We first investigate a rotation property of Wiener measure on the product of Wiener spaces. Next, using the concept of the generalized analytic Feynman integral, we define a generalized Fourier-Feynman transform and a generalized convolution product for functionals on Wiener space. We then proceed to establish a fundamental result involving the generalized transform and the generalized convolution product.

1. Introduction

Let 𝐶0[0,𝑇] denote one-parameter Wiener space, that is, the space of all real-valued continuous functions 𝑥 on [0,𝑇] with 𝑥(0)=0. Let ℳ denote the class of all Wiener measurable subsets of 𝐶0[0,𝑇], and let 𝑚 denote Wiener measure. Then (𝐶0[0,𝑇],ℳ,𝑚) is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional 𝐹 by 𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥).(1.1)

In [1], Bearman gave a significant theorem for Wiener integral on product Wiener space. It can be summarized as follows.

Theorem 1.1 (Bearman's Rotation Theorem). Let 𝐺(𝑤,𝑧) be an 𝑚×𝑚-integrable functional on 𝐶20[0,𝑇], the product of 2 copies of 𝐶0[0,𝑇], and let 𝜃 be a function of bounded variation on [0,𝑇]. Let 𝑇𝜃∶𝐶20[0,𝑇]→𝐶20[0,𝑇] be the transformation defined by 𝑇𝜃(𝑤,𝑧)=(ğ‘¤î…ž,ğ‘§î…ž) with ğ‘¤î…ž(𝑡)=𝑡0cos𝜃(𝑠)𝑑𝑤(𝑠)−𝑡0𝑧sin𝜃(𝑠)𝑑𝑧(𝑠),(𝑡)=𝑡0sin𝜃(𝑠)𝑑𝑤(𝑠)+𝑡0cos𝜃(𝑠)𝑑𝑧(𝑠).(1.2) Then the transform 𝑇𝜃 is measure preserving and 𝐶20[0,𝑇]𝐺(𝑤,𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶20[0,𝑇]𝐺𝑇𝜃(𝑤,𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧).(1.3)

As a special case of Theorem 1.1, one can obtain the following corollary.

Corollