{1,…,n}. Let θ:Q×Rn→C be given by θ(η ·)=νη(·) where νη is in M(Rn) for every η∈Q and where the family {νη:η∈Q} satisfies:

• (i)

  νη(B) is a Σ-measurable function of η for every B∈B(Rn),

• (ii)

  ∣νη∣∈L1(Q,Σ,γ).

Under these conditions, the function F:Ca,b[0,T]→C given by

(3.10)F(x)=∫Qθ(η (φ1(η),x)∼,…,(φn(η),x)∼)dγ(η)

is in the class F(Ca,b[0,T]) and satisfies the inequality ∣F∣≤∫Q∣νη∣dγ(η).

Proof.

Using the techniques similar to those used in [7], we can show that ∣νη∣ is measurable as a function of η, that θ is Σ×B(Rn)-measurable, and that the integrand in equation (3.10) is a measurable function of η for every x∈Ca,b[0,T].

We define a measure τ on Σ×B(Rn) by

(3.11)τ(B)=∫Qνη(B(η))dγ(η)forB∈Σ×B(Rn).

Then by the first assertion of [17, Theorem 3.1] with the current condition (ii), τ satisfies ∣τ∣≤∫Q∣νη∣dγ(η). Now let Φ:Q×Rn→Ca,b'[0,T] be defined by

(3.12)Φ(η v1,…,vn)=∑l=1nvlφl(η).

Then Φ is Σ×B(Rn)–B(Ca,b'[0,T])-measurable using the hypothesis for φl, l∈{1,…,n}. Let σ=τ∘Φ-1.

Then clearly σ∈M(Ca,b'[0,T]) and satisfies ∣σ∣≤∣τ∣.

From the change of variables theorem and the second assertion of [17, Theorem 3.1], it follows that for a.e. x∈Ca,b[0,T] and for every ρ>0,

(3.13)F(ρx)=∫Q ν^η ((φ1(η),ρx)~,…,(φn(η),ρx)~)dγ(η)=∫Q[∫ℝ n exp{i∑ l=1nvl (φl(η),ρx)~}dνη(v1,…,vn)]dγ(η)=∫Q×ℝ n exp{i∑ l=1nvl (φl(η),ρx)~}dτ(η;v1,…,vn)=∫Q×ℝ n exp{i(Φ(η; v1 ,…, vn),ρx)~}dτ(η;v1,…,vn)=∫Ca,b′ [0,T] exp{i(w,ρx)~}dτ°Φ−1(w)=∫Ca,b′ [0,T] exp{i(w,ρx)~}dσ(w).

Clearly, σ is a complex measure in M(Ca,b'[0,T]). Thus the function F given by equation (3.10) belongs to F(Ca,b[0,T]) and satisfies the inequality

∣F∣=∣σ∣≤∣τ∣≤∫Q∣νη∣dγ(η)

as desired.

The following corollaries are relevant to Feynman integration theories and quantum mechanics where exponential functions play an important role.

Our next corollary comes from the fact that F(Ca,b[0,T]) is a Banach algebra

Corollary 3.5. Let F be given by equation (3.10), and let Ξ:C→C be an entire function.

Then (Ξ∘F)(x) is in F(Ca,b[0,T]). In particular, exp{F(x)}∈F(Ca,b[0,T]).

Corollary 3.6 (Necessary condition of Theorem 3.3 with weaker condition). Let {g1,…,gn} be a finite (not necessarily linearly independent) subset of Ca,b'[0,T]. Given ν with ν∈M(Rn), define a function F:Ca,b[0,T]→C by

F(x)=Θ((g1,x)∼,…,(gn,x)∼).

Then F is in the class F(Ca,b[0,T]).

Proof. Let (Q,Σ,γ) be a probability space and for l∈{1,…,n}, let φl(η)≡gl. Take ν(·). Then for all ρ>0 and for a.e. x∈Ca,b[0,T],

(3.14)∫Qθ(η;(φ1(η),ρx)~,…,(φn(η),ρx)~)dγ(η)=∫QΘ((g1,ρx)~,…,(gn,ρx)~)dγ(η)=Θ((g1,ρx)~,…,(gn,ρx)~)=F(ρx).

Hence F∈F(Ca,b[0,T]).

In this section, we obtain an explicit formula for the Lp analytic GFFT of the cylinder functions in F(Ca,b[0,T]). Let C+={λ∈C:Re(λ)>0} and let C˜+={λ∈C∖{0}:Re(λ)≥0}. Throughout the rest of this paper, λ-1/2(or λ1/2) always is chosen to have positive real part for all λ∈C˜+.

Let F be a s.i.m. function on Ca,b[0,T] such that JF(λ)=∫Ca,b[0,T]F(λ-1/2x)dμ(x)

exists and is finite for all λ>0.

If there exists a function JF*(λ) analytic in C+ such that JF*(λ)=JF(λ) for all λ>0, then JF*(λ) is defined to be the analytic function space integral of F over Ca,b[0,T] with parameter λ, and for λ∈C+ we write Eanλ[F]≡JF*athrmanλ​x[F(x)]=JF*(λ).

Let q∈R∖{0} and let F be a s.i.m. function whose analytic function space integral JF*(λ) exists for all λ∈C+.

If the following limit exists, we call it the analytic generalized Feynman integral of F with parameter q, and we write

(4.1)Eanfq[F]≡Exanfq[F(x)]=limλ→-iqExanλ[F(x)]

where λ→-iq through C+.

We are now ready to state the definition of the analytic GFFT of functions F on Ca,b[0,T].

Definition 4.1. Let F be a s.i.m. function on Ca,b[0,T]. For λ∈C+ and y∈Ca,b[0,T], let Tλ(F)(y)=Exanλ[F(y+x)]. For p∈(1,2], we define the Lp analytic GFFT, Tq(p)(F) of F, by the formula

Tq(p)(F)(y)=*l.i.m.λ→−iqλ∈ℂ+Tλ(F)(y)

if it exists; i.e., for each ρ>0,

limλ→−iqλ∈ℂ+∫Ca,b[0,T]|Tλ(F)(ρy)−Tq(p)(F)(ρy)|p′dμ(y)=0

where 1/p+1/p'=1. We define the L1 analytic GFFT, Tq(1)(F) of F, by the formula

(4.2)Tq(1)(F)(y)=limλ→−iqλ∈ℂ+Tλ(F)(y)=limλ→−iqλ∈ℂ+Exanλ[F(y+x)],

for s-a.e. y∈Ca,b[0,T], if the limit exists.

Remark 4.2. In [2, pp. 5–7], Cameron and Storvick exhibited two measurable functions F and G on the classical Wiener space C0[0,T] such that F(x)=G(x) for a.e. x∈C0[0,T] and yet their Fourier–Feynman transforms are unequal a.e.. Based on this fact, Johnson and Skoug [15] defined the Lp analytic Fourier–Feynman transform for functions on C0[0,T] under the concept of the scale-invariant measurability. In fact, it was pointed out in [16] that the concept of `scale-invariant measurability' is correct for the analytic Fourier–Feynman transform and the analytic Feynman integration theories. For more details, see [18, pp. 1155–1157].

We note that for 1≤p≤2, Tq(p)(F) is defined only s-a.e.. If Tq(p)(F) exists and if F≈G, then Tq(p)(G) exists and Tq(p)(G)≈Tq(p)(F). For more detailed studies of the GFFT of functions on Ca,b[0,T], see [9, 11].

In view of (4.1) and (4.2), we set

(4.3)Tq(1)(F)(0)=Exanfq[F(x)].

Theorem 4.3 below is a simple modification of the result [12, Theorem 9]. The condition (4.4) below will guarantee the existence of the right-hand side of (4.5) below.

