Sequential Generalized Transforms on Function Space

Abstract

We define two sequential transforms on a function space 𝐶_𝑎,𝑏[0,𝑇] induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals on 𝐶_𝑎,𝑏[0,𝑇]. We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms on 𝐶_𝑎,𝑏[0,𝑇].

1. Introduction and Preliminaries

Let 𝐶₀[0,𝑇] denote one-parameter Wiener space; that is, the space of all real-valued continuous functions 𝑥(𝑡) on [0,𝑇] with 𝑥(0)=0. The study of the Fourier-Wiener transform of functionals on 𝐶₀[0,𝑇] was initiated by Cameron and Martin [1–3]. This transform and its properties are similar in many respects to the ordinary Fourier function transform. Since then, many transforms which were somewhat analogous to the Fourier-Wiener transform have been defined and developed in the literature. There are two well-known transforms on 𝐶₀[0,𝑇]. One of them is the analytic Fourier-Feynman transform [4–6] and the other is the integral transform [7–10]. Each of the transforms on Wiener space has an inverse transform. For an elementary survey, see [11].

In [12–16], the authors studied the generalized analytic Fourier-Feynman transform and the generalized integral transform for functionals defined on a more general function space 𝐶_𝑎,𝑏[0,𝑇]. The function space 𝐶_𝑎,𝑏[0,𝑇], induced by generalized Brownian motion process, was introduced by Yeh [17, 18] and was used extensively by Chang and Chung [19]. The Wiener process used in [1–10] is stationary in time and is free of drift, while the stochastic process used in this paper, as well as in [12–17, 19], is nonstationary in time and is subject to a drift 𝑎(𝑡). In case 𝑎(𝑡)≡0 and 𝑏(𝑡)=𝑡 on [0,𝑇], the general function space 𝐶_𝑎,𝑏[0,𝑇] reduces to the Wiener space 𝐶₀[0,𝑇] and so most of the results in [4–6, 9] follow immediately from the results in [12, 13, 15, 16].

However, the existence of an inverse transform of each of the two generalized transforms on 𝐶_𝑎,𝑏[0,𝑇] has not yet been established. It is a critical point that the generalized transforms on 𝐶_𝑎,𝑏[0,𝑇] are essentially different from the transforms on Wiener space 𝐶₀[0,𝑇]. The main purpose of this paper is to define a transform on 𝐶_𝑎,𝑏[0,𝑇] which has an inverse transform.

In this paper, we define two sequential transforms on the function space 𝐶_𝑎,𝑏[0,𝑇]. To do this, we investigate a representation for sample paths of the generalized Brownian motion process and introduce the concept of the 𝒢-s-continuity for functionals on 𝐶_𝑎,𝑏[0,𝑇]. We then proceed to establish the existence of the sequential transforms for functionals in a Banach algebra ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) of functionals on 𝐶_𝑎,𝑏[0,𝑇]. Next, we establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we examine certain aspects of the generalized analytic Fourier-Feynman transform, the generalized integral transform, and the sequential transforms.

We briefly list some of the preliminaries from [12, 13, 17] that we will need in order to establish the results in this paper.

Let 𝑎(𝑡) be an absolutely continuous real-valued function on [0,𝑇] with 𝑎(0)=0, 𝑎^′(𝑡)∈𝐿²[0,𝑇], and let 𝑏(𝑡) be a strictly increasing, continuously differentiable real-valued function with 𝑏(0)=0 and 𝑏^′(𝑡)>0 for each 𝑡∈[0,𝑇]. The generalized Brownian motion process 𝑌 determined by 𝑎(𝑡) and 𝑏(𝑡) is a Gaussian process with mean function 𝑎(𝑡) and covariance function 𝑟(𝑠,𝑡)=min{𝑏(𝑠),𝑏(𝑡)}. For more details, see [12, 13]. By Theorem  14.2 in [18], the probability measure 𝜇 induced by 𝑌, taking a separable version, is supported by 𝐶_𝑎,𝑏[0,𝑇] (which is equivalent to the Banach space of continuous functions 𝑥 on [0,𝑇] with 𝑥(0)=0 under the sup norm). Hence, (𝐶_𝑎,𝑏[0,𝑇],ℬ(𝐶_𝑎,𝑏[0,𝑇]),𝜇) is the function space induced by 𝑌 where ℬ(𝐶_𝑎,𝑏[0,𝑇]) is the Borel 𝜎-algebra of 𝐶_𝑎,𝑏[0,𝑇]. We then complete this function space to obtain (𝐶_𝑎,𝑏[0,𝑇],𝒲(𝐶_𝑎,𝑏[0,𝑇]),𝜇), where 𝒲(𝐶_𝑎,𝑏[0,𝑇]) is the set of all Wiener measurable subsets of 𝐶_𝑎,𝑏[0,𝑇].

A subset 𝐵 of 𝐶_𝑎,𝑏[0,𝑇] is said to be scale-invariant measurable provided 𝜌𝐵 is 𝒲(𝐶_𝑎,𝑏[0,𝑇])-measurable for all 𝜌>0, and a scale-invariant measurable 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 functional 𝐹 is said to be scale-invariant measurable provided 𝐹 is defined on a scale-invariant measurable set, and 𝐹(𝜌⋅) is 𝒲(𝐶_𝑎,𝑏[0,𝑇])-measurable for every 𝜌>0. If two functionals 𝐹 and 𝐺 defined on 𝐶_𝑎,𝑏[0,𝑇] are equal s-a.e., we write 𝐹≈𝐺.

Let 𝐿²_𝑎,𝑏[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(⋅); that is, 𝐿²_𝑎,𝑏[]=0,𝑇𝑣∶^𝑇₀𝑣²(𝑠)𝑑𝑏(𝑠)<+∞,^𝑇₀𝑣²(𝑠)𝑑|𝑎|(𝑠)<+∞, (1) where |𝑎|(⋅) is the total variation function of 𝑎(⋅). Then, (𝐿²_𝑎,𝑏[0,𝑇],‖⋅‖_𝑎,𝑏) is a separable Hilbert space with inner product defined by (𝑢,𝑣)_𝑎,𝑏=^𝑇₀𝑢(𝑡)𝑣(𝑡)𝑑[𝑏(𝑡)+|𝑎|(𝑡)]. (2) Note that ‖𝑢‖_𝑎,𝑏=(𝑢,𝑢)_𝑎,𝑏=0 if and only if 𝑢(𝑡)=0 a.e. on [0,𝑇]. Also note that all functions of bounded variation on [0,𝑇] are elements of 𝐿²_𝑎,𝑏[0,𝑇]. If 𝑎(𝑡)≡0 and 𝑏(𝑡)=𝑡, then 𝐿²_𝑎,𝑏[0,𝑇]=𝐿²[0,𝑇]. In fact, 𝐿²_𝑎,𝑏[0,𝑇],‖⋅‖_𝑎,𝑏⊂𝐿²_0,𝑏[0,𝑇],‖⋅‖_0,𝑏=𝐿²[0,𝑇],‖⋅‖₂ (3) because the norms ‖⋅‖_0,𝑏 and ‖⋅‖₂ are equivalent.

Let 𝐶^′_𝑎,𝑏[0,𝑇]=𝑤∈𝐶_𝑎,𝑏[](𝑡)=0,𝑇∶𝑤_𝑡0𝑧(𝑠)𝑑𝑏(𝑠)forsome𝑧∈𝐿²_𝑎,𝑏[0,𝑇]. (4) For 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], with 𝑤(𝑡)=∫^𝑡₀𝑧(𝑠)𝑑𝑏(𝑠) for 𝑡∈[0,𝑇], let 𝐷∶𝐶^′_𝑎,𝑏[0,𝑇]→𝐿²_𝑎,𝑏[0,𝑇] be defined by the formula (𝑡)(𝑡)=𝑤𝐷𝑤=𝑧^′(𝑡)𝑏^′(𝑡). (5) Then 𝐶^′_𝑎,𝑏≡𝐶^′_𝑎,𝑏[0,𝑇] with inner product 𝑤₁,𝑤_{2𝐶′𝑎,𝑏}=^𝑇₀𝐷𝑤₁(𝑡)𝐷𝑤₂(𝑡)(𝑡)=𝑑𝑏_𝑇0𝑧₁(𝑡)𝑧₂(𝑡)𝑑𝑏(𝑡)=𝑧₁,𝑧_20,𝑏 (6) is a separable Hilbert space.

Note that the two separable Hilbert spaces 𝐿²_𝑎,𝑏[0,𝑇] and 𝐶^′_𝑎,𝑏[0,𝑇] are homeomorphic under the linear operator given by (5). The inverse operator of 𝐷 is given by 𝐷⁻¹𝑧(𝑡)=^𝑡₀𝑧(𝑠)𝑑𝑏(𝑠). (7)

Recall that above, as well as in papers [12–16], we require that 𝑎∶[0,𝑇]→ℝ is an absolutely continuous function with 𝑎(0)=0 and with ∫^𝑇₀|𝑎^′(𝑡)|²𝑑𝑡<+∞. Our conditions on 𝑏∶[0,𝑇]→ℝ imply that 𝛿<𝑏^′(𝑡)<Δ for some positive real numbers 𝛿 and Δ, and 𝑏^′(𝑡) is continuous on [0,𝑇]. Hence, we have ^𝑇₀𝑎^′(𝑡)²𝑑𝑏(𝑡)=^𝑇₀𝑎^′(𝑡)²𝑏^′(𝑡)𝑑𝑡<+∞. (8) But we cannot ensure that ^𝑇₀𝑎^′(𝑡)²𝑑|𝑎|(𝑡)<+∞ (9) under current conditions. Note that the function 𝑎(𝑡)=𝑡^2/3, 0≤𝑡≤𝑇, does not satisfy condition (9) even though its derivative is an element of 𝐿²[0,𝑇].

In this paper, we add the requirement (9). Then we obtain the following lemma.

Lemma 1. The function 𝑎∶[0,𝑇]→ℝ satisfies the requirement (9) if and only if 𝑎 is an element of 𝐶^′_𝑎,𝑏[0,𝑇].

Under the requirement (9), we observe that for each 𝑧∈𝐿²_𝑎,𝑏[0,𝑇], ^𝑇₀𝑧(𝑡)𝑑𝑎(𝑡)=^𝑇₀𝐷𝑤(𝑡)𝐷𝑎(𝑡)𝑑𝑏(𝑡)=(𝑤,𝑎)_{𝐶′𝑎,𝑏}, (10) where 𝑤(𝑡)=∫^𝑡₀𝑧(𝑠)𝑑𝑏(𝑠) for 𝑡∈[0,𝑇].

The following integration formula is used several times in this paper: _ℝexp−𝛼𝑢²+𝛽𝑢𝑑𝑢=𝜋𝛼𝛽exp²4𝛼 (11) for complex numbers 𝛼 and 𝛽 with Re(𝛼)>0.

2. A Representation for Paths 𝑥 in 𝐶_𝑎,𝑏[0,𝑇]

In this section, we investigate a representation for paths 𝑥 in 𝐶_𝑎,𝑏[0,𝑇]. To do this, we first define a Paley-Wiener-Zygmund (PWZ) type stochastic integral.

Let {𝑔₁,𝑔₂,…} be a complete orthonormal set of functions in the separable Hilbert space 𝐶^′_𝑎,𝑏[0,𝑇], such that for each 𝑗=1,2,…, 𝐷𝑔_𝑗(𝑡)=𝛼_𝑗(𝑡) is of bounded variation on [0,𝑇]. Then, for each 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇] with 𝐷𝑤=𝑧∈𝐿²_𝑎,𝑏[0,𝑇], we can write 𝑤(𝑡)=^∞_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝑔_𝑗(𝑡)=^∞_𝑗=1𝑧,𝛼_𝑗0,𝑏𝑔_𝑗(𝑡) (12) on [0,𝑇].

For each 𝑤=𝐷⁻¹𝑧∈𝐶^′_𝑎,𝑏[0,𝑇], the PWZ type stochastic integral (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏} is defined by the formula (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}=lim_𝑛→∞^𝑇₀^𝑛_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐷𝑔_𝑗(𝑡)𝑑𝑥(𝑡)=lim_𝑛→∞^𝑇₀^𝑛_𝑗=1𝑧,𝛼_𝑗0,𝑏𝛼_𝑗(𝑡)𝑑𝑥(𝑡) (13) for all 𝑥∈𝐶_𝑎,𝑏[0,𝑇] for which the limit exists.

The following are some basic properties of the PWZ type stochastic integral. They are nontrivial, but straightforward to prove.(1)For each 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], the PWZ type stochastic integral (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏} exists for 𝜇-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇].(2)The PWZ type stochastic integral (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏} is essentially independent of the choice of the complete orthonormal set {𝑔_𝑗}^∞_𝑗=1 in 𝐶^′_𝑎,𝑏[0,𝑇].(3)It follows from the definition of the PWZ type stochastic integral and from Parseval’s equality that if 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇] and 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], then (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}exists and we have (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}=(𝑤,𝑥)_{𝐶′𝑎,𝑏}.(4)If 𝐷𝑤=𝑧∈𝐿_𝑎,𝑏[0,𝑇] is of bounded variation on [0,𝑇], then the PWZ type stochastic integral (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏} equals the Riemann-Stieltjes integral ∫^𝑇₀𝑧(𝑡)𝑑𝑥(𝑡) for 𝜇-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇].(5)The PWZ type stochastic integral has the expected linearity properties. That is, for any real number 𝑐, 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇] and 𝑥∈𝐶_𝑎,𝑏[0,𝑇], we have (𝑤,c𝑥)^∼_{𝐶′𝑎,𝑏}=𝑐(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}=(𝑐𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}. (14) Thus, the assertions (1) and (4) hold for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇].(6)For each 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏} is a Gaussian random variable with mean (𝑤,𝑎)_{𝐶′𝑎,𝑏} and variance ‖𝑤‖²_{𝐶′𝑎,𝑏}. For all 𝑤₁,𝑤₂∈𝐶^′_𝑎,𝑏[0,𝑇], we have _{𝐶𝑎,𝑏[]0,𝑇}𝑤₁,𝑥^∼_{𝐶′𝑎,𝑏}𝑤₂,𝑥^∼_{𝐶′𝑎,𝑏}(𝑥)=𝑤𝑑𝜇₁,𝑤_{2𝐶′𝑎,𝑏}+𝑤₁,𝑎_{𝐶′𝑎,𝑏}𝑤₂,𝑎_{𝐶′𝑎,𝑏}. (15)

Now, we are ready to examine a representation for paths 𝑥 in 𝐶_𝑎,𝑏[0,𝑇]. Throughout the rest of this paper, we will use the symbol 𝒢 for a complete orthonormal set {𝑔_𝑗}^∞_𝑗=1 in 𝐶^′_𝑎,𝑏[0,𝑇], such that for each 𝑗=1,2,…, 𝐷𝑔_𝑗=𝛼_𝑗 is of bounded variation on [0,𝑇]. Then, for each 𝑗=1,2,…, 𝛾_𝑗(𝑥)≡𝑔_𝑗,𝑥^∼_{𝐶′𝑎,𝑏}=^𝑇₀𝛼_𝑗(𝑡)𝑑𝑥(𝑡) (16) is a Gaussian random variable with mean 𝐴_𝑗≡𝑔_𝑗,𝑎_{𝐶′𝑎,𝑏}=^𝑇₀𝛼_𝑗(𝑡)𝑑𝑎(𝑡) (17) and variance 𝑔_𝑗²_{𝐶′𝑎,𝑏}=^𝑇₀𝛼²_𝑗(𝑡)𝑑𝑏(𝑡)=1. (18) We note that the set {𝛾₁(𝑥),𝛾₂(𝑥),…} forms a set of independent Gaussian random variables on 𝐶_𝑎,𝑏[0,𝑇].

Let 𝒢≡{𝑔_𝑗}^∞_𝑗=1 be as above, and let 𝑥∈𝐶_𝑎,𝑏[0,𝑇]. For each 𝑛=1,2,…, define 𝑍_𝑛(𝑥)(𝑡)≡^𝑛_𝑗=1𝛾_𝑗(𝑥)𝑔_𝑗(𝑡). (19) Then, 𝑍_𝑛(𝑥) is an element of 𝐶^′_𝑎,𝑏[0,𝑇] for all 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑛=1,2,….

For each 𝑡∈[0,𝑇], let Ψ_𝑡(𝑠)=∫^𝑠₀𝜒_[0,𝑡](𝜏)𝑑𝑏(𝜏). Then, we observe that for each 𝑡∈[0,𝑇], 𝑥(𝑡)=^𝑇₀𝜒_[0,𝑡](𝑠)(𝑠)=Ψ𝑑𝑥_𝑡,𝑥^∼_{𝐶′𝑎,𝑏}=lim^𝑛_{𝑛→∞𝑗=1}Ψ_𝑡,𝑔_{𝑗𝐶′𝑎,𝑏}^𝑇₀𝛼_𝑗(𝑠)𝑑𝑥(𝑠)=lim^𝑛_{𝑛→∞𝑗=1}^𝑡₀𝛼_𝑗(𝑠)𝑑𝑏(𝑠)𝛾_𝑗(𝑥)=lim^𝑛_{𝑛→∞𝑗=1}𝛾_𝑗(𝑥)𝑔_𝑗(𝑡)=lim_𝑛→∞𝑍_𝑛(𝑥)(𝑡). (20) By the property of the PWZ type stochastic integral, the last expression of (20) converges for 𝜇-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇].

Remark 2. By the definition of the PWZ type stochastic integral, the last expression of (20) is independent of the choice of the complete orthonormal set 𝒢 in 𝐶^′_𝑎,𝑏[0,𝑇]. If we choose the complete orthonormal sine sequence 𝒢≡{𝑔_𝑗}^∞_𝑗=1 in 𝐶^′_𝑎,𝑏[0,𝑇], where 𝑔_𝑗(𝑡)=2𝑏(𝑇)𝜋(𝑡)𝑗−1/2sin𝑗−1/2𝜋𝑏𝑏(𝑇); (21) that is, 𝛼_𝑗(𝑡)=𝐷𝑔_𝑗(𝑡)=2𝑏(𝑇)(𝑡)cos𝑗−1/2𝜋𝑏𝑏(𝑇) (22) for 𝑗=1,2,…, then the corresponding 𝑍_𝑛(𝑥)(𝑡) converges to 𝑥(𝑡) uniformly in 𝑡 with probability one. For more details, see [20] and the references therein.
We now state a fundamental integration formula on the function space 𝐶_𝑎,𝑏[0,𝑇]. Let 𝛾_𝑗(𝑥) and 𝐴_𝑗 be as above, let ℎ∶ℝ^𝑛→ℂ be Lebesgue measurable, and let 𝐻(𝑥)=ℎ(𝛾₁(𝑥),…,𝛾_𝑛(𝑥)). Then _{𝐶𝑎,𝑏[]0,𝑇}𝐻(𝑥)(𝑥)≡𝑑𝜇_{𝐶𝑎,𝑏[]0,𝑇}ℎ𝛾₁(𝑥),…,𝛾_𝑛(𝑥(𝑥)=())𝑑𝜇2𝜋^−𝑛/2_ℝ𝑛ℎ𝑢₁,…,𝑢_𝑛×exp−^𝑛_𝑗=1𝑢_𝑗−𝐴_𝑗²2𝑑𝑢₁…𝑑𝑢_𝑛 (23) in the sense that if either side of (23) exists, both sides exist and equality holds.
Let 𝐹 be a functional on 𝐶_𝑎,𝑏[0,𝑇] and let 𝒢≡{𝑔_𝑗}^∞_𝑗=1 be a complete orthonormal set in 𝐶^′_𝑎,𝑏[0,𝑇]. Then, we say that 𝐹 is 𝒢-continuous at 𝑥 if lim_𝑛→∞𝐹𝑍_𝑛(𝑥)=𝐹(𝑥), (24) where 𝑍_𝑛(𝑥) is given by (19).

Example 3. For each 𝑡∈[0,𝑇], let 𝐹∶𝐶_𝑎,𝑏[0,𝑇]→ℝ be given by 𝐹(𝑥)=𝑥(𝑡). Then, using (20) and (14), we see that 𝐹 is 𝒢-continuous for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇].

Example 4. Given 𝑤=𝐷⁻¹𝑧∈𝐶^′_𝑎,𝑏[0,𝑇], let 𝐹∶𝐶_𝑎,𝑏[0,𝑇]→ℝ be given by 𝐹(𝑥)=(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}. Since (𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}=(𝑤,𝑥)_{𝐶′𝑎,𝑏} for every 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], by the definition of PWZ type stochastic integral, we obtain that lim_𝑛→∞𝐹𝑍_𝑛(𝑥)=lim_𝑛→∞𝑤,𝑍_𝑛(𝑥)_{𝐶′𝑎,𝑏}=lim_𝑛→∞^𝑇₀𝑧(𝑡)𝑑𝑍_𝑛(𝑥)(𝑡)=lim_𝑛→∞^𝑇₀𝑧(𝑡)𝑑^𝑛_𝑗=1𝛾_𝑗(𝑥)𝑔_𝑗(𝑡)=lim^𝑛_{𝑛→∞𝑗=1}𝛾_𝑗(𝑥)^𝑇₀𝑧(𝑡)𝑑𝑔_𝑗(𝑡)=lim^𝑛_{𝑛→∞𝑗=1}𝑧,𝛼_𝑗0,𝑏𝛾_𝑗(𝑥)=(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}=𝐹(𝑥) (25) for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇].

Proposition 5. Let 𝐹∶𝐶_𝑎,𝑏[0,𝑇]→ℂ be 𝜇-integrable. Assume that 𝐹 is 𝒢-continuous for 𝜇-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇]. Then _{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑𝜇(𝑥)=lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑑𝜇(𝑥), (26) where 𝑍_𝑛(𝑥) is given by (19).

Proof. For each 𝑛=1,2,…, let 𝐹_𝑛(𝑥)=𝐹(𝑍_𝑛(𝑥)). Then 𝐹_𝑛 is 𝜇-integrable. By our assumption, we observe that lim_𝑛→∞𝐹_𝑛(𝑥)=lim_𝑛→∞𝐹𝑍_𝑛(𝑥)=𝐹(𝑥) (27) for 𝜇-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇]. Thus, by the dominated convergence theorem, it follows that lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑑𝜇(𝑥)=lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹_𝑛(𝑥)𝑑𝜇(𝑥)=_{𝐶𝑎,𝑏[0,𝑇]}lim_𝑛→∞𝐹_𝑛(𝑥)(𝑥)=𝑑𝜇_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑𝜇(𝑥) (28) as desired.

3. Sequential Function Space Integrals

In [21], Cameron and Storvick defined the “sequential” Feynman integral by means of finite dimensional approximations for functionals on Wiener space 𝐶₀[0,𝑇]. The sequential definition for the Feynman integral was intended to interpret the Feynman’s uniform measure [22] on path space 𝐶₀[0,𝑇], because there is no countably additive measure which weights all paths 𝑥 in 𝐶₀[0,𝑇] equally in much the same way as Lebesgue measure weights all points in ℝ equally. Thus, the Cameron and Storvick’s sequential Feynman integral is a rigorous mathematical formulation of the Feynman’s path integral.

The Cameron and Storvick’s sequential Feynman integral is related by sequential Wiener integral [23]; that is, the integral is based on polygonal path approximations. In this section, we define different kinds of sequential function space integrals for functionals on the function space 𝐶_𝑎,𝑏[0,𝑇]. In Section 4 below, we also adopt sequential approaches to define our function space transforms on 𝐶_𝑎,𝑏[0,𝑇]. The sequential definition for the Feynman integral in [21] was defined as the limit of a sequence of finite dimensional Lebesgue integrals. Essentially, our sequential function space integrals and transforms are defined in terms of a sequence of complex measures on the function space 𝐶_𝑎,𝑏[0,𝑇].

Next, we introduce two sequential definitions for certain function space integrals on 𝐶_𝑎,𝑏[0,𝑇]. Throughout the rest of this paper, let ℂ₊ and ℂ₊ denote the set of complex numbers with positive real part and nonzero complex numbers with nonnegative real part, respectively. Furthermore, for all 𝜆∈ℂ, 𝜆^1/2 is always chosen to have nonnegative real part.

Let 𝒢≡{𝑔_𝑗}^∞_𝑗=1, 𝛾_𝑗(𝑥), and 𝐴_𝑗 be as in Section 2. For 𝜆∈ℂ₊, 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and each 𝑛=1,2,…, let 𝑊⁺_𝑛(𝜆;𝑥)≡𝜆^𝑛/2exp1−𝜆2^𝑛_𝑗=1𝛾²_𝑗(𝑥)+𝜆^1/2−1^𝑛_𝑗=1𝐴_𝑗𝛾_𝑗(𝑥) (29) and let 𝑊⁻_𝑛(𝜆;𝑥)≡𝜆^𝑛/2exp1−𝜆2^𝑛_𝑗=1𝛾²_𝑗(𝑥)+(−𝜆)^1/2−1^𝑛_𝑗=1𝐴_𝑗𝛾_𝑗(𝑥)+^𝑛_𝑗=1𝐴²_𝑗. (30)

Using (23) and (11), we observe that for all 𝜆∈ℂ₊ and every 𝑛=1,2,…, _{𝐶𝑎,𝑏[0,𝑇]}𝑊⁺_𝑛()(𝑥)=𝜆;𝑥𝑑𝜇_{𝐶𝑎,𝑏[0,𝑇]}𝑊⁻_𝑛(𝜆;𝑥)𝑑𝜇(𝑥)=1. (31)

We are now ready to state the definition of the sequential function space integrals.

Definition 6. Let 𝐹 be a measurable functional on 𝐶_𝑎,𝑏[0,𝑇]. Let 𝑞≠0 be a real number and let {𝜆_𝑛}^∞_𝑛=1 be a sequence of complex numbers in ℂ₊ such that 𝜆_𝑛→−𝑖𝑞. If the following limit exists, one calls it the sequential 𝒫-function space integral of 𝐹 with parameter 𝑞, and we write ^𝑃𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹(𝑥)𝑑^se𝑥=lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑊⁺_𝑛𝜆_𝑛;𝑥𝑑𝜇(𝑥). (32)
We also define the sequential 𝒩-function space integral of 𝐹 with parameter 𝑞 by the formula ^𝑁𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹(𝑥)𝑑^se𝑥=lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑊⁻_𝑛𝜆_𝑛;𝑥𝑑𝜇(𝑥) (33) if it exists.
Let 𝐹₁(𝑥)≡1 on 𝐶_𝑎,𝑏[0,𝑇] and let 𝐹₂(𝑥)=0,𝑥∈𝐶^′_𝑎,𝑏[0,𝑇],1,𝑥∈𝐶_𝑎,𝑏[0,𝑇]−𝐶^′_𝑎,𝑏[0,𝑇]. (34) Then, since 𝐶^′_𝑎,𝑏[0,𝑇] is a scale invariant null set, we have 𝐹₁≈𝐹₂. But by the definition of the sequential function space integrals, we see that ^𝑃𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹₁(𝑥)𝑑^se𝑥=^𝑁𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹₁(𝑥)𝑑^se𝑥=1,_{𝑃𝑞𝐶𝑎,𝑏[]0,𝑇}𝐹₂(𝑥)𝑑^se𝑥=^𝑁𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹₂(𝑥)𝑑^se𝑥=0. (35)
Given two complex-valued measurable functionals 𝐹 and 𝐺 on 𝐶_𝑎,𝑏[0,𝑇], we will write 𝐹≊𝐺 if 𝐹≈𝐺 and furthermore if 𝐹(𝑥)=𝐺(𝑥) for all 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇]. The relation ≊ is clearly an equivalence relation.

Definition 7. Let 𝐹 be a functional on 𝐶_𝑎,𝑏[0,𝑇]. If 𝐹 is 𝒢-continuous for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and every 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], then one says that 𝐹 is 𝒢-s-continuous.
The functionals discussed in Examples 3 and 4 above are 𝒢-s-continuous. Next we introduce a class of functionals which are 𝒢-s-continuous.
Let ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) be the space of complex-valued, countably additive (and hence finite) Borel measures on 𝐶^′_𝑎,𝑏[0,𝑇]. ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) is a Banach algebra under the total variation norm and with convolution as multiplication.

Definition 8. The Fresnel type class ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) is the space of functionals on 𝐶_𝑎,𝑏[0,𝑇] expressible in the form 𝐹(𝑥)≊_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤) (36) for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and every 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], where the associated measure 𝑓 is an element of ℳ(𝐶^′_𝑎,𝑏[0,𝑇]). More precisely, since we will identify functionals which coincide under the relation ≊ on 𝐶_𝑎,𝑏[0,𝑇], ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) can be regarded as the space of all ≊-equivalence classes of functionals of the form (36).
The Fresnel type class ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) is a Banach algebra with norm ‖𝐹‖=𝑓=_{𝐶′𝑎,𝑏[0,𝑇]}𝑑𝑓(𝑤). (37) In fact, the correspondence 𝑓↦𝐹 is injective, carries convolution into pointwise multiplication, and is a Banach algebra isomorphism where 𝑓 and 𝐹 are related by (36).

Proposition 9. Let 𝐹∈ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]). Then 𝐹 is 𝒢-s-continuous.

Proof. Let 𝐹 be given by (36), and for each 𝑛=1,2,…, let 𝑍_𝑛(𝑥) be given by (19). Then substituting 𝑍_𝑛(𝑥) for 𝑥, we have 𝐹𝑍_𝑛(𝑥)=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑍_𝑛(𝑥)^∼_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤). (38) By Example 4, the exponential in (38) approaches the exponential in equation (36) as 𝑛→∞. Note that for each 𝑛∈ℕ, the exponential in (38) is measurable in (𝑤,𝑥) on 𝐶^′_𝑎,𝑏[0,𝑇]×𝐶_𝑎,𝑏[0,𝑇]. Thus, by the bounded convergence theorem, we observe 𝐹(𝑍_𝑛(𝑥))→𝐹(𝑥) for s-a.e 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and every 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], and the proposition is proved.

The functional 𝐹₂ defined by equation (34) above is not 𝒢-s-continuous. Thus, 𝐹₂∉ℱ^∗(𝐶^′_𝑎,𝑏[0,𝑇]).

The following lemma plays a key role in the proofs of Propositions 12 and 13 below.

Lemma 10. For each 𝑛=1,2,…, let 𝑍_𝑛(𝑥) be given by (19) and let 𝑊⁺_𝑛 and 𝑊⁻_𝑛 be given by (29) and (30), respectively. Then for all 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], all 𝜆∈ℂ₊, and each 𝑛=1,2,…, the function space integrals exp𝑖_{𝐶𝑎,𝑏[0,𝑇]}𝑤,𝑍_𝑛(𝑥)^∼_{𝐶′𝑎,𝑏}𝑊⁺_𝑛(𝜆;𝑥)𝑑𝜇_{𝐶𝑎,𝑏[0,𝑇]}(𝑥),exp𝑖𝑤,𝑍_𝑛(𝑥)^∼_{𝐶′𝑎,𝑏}𝑊⁻_𝑛(𝜆;𝑥)𝑑𝜇(𝑥) (39) exist and are given by (40) and (41) below, respectively.