Theorem 4.3. Let q0∈R∖{0} and let F be given by equation (3.1). Suppose that the associated measure σ of F satisfies the condition

(4.4)∫Ca,b′[0,T] exp{1|2q0| ‖w‖Ca,b′ ‖a‖Ca,b′ }d|σ|(w)<+∞.

Then, for all p∈[1,2] and all q∈R∖[-q0,q0], the Lp analytic GFFT Tq(p)(F) exists and is given by the formula

(4.5)Tq(p)(F)(y)=∫Ca,b′[0,T] exp{i(w,y)~−i2q‖w‖Ca,b′ 2+i(−iq)−1/2(w,a)Ca,b′ }dσ(w)

for s-a.e. y∈Ca,b[0,T].

In view of Theorems 3.4 and 4.3, we can provide the following evaluation formula for the Lp analytic GFFT of functions F in F(Ca,b[0,T]).

Theorem 4.4. Let (Q,Σ,γ), {φ1,…,φn}, {νη:η∈Q}, θ, and F be as in Theorem 3.4. Suppose that given a positive real q0,

(4.6)∫Q×ℝn exp{‖a‖Ca,b′ 2|q0| ∑ l=1n‖φl(η)‖Ca,b′ |vl|}d(|νη|×γ)(η,v→)=∫Q[∫ℝn exp{‖a‖Ca,b′ 2|q0| ∑ l=1n‖φl(η)‖Ca,b′ |vl|}d|νη|(v→)]dγ(η)<+∞.

Then for all p∈[1,2] and all q∈R∖[-q0,q0], the Lp analytic GFFT Tq(p)(F) of F exists and is given by the formula

(4.7)Tq(p)(F)(y)=∫Q[∫ℝn exp{i∑ l=1nvl (φl(η),y)~−i2q‖∑ l=1nvl φl(η)‖Ca,b′ 2+i(−iq)−1/2∑ l=1nvl (φl(η),a)Ca,b′ }dνη(v1,…,vn)]dγ(η)

for s-a.e. y∈Ca,b[0,T]. In particular, if {φ1(η),…,φn(η)} is an orthogonal set of functions in Ca,b'[0,T], then it follows that

(4.8)Tq(p)(F)(y)=∫Q[∫ℝn exp{i∑ l=1nvl (φl(η),y)~−i2q∑ l=1nvl2 ‖φl(η)‖Ca,b′ 2+i(−iq)−1/2∑ l=1nvl (φl(η),a)Ca,b′ }dνη(v1,…,vn)]dγ(η)

for s-a.e. y∈Ca,b[0,T].

Proof. From (3.13) with ρ=1, we see that the function F given by (3.10) is rewritten by

F(x)=∫Q[∫ℝn exp{i∑ l=1nvl (φl(η),x)~}dνη(v1,…,vn)]dγ(η)=∫Ca,b′ [0,T] exp{i(w,x)~}dτ°Φ−1(w)

for s-a.e. y∈Ca,b[0,T], where τ and Φ are given by (3.11) and (3.12) respectively. Thus the condition (4.6) implies the condition (4.4) with σ=τ∘Φ-1, and by Theorem 4.3, the Lp analytic GFFT of F given by (3.10) exists and is given by the formula

Tq(p)(F)(y)=∫Ca,b′[0,T] exp{i(w,y)~−i2q‖w‖Ca,b′ 2+i(−iq)−1/2(w,a)Ca,b′ }dτ°Φ−1(w)=∫Q×ℝn exp{i(Φ(η;v1 ,…,vn ),y)~−i2q‖Φ(η;v1,…,vn)‖Ca,b′ 2+i(−iq)−1/2(Φ(η;v1 ,…,vn ),a)Ca,b′ }dτ(η;v1,…,vn)=∫Q[∫ℝn exp{i∑ l=1n vl (φl (η),y)~−i2q‖∑ l=1n vl φl(η)‖Ca,b′ 2+i(−iq)−1/2∑ l=1n vl (φl (η),a)Ca,b′ }dνη(v1,…,vn)]dγ(η)

for s-a.e. y∈Ca,b[0,T]. From this, we also have (4.8).

From (4.3) and (4.7) with p=1, we have the following corollary.

Corollary 4.5. Let (Q,Σ,γ), {φ1,…,φn}, {νη:η∈Q}, θ, and F be as in Theorem 4.4. Then, for all q∈R∖[-q0,q0], the generalized analytic Feynman integral Eanfq[F] of F exists and is given by the formula

Exanfq[F(x)]=∫Q[∫ℝn exp{−i2q‖∑ l=1nvlφl(η)‖Ca,b′2+i(−iq)−1/2∑ l=1nvl(φl(η),a)Ca,b′}dνη(v1,…,vn)]dγ(η).

under the condition (4.6).

Given an orthonormal set {g1,…,gn} of functions in Ca,b[0,T], let the function F:Ca,b[0,T]→C be given by

(4.9)ν((g1,x)∼,…,(gn,x)∼), x∈Ca,b[0,T],

where ν is the Fourier transform defined by equation ((3.9) for a complex-valued Borel measure ν in M(Rn). Then F is a bounded cylinder function, since ν(u→)|≤∣ν∣<+∞. In [8], Chang and Choi studied an inverse transform corresponding to the Lp analytic GFFT of the function given by 3.9" ref-type="disp-formula">3.9) for a complex-valued Borel measure ν in M(Rn). Then F is a bounded cylinder function, since ν(u→)|≤∣ν∣<+∞. In [8], Chang and Choi studied an inverse transform corresponding to the Lp analytic GFFT of the function given by (4.9). One of the main results in [8] is to establish the existence of the GFFT of the functions F given by (4.9).

Corollary 4.6. Let q0∈R∖{0} and let F be given by equation (4.9). Suppose that the associated measure ν of F satisfies the condition

(4.10)∫ℝn exp{‖a‖Ca,b′ |2q0| ∑ l=1n|vl|}d|ν|(v→)<+∞.

Then, for each p∈[1,2] and any q∈R∖[-q0,q0], the Lp analytic GFFT Tq(p)(F) exists and is given by the formula

(4.11)Tq(p)(F)(y)=∫ℝn exp{i∑ l=1nvl (gl,y)~−i2q∑ l=1nvl2 +i(−iq)−1/2∑ l=1nvl (gl,a)Ca,b′ }dν(v→)

for s-a.e. y∈Ca,b[0,T].

Proof. From equation (3.14), we already observe that

ν((g1,x)∼,…,(gn,x)∼)=∫Qθ(η (φ1(η),x)∼,…,(φn(η),x)∼)dγ(η)

for s-a.e. x∈Ca,b[0,T], where (Q,Σ,γ) is any probability space, φl(η)≡gl for each l∈{1,…,n}, and ν(·). Also, the condition (4.6) implies the condition

∫Q×ℝn exp{‖a‖Ca,b′ 2|q0| ∑ l=1n‖gl‖Ca,b′ |vl|}d(|νη|×γ)(η,v→)=∫Q[∫ℝn exp{‖a‖Ca,b′ 2|q0| ∑ l=1n|vl|}d|νη|(v→)]dγ(η)=∫ℝn exp{‖a‖Ca,b′ 2|q0| ∑ l=1n|vl|}d|ν|(v→)<+∞.

Thus, in view of Theorem 4.4 with these setting, equation (4.8) yields the formula (4.11) as desired.

From (4.3) and (4.11) with p=1, we have the following corollary.

Corollary 4.7. Let q0 and F be as in Corollary 4.6.

Then, for any q∈R∖[-q0,q0], the generalized Feynman integral Eanfq[F] exists and is given by the formula

Exanfq[F(x)]=∫ℝn exp{−i2q∑ l=1nvl2+i(−iq)−1/2∑ l=1nvl(gl,a)Ca,b′}dν(v→).