Proof. Using (19), (29), (23), the Fubini theorem, and (11), we obtain exp𝑖_{𝐶𝑎,𝑏[0,𝑇]}𝑤,𝑍_𝑛(𝑥)^∼_{𝐶′𝑎,𝑏}𝑊⁺_𝑛()(𝑥)=𝜆𝜆;𝑥𝑑𝜇2𝜋^𝑛/2_ℝ^𝑛−𝜆exp2^𝑛_𝑗=1𝑢²_𝑗+^𝑛_𝑗=1𝑖𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}+𝜆^1/2𝐴_𝑗𝑢_𝑗−12^𝑛_𝑗=1𝐴²_𝑗𝑑𝑢₁…𝑑𝑢_𝑛−1=exp2𝜆^𝑛_𝑗=1𝑤,𝑔_𝑗²_{𝐶′𝑎,𝑏}+𝑖𝜆^𝑛−1/2_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐴_𝑗. (40)
Also, using (19), (30), (23), the Fubini theorem, and (11), we obtain exp𝑖_{𝐶𝑎,𝑏[]0,𝑇}𝑤,𝑍_𝑛(𝑥𝐶)_′𝑎,𝑏𝑊⁻_𝑛()(𝑥)−1𝜆;𝑥𝑑𝜇=exp2𝜆^𝑛_𝑗=1𝑤,𝑔_𝑗²_{𝐶′𝑎,𝑏}−𝑖(−𝜆)^𝑛−1/2_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐴_𝑗. (41)

Let 𝐹 be given by (36). Proceeding formally using (36), (40), and (41), we see that the sequential function space integrals ∫^𝑃𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^se𝑥 and ∫^𝑁𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^se𝑥 are given by the formulas ^𝑃𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹(𝑥)𝑑^se𝑥=_{𝐶′𝑎,𝑏[0,𝑇]}−𝑖exp‖𝑤‖2𝑞^2𝐶′_𝑎,𝑏+𝑖−𝑖𝑞^−1/2(𝑤,𝑎)_{𝐶′𝑎,𝑏}𝑑𝑓_{𝑁𝑞𝐶𝑎,𝑏[]0,𝑇}(𝑤),𝐹(𝑥)𝑑^se𝑥=_{𝐶′𝑎,𝑏[0,𝑇]}−𝑖exp‖𝑤‖2𝑞^2𝐶′_𝑎,𝑏−𝑖𝑖𝑞^−1/2(𝑤,𝑎)_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤), (42) respectively.

For 𝜆∈ℂ₊ and 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇], let 𝜓_𝑃()−1𝜆;𝑤≡exp‖𝑤‖2𝜆^2𝐶′_𝑎,𝑏+𝑖𝜆^−1/2(𝑤,𝑎)_{𝐶′𝑎,𝑏}𝜓_𝑁,()−1𝜆;𝑤≡exp‖𝑤‖2𝜆^2𝐶′_𝑎,𝑏−𝑖(−𝜆)^−1/2(𝑤,𝑎)_{𝐶′𝑎,𝑏}. (43)(44) Then, for each 𝜆∈ℂ₊, |𝜓_𝑃(𝜆;𝑤)| and |𝜓_𝑁(𝜆;𝑤)| are unbounded functions of 𝑤. Hence, ∫^𝑃𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^se𝑥 and ∫^𝑁𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^se𝑥 might not exist. From this observation, we clearly need to impose additional restrictions, such as (45) below, on the functionals 𝐹 in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) for the existence of the sequential 𝒫 and 𝒩-function space integrals of 𝐹.

Let 𝑞₀ be a positive real number. We define a subclass ℱ^∗_𝑞0≡ℱ^∗_𝑞0(𝐶_𝑎,𝑏[0,𝑇]) of ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) by 𝐹∈ℱ^∗_𝑞0 if and only if _{𝐶′𝑎,𝑏[0,𝑇]}1exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤)<+∞, (45) where 𝑓 and 𝐹 are related by (36).

The following example suggests the necessity of the condition (45) to ensure the existence of the sequential function space integrals of functionals in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]).

For each 𝑛∈ℕ, let 𝑤_𝑛(𝑡)=𝑛𝑏(𝑡)=∫^𝑡₀𝑛𝑑𝑏(𝑠) for 𝑡∈[0,𝑇]. Consider a measure 𝜂 on 𝐶^′_𝑎,𝑏[0,𝑇] which is concentrated on the set {𝑤_𝑛∶𝑛∈ℕ} and 𝜂({𝑤_𝑛})=1/(𝑛²) for each 𝑛∈ℕ. Then, 𝜂 is an element of ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) and it follows that _{𝐶′𝑎,𝑏[0,𝑇]}1exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}𝑑𝜂𝑤)=_∞𝑛=1(1exp2𝑞₀𝑤_{𝑛𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}1𝑛²=^∞_𝑛=1exp𝑏(𝑇)‖𝑎‖_{𝐶′𝑎,𝑏}2𝑞₀𝑛1𝑛²≥^∞_𝑛=1𝑏(𝑇)‖𝑎‖_{𝐶′𝑎,𝑏}2𝑞₀1𝑛=+∞. (46) Using the same method, we can find an example for the functional in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) that the sequential integrals do not exist.

Given a positive real number 𝑞₀, let Γ_𝑞0=𝜆∈ℂ₊∣𝜆=𝛼+𝑖𝛽,Im𝜆^−1/2<12𝑞₀. (47) Let 𝑞 be a real number with |𝑞|>𝑞₀. Then, since −𝑖𝑞^−1/2=12𝑞𝑞+𝑖sign2𝑞,𝑖𝑞^−1/2=12𝑞𝑞−𝑖sign2𝑞, (48) we can see that −𝑖𝑞 and 𝑖𝑞 are elements of the domain Int(Γ_𝑞0).

We also need the following lemma to obtain Propositions 12 and 13.

Lemma 11. Let 𝒢≡{𝑔_𝑗}^∞_𝑗=1 and 𝐴_𝑗 be as in Section 2, and let 𝑞₀ be a positive real number. Let {𝜆_𝑛}^∞_𝑛=1 be a sequence in ℂ₊ such that 𝜆→−𝑖𝑞, where 𝑞 is a real number with |𝑞|>𝑞₀. Then there exists a sufficiently large 𝐾∈ℕ such that for all 𝑛≥𝐾, expIm𝜆_𝑛^−1/2𝑛_𝑗=1𝑔_𝑗,𝑤_{𝐶′𝑎,𝑏}𝐴_𝑗1<exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏},expIm−𝜆_𝑛^−1/2𝑛_𝑗=1𝑔_𝑗,𝑤_{𝐶′𝑎,𝑏}𝐴_𝑗1<exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}. (49)(50)

Proof. Let {𝜆_𝑛}^∞_𝑛=1={𝛼_𝑛+𝑖𝛽_𝑛}^∞_𝑛=1 be a sequence in ℂ₊ such that 𝜆_𝑛→−𝑖𝑞, and let Γ_𝑞0 be given by (47). Then, we observe that for each 𝑛∈ℕ, Im𝜆_𝑛^−1/2=−sign𝛽_𝑛𝛼²_𝑛+𝛽²_𝑛−𝛼_𝑛2𝛼²_𝑛+𝛽²_𝑛,Im−𝜆_𝑛^−1/2=sign𝛽_𝑛𝛼²_𝑛+𝛽²_𝑛+𝛼_𝑛2𝛼²_𝑛+𝛽²_𝑛. (51) Since 𝛼_𝑛→0, there exists a sufficiently large 𝐾₁∈ℕ such that for every 𝑛≥𝐾₁, 𝜆_𝑛∈Int(Γ_𝑞0) and 𝛿𝑞=supIm𝜆_𝑛^−1/2∶𝑛≥𝐾₁∪Im−𝜆_𝑛^−1/2∶𝑛≥𝐾₁∪12𝑞<12𝑞₀. (52) Also, there exists a positive real number 𝜀₀>1 such that 𝛿(𝑞)<1/(𝜀₀2𝑞₀).
Let 𝒢≡{𝑔_𝑗}^∞_𝑗=1 and 𝐴_𝑗 be as in Section 2 above. Using Parseval’s identity, we observe (𝑤,𝑎)_{𝐶′𝑎,𝑏}=^∞_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝑔_𝑗,𝑎_{𝐶′𝑎,𝑏}=^∞_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐴_𝑗 (53) for 𝑤∈𝐶^′_𝑎,𝑏[0,𝑇]. Also, using the Cauchy-Schwartz inequality, we have −𝜀₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}<(𝑤,𝑎)_{𝐶′𝑎,𝑏}<𝜀₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}. (54) Thus, there exists a sufficiently large 𝐾₂∈ℕ such that for every 𝑛≥𝐾₂, ^𝑛_𝑗=1𝑔_𝑗,𝑤_{𝐶′𝑎,𝑏}𝐴_𝑗<𝜀₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}. (55)
Using these facts, we obtain the inequality (49) and (50).

Proposition 12. Let 𝑞₀ be a positive real number and let 𝐹∈ℱ^∗_𝑞0 be given by (36). Then for all real 𝑞 with |𝑞|>𝑞₀, the sequential 𝒫-function space integral ∫^𝑃𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^se𝑥 exists and is given by the formula ^𝑃𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹(𝑥)𝑑^se𝑥=_{𝐶′𝑎,𝑏[0,𝑇]}𝜓_𝑃−𝑖𝑞;𝑤𝑑𝑓(𝑤), (56) where 𝜓_𝑃 is given by (43) above.

Proof. Let 𝐹 be given by (36). First note that the equality in (36) holds for all 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇]. Let 𝑞 be a real number with |𝑞|>𝑞₀, and let {𝜆_𝑛}^∞_𝑛=1 be a sequence in ℂ₊ which converges to −𝑖𝑞. Using (36), (19), (29), and the Fubini theorem, we obtain that for each 𝑛=1,2,…, _{𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑊⁺_𝑛𝜆_𝑛(𝑥)=;𝑥𝑑𝜇_{𝐶′𝑎,𝑏[0,𝑇]𝐶𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑍_𝑛(𝑥)^∼_{𝐶′𝑎,𝑏}×𝑊⁺_𝑛𝜆_𝑛;𝑥𝑑𝜇(𝑥)𝑑𝑓(𝑤). (57) But, by (40) and (49), we know that the right hand side of (57) is dominated by (45) for all but a finite number of values of 𝑛. Next, using (57), (40) with 𝜆 replaced with 𝜆_𝑛, the dominated convergence theorem, Parseval’s relation, and (43), we obtain lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑍_𝑛(𝑥)𝑊⁺_𝑛𝜆_𝑛(𝑥)=;𝑥𝑑𝜇_{𝐶′𝑎,𝑏[0,𝑇]}lim_𝑛→∞−1exp2𝜆_𝑛^𝑛_𝑗=1𝑤,𝑔_𝑗²_{𝐶′𝑎,𝑏}+𝑖𝜆_𝑛^𝑛−1/2_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐴_𝑗(𝑤)=𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}−𝑖exp‖𝑤‖2𝑞^2𝐶′_𝑎,𝑏+𝑖−𝑖𝑞^−1/2(𝑤,𝑎)_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤), (58) which concludes the proof of Proposition 12.

We establish our next proposition after careful examination of the proof of Proposition 12, and by using (30), (41), (44), and (50) instead of (29), (40), (43), and (49), respectively.

Proposition 13. Let 𝑞₀ and 𝐹∈ℱ^∗_𝑞0 be as in Proposition 12. Then for all real 𝑞 with |𝑞|>𝑞₀, the sequential 𝒩-function space integral ∫^𝑁𝑞_{𝐶𝑎,𝑏[0,𝑇]}𝐹(𝑥)𝑑^𝑠𝑒𝑥 exists and is given by the formula ^𝑁𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹(𝑥)𝑑^se𝑥=_{𝐶′𝑎,𝑏[0,𝑇]}𝜓_𝑁−𝑖𝑞;𝑤𝑑𝑓(𝑤), (59) where 𝜓_𝑁 is given by (44).

4. Sequential Function Space Transforms

In this section, we introduce two sequential transforms on the function space 𝐶_𝑎,𝑏[0,𝑇]. We then establish that each of these transforms acts like an inverse transform of the other transform. Our definitions of the sequential transforms are based on the sequential function space integrals defined in Section 3 above.

Definition 14. Let 𝑞 be a nonzero real number. For 𝑦∈𝐶_𝑎,𝑏[0,𝑇], we define the sequential 𝒫-function space transform 𝒫_𝑞(𝐹) of 𝐹 with parameter 𝑞 by the formula 𝒫_𝑞(𝐹)𝑦=^𝑃𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹𝑦+𝑥𝑑^se𝑥 (60) if it exists. Also, we define the sequential 𝒩-function space transform 𝒩_𝑞(𝐹) of 𝐹 with parameter 𝑞 by the formula 𝒩_𝑞(𝐹)𝑦=^𝑁𝑞_{𝐶𝑎,𝑏[]0,𝑇}𝐹𝑦+𝑥𝑑^se𝑥 (61) if it exists.
In Theorem 15 below, we establish the existence of the sequential 𝒫-function space transform of functionals in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]).

Theorem 15. Let 𝑞₀ and 𝐹∈ℱ^∗_𝑞0 be as in Proposition 12. Then for all real 𝑞 with |𝑞|>𝑞₀, the sequential 𝒫-function space transform of 𝐹, 𝒫_𝑞(𝐹) exists and is an element of ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) with associated measure 𝑓^𝑃_𝑞 defined by 𝑓^𝑃_𝑞(𝐸)=_𝐸𝜓_𝑃−𝑖𝑞;𝑤𝑑𝑓(𝑤),𝐸∈ℬ𝐶^′_𝑎,𝑏[0,𝑇], (62) where 𝜓_𝑃 is given by (43), ℬ(𝐶^′_𝑎,𝑏[0,𝑇]) is the Borel 𝜎-algebra, and 𝑓 is the associated measure of 𝐹 by (36). Furthermore, one sees that for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇], 𝒫_𝑞(𝐹)𝑦=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}𝜓_𝑃(𝑤)=−𝑖𝑞;𝑤𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}𝑑𝑓^𝑃_𝑞(𝑤), (63) with 𝑓^𝑃_𝑞<_{𝐶′𝑎,𝑏[0,𝑇]}1exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤)<+∞. (64)

Proof. Let 𝐹 be given by (36), and for a real 𝑞 with |𝑞|>𝑞₀, let {𝜆_𝑛}^∞_𝑛=1 be a sequence in ℂ₊ which converges to −𝑖𝑞. Proceeding as in the proof of Proposition 12, we obtain that for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇], 𝒫_𝑞(𝐹)𝑦=lim_{𝑛→∞𝐶𝑎,𝑏[0,𝑇]}𝐹𝑦+𝑍_𝑛(𝑥)𝑊⁺_𝑛𝜆_𝑛(𝑥)=;𝑥𝑑𝜇_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}×lim_𝑛→∞−1exp2𝜆_𝑛^𝑛_𝑗=1𝑤,𝑔_𝑗²_{𝐶′𝑎,𝑏}+𝑖𝜆_𝑛^𝑛−1/2_𝑗=1𝑤,𝑔_{𝑗𝐶′𝑎,𝑏}𝐴_𝑗𝑑𝑓(𝑤)=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}𝜓_𝑃−𝑖𝑞;𝑤𝑑𝑓(𝑤). (65)
Given 𝑞 with |𝑞|>𝑞₀, let the set function 𝑓^𝑃_𝑞∶ℬ(𝐶^′_𝑎,𝑏[0,𝑇])→ℂ be given by (62). Then, by the definition of the class ℱ^∗_𝑞0, the inequality (64) holds. From this, we see that 𝑓^𝑃_𝑞 belongs to ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) and that 𝒫_𝑞(𝐹) can be expressed as the third expression of (63) for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇]. Thus, we conclude the proof of Theorem 15.

In Theorem 16 below, we also establish the existence of the sequential 𝒩-function space transform of functionals in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]).

Theorem 16. Let 𝑞₀ and 𝐹∈ℱ^∗_𝑞0 be as in Proposition 12. Then for all real 𝑞 with |𝑞|>𝑞₀, the sequential 𝒩-function space transform of 𝐹, 𝒩_𝑞(𝐹) exists and is an element of ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) with associated measure 𝑓^𝑁_𝑞 defined by 𝑓^𝑁_𝑞(𝐸)=_𝐸𝜓_𝑁−𝑖𝑞;𝑤𝑑𝑓(𝑤),𝐸∈ℬ𝐶^′_𝑎,𝑏[0,𝑇], (66) where 𝜓_𝑁 is given by (44) and 𝑓 is the associated measure of 𝐹 by (36). Furthermore, one sees that for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇], 𝒩_𝑞(𝐹)𝑦=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}𝜓_𝑁(𝑤)=−𝑖𝑞;𝑤𝑑𝑓_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖𝑤,𝑦^∼_{𝐶′𝑎,𝑏}𝑑𝑓^𝑁_𝑞(𝑤) (67) with 𝑓^𝑁_𝑞<_{𝐶′𝑎,𝑏[0,𝑇]}1exp2𝑞₀‖𝑤‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤)<+∞. (68)

In view of Theorems 15 and 16, we obtain the following assertion.

Theorem 17. Let 𝑞₀ be a positive real number and let 𝐹∈ℱ^∗_𝑞0/4 be given by (36). Then(i)for all real 𝑞 with |𝑞|>𝑞₀, 𝒫_𝑞(𝐹) and 𝒫_−𝑞(𝐹) are elements of ℱ^∗_𝑞0, and one has 𝒩_−𝑞𝒫_𝑞(𝐹)≊𝐹≊𝒩_𝑞𝒫_−𝑞(𝐹); (69) and(ii)for all real 𝑞 with |𝑞|>𝑞₀, 𝒩_𝑞(𝐹) and 𝒩_−𝑞(𝐹) are elements of ℱ^∗_𝑞0, and one has 𝒫_−𝑞𝒩_𝑞(𝐹)≊𝐹≊𝒫_𝑞𝒩_−𝑞(𝐹). (70)

We finish this section with some examples to apply our results. Let ℳ(ℝ) be the class of complex-valued, countably additive Borel measures on ℬ(ℝ), the Borel class of ℝ. For 𝜏∈ℳ(ℝ), the Fourier transform 𝜏 of 𝜏 is a complex-valued function defined on ℝ by the formula 𝜏(𝑢)=_ℝexp{𝑖𝑢𝑣}𝑑𝜏(𝑣). (71)

Let ℎ∈𝐶^′_𝑎,𝑏[0,𝑇] and let 𝜏∈ℳ(ℝ). Define 𝐹_ℎ,𝜏∶𝐶_𝑎,𝑏[0,𝑇]→ℂ by 𝐹_ℎ,𝜏(𝑥)=𝜏(ℎ,𝑥)^∼_{𝐶′𝑎,𝑏}. (72) Define a function 𝜙∶ℝ→𝐶^′_𝑎,𝑏[0,𝑇] by 𝜙(𝑣)=𝑣ℎ and let 𝑓=𝜏∘𝜙⁻¹. It is quite clear that 𝑓 is in ℳ(𝐶^′_𝑎,𝑏[0,𝑇]) and is supported by [ℎ], the subspace of 𝐶^′_𝑎,𝑏[0,𝑇] spanned by {ℎ}. Now, for s-a.e. 𝑥∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑥∈𝐶^′_𝑎,𝑏[0,𝑇], 𝐹_ℎ,𝜏(𝑥)=_ℝexp𝑖(ℎ,𝑥)^∼_{𝐶′𝑎,𝑏}𝑣(𝑣)=𝑑𝜏_ℝexp𝑖𝜙(𝑣),𝑥^∼_{𝐶′𝑎,𝑏}(𝑣)=𝑑𝜏_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}𝑑𝜏∘𝜙⁻¹(𝑤)=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}𝑑𝑓(𝑤). (73) Thus, 𝐹_ℎ,𝜏 is an element of ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]).

Suppose that for a fixed positive real number 𝑞₀>0, _ℝexp2𝑞₀^−1/2‖ℎ‖_{𝐶′𝑎,𝑏}‖𝑎‖_{𝐶′𝑎,𝑏}|𝑣|𝑑|𝜏|(𝑣)<+∞. (74) It is easy to show that condition (74) is equivalent to condition (45) above. Thus, applying Theorems 15 and 16, the sequential function space transforms 𝒫_𝑞(𝐹_ℎ,𝜏) and 𝒩_𝑞(𝐹_ℎ,𝜏) of 𝐹_ℎ,𝜏 exist for all 𝑞∈ℝ with |𝑞|>𝑞₀ if the complex measure 𝜏 corresponding to 𝐹_ℎ,𝜏 satisfies condition (74).

Next, we present functionals 𝐹_ℎ,𝜏 in ℱ^∗(𝐶_𝑎,𝑏[0,𝑇]) whose associated measures satisfy the condition (74).

Example 18. Let 𝑆∶𝐶^′_𝑎,𝑏[0,𝑇]→𝐶^′_𝑎,𝑏[0,𝑇] be the linear operator defined by 𝑆𝑤(𝑡)=∫^𝑡₀𝑤(𝑠)𝑑𝑏(𝑠). Then, the adjoint operator 𝑆^∗ of 𝑆 is given by 𝑆^∗𝑤(𝑡)=∫^𝑡₀(𝑤(𝑇)−𝑤(𝑠))𝑑𝑏(𝑠). Using an integration by parts formula, we have 𝑆^∗𝑏,𝑥^∼_{𝐶′𝑎,𝑏}=^𝑇₀𝑥(𝑡)𝑑𝑏(𝑡). (75)
Given real numbers 𝑚 and 𝜎² with 𝜎²>0, let 𝜏_𝑚,𝜎2 be the Gaussian measure given by 𝜏_𝑚,𝜎2(𝐵)=2𝜋𝜎^2−1/2_𝐵−()exp𝑣−𝑚²2𝜎²(ℝ).𝑑𝑣,𝐵∈ℬ (76) Then 𝜏_𝑚,𝜎2∈ℳ(ℝ) and 𝜏_𝑚,𝜎2(𝑢)=_ℝexp{𝑖𝑢𝑣}𝑑𝜏_𝑚,𝜎2(𝑣)−1=exp2𝜎²𝑢²+𝑖𝑚𝑢,𝑢∈ℝ. (77) The complex measure 𝜏_𝑚,𝜎2 given by (76) satisfies condition (74) for all real 𝑞₀>0. Thus, we can apply the results in Section 4 to the functional of the form 𝐹_{ℎ,𝜏2𝑚,𝜎}(𝑥)=𝜏_𝑚,𝜎2(ℎ,𝑥)^∼_{𝐶′𝑎,𝑏}−1=exp2𝜎²(ℎ,𝑥)^∼_{𝐶′𝑎,𝑏}²+𝑖𝑚(ℎ,𝑥)^∼_{𝐶′𝑎,𝑏}. (78)
We note that 𝐹_{ℎ,𝜏2𝑚,𝜎}∈∪_𝑞0>0ℱ^∗_𝑞0 and so that for every 𝑞∈ℝ−{0}, the sequential transforms of 𝐹_{ℎ,𝜏2𝑚,𝜎}, 𝒫_𝑞(𝐹_{ℎ,𝜏2𝑚,𝜎}), and 𝒩_𝑞(𝐹_{ℎ,𝜏2𝑚,𝜎}) exist and are elements of ℱ(𝐶_𝑎,𝑏[0,𝑇]) by Theorems 15 and 16 above. Furthermore, we have that 𝒩_−𝑞𝒫_𝑞𝐹_{ℎ,𝜏2𝑚,𝜎}≊𝐹_{ℎ,𝜏2𝑚,𝜎} (79) for all real 𝑞∈ℝ−{0} by Theorem 17. In fact, by a simple calculation, we obtain that for all 𝑞∈ℝ−{0}, 𝒫_𝑞𝐹_{ℎ,𝜏2𝑚,𝜎}𝑦=_ℝexp𝑖ℎ,𝑦^∼_{𝐶′𝑎,𝑏}𝑖𝑣−‖ℎ‖2𝑞^2𝐶′_𝑎,𝑏𝑣²+𝑖−𝑖𝑞^−1/2(ℎ,𝑎)_{𝐶′𝑎,𝑏}𝑣𝑑𝜏_𝑚,𝜎2(𝑣),𝒩_−𝑞𝐹_{ℎ,𝜏2𝑚,𝜎}𝑦=_ℝexp𝑖ℎ,𝑦^∼_{𝐶′𝑎,𝑏}𝑖𝑣+‖ℎ‖2𝑞^2𝐶′_𝑎,𝑏𝑣²−𝑖−𝑖𝑞^−1/2(ℎ,𝑎)_{𝐶′𝑎,𝑏}𝑣𝑑𝜏_𝑚,𝜎2(𝑣) (80) for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇].
In particular, if we choose ℎ=𝑆^∗𝑏, 𝑚=0, and 𝜎²=2 in (78), we have 𝐹_𝑆^∗_𝑏,𝜏0,2(𝑥)=exp−^𝑇₀𝑥(𝑡)𝑑𝑏(𝑡)². (81)

Example 19. The functional 𝐹_{𝛿𝑆∗𝑏}(𝑥)=exp𝑖^𝑇₀𝑥(𝑡)𝑑𝑏(𝑡),𝑥∈𝐶_𝑎,𝑏[0,𝑇] (82) also is a functional under our consideration because 𝐹_{𝛿𝑆∗𝑏}(𝑥)=exp𝑖𝑆^∗𝑏,𝑥^∼_{𝐶′𝑎,𝑏}=_{𝐶′𝑎,𝑏[0,𝑇]}exp𝑖(𝑤,𝑥)^∼_{𝐶′𝑎,𝑏}𝑑𝛿_𝑆^∗_𝑏(𝑤), (83) where 𝛿_{𝛿𝑆∗𝑏} is the Dirac measure concentrated at 𝑆^∗𝑏 in 𝐶^′_𝑎,𝑏[0,𝑇]. The Dirac measure 𝛿_{𝛿𝑆∗𝑏} also satisfies condition (45) for all real 𝑞₀>0; that is, 𝐹_{𝛿𝑆∗𝑏}∈∪_𝑞0>0ℱ^∗_𝑞0.

5. Concluding Remarks

In this section, we examine certain aspects among the generalized analytic Fourier-Feynman transform [12, 13], the generalized integral transform [15, 16], and the sequential transforms for functionals on 𝐶_𝑎,𝑏[0,𝑇]. As possible, we adopt the definitions and notation of [11–13, 15, 16] for the (generalized) analytic Fourier-Feynman transform and the (generalized) integral transform.

A major goal of the authors in [12, 13, 15, 16] was to generalize the concepts of the analytic Fourier-Feynman transform and the integral transform of the functionals of paths for the generalized Brownian motion process.

In [5, 6], the authors obtained the existence of the 𝐿₂ analytic Fourier-Feynman transform 𝑇_𝑞⁽²⁾(𝐹) for several large classes of functionals 𝐹 on Wiener space 𝐶₀[0,𝑇]. In particular, they showed that for all real 𝑞≠0, 𝑇⁽²⁾_−𝑞𝑇_𝑞⁽²⁾(𝐹)𝑦=𝐹𝑦 (84) for s-a.e. 𝑦∈𝐶₀[0,𝑇]. Thus, 𝐿₂ analytic Fourier-Feynman transform “𝑇_𝑞⁽²⁾” with parameter 𝑞 has the inverse transform “𝑇⁽²⁾_−𝑞”. However, in view of the results in [12, 13], (84) does not hold for functionals 𝐹 on 𝐶_𝑎,𝑏[0,𝑇]. As mentioned in Section 1, the existence of an inverse generalized Fourier-Feynman transform on 𝐶_𝑎,𝑏[0,𝑇] has not yet been established. It is not easy to verify the existence of the inverse generalized Fourier-Feynman transform because the generalized Brownian motion process has a drift term 𝑎(𝑡).

In [9, 10], the authors studied the integral transform of functionals 𝐹 in 𝐿₂(𝐶₀[0,𝑇]). They showed that for 𝐹∈𝐿₂(𝐶₀[0,𝑇]) and nonzero complex numbers 𝛼 and 𝛽 with |𝛽|≤1, 𝛽≠±1, Re(1−𝛽²)>0, 𝛼=1−𝛽², and −𝜋/4<arg(𝛼)<𝜋/4, ℱ_{𝛼′,1/𝛽}ℱ_𝛼,𝛽𝐹𝑦=𝐹𝑦,𝑦∈𝐶₀[0,𝑇], (85) where 𝛼^′=1−1/𝛽². That is to say, “ℱ⁻¹_𝛼,𝛽” is given by “ℱ_{𝑖𝛼/𝛽,1/𝛽}”. In [16], Chang et al. presented a version of inverse transform of the generalized integral transform ℱ_𝛼,𝛽 as follows: for appropriate functionals 𝐹 on 𝐾_𝑎,𝑏[0,𝑇], the complexification of 𝐶_𝑎,𝑏[0,𝑇], ℱ_{−𝑖𝛼,1}ℱ_𝑖𝛼,1ℱ_{−𝛼/𝛽,1/𝛽}ℱ_𝛼,𝛽𝐹𝑦=𝐹𝑦 (86) for 𝑦∈𝐾_𝑎,𝑏[0,𝑇]; that is, ℱ⁻¹_𝛼,𝛽=ℱ_{−𝑖𝛼,1}ℱ_𝑖𝛼,1ℱ_{−𝛼/𝛽,1/𝛽}. (87)

But they pointed out that for any nonzero complex numbers 𝛼₁,𝛼₂,𝛽₁ and 𝛽₂, there are no nonzero complex numbers 𝛼^′, and 𝛽^′ such that ℱ_𝛼1,𝛽1ℱ_𝛼2,𝛽2=ℱ_{𝛼′,𝛽′}. (88) Hence, the inverse generalized integral transform, ℱ⁻¹_𝛼,𝛽, cannot be expressed as a simple integral transform.

Let us return to the singular integral transforms discussed in this paper. Theorem 17 tells us that the sequential function space transforms 𝒫_𝑞 and 𝒩_𝑞 have inverse transforms 𝒫_𝑞⁻¹=𝒩_−𝑞 and 𝒩_𝑞⁻¹=𝒫_−𝑞, respectively. By choosing 𝑎(𝑡)≡0 and 𝑏(𝑡)=𝑡 on [0,𝑇], the function space 𝐶_𝑎,𝑏[0,𝑇] reduces to the Wiener space 𝐶₀[0,𝑇] and the sequential transforms 𝒫_𝑞 and 𝒩_𝑞 coincide.

Let 𝐹 be as in Theorem 15. Then, by the definition of the 𝐿₂ generalized analytic Fourier-Feynman transform, we can see that 𝑇_𝑞⁽²⁾(𝐹) of 𝐹 exists and 𝑇_𝑞⁽²⁾(𝐹)𝑦=𝒫_𝑞(𝐹)𝑦 (89) for s-a.e. 𝑦∈𝐶_𝑎,𝑏[0,𝑇] and all 𝑦∈𝐶^′_𝑎,𝑏[0,𝑇]. This fact tells us that the sequential 𝒩-function space transform 𝒩_−𝑞 plays a prominent role as an inverse transform of the generalized analytic Fourier-Feynman transform 𝑇_𝑞⁽²⁾.

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions which have improved the original paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2011-0014552) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2012R1A1A1004774). The authors declare that there is no conflict of interests regarding the publication of this paper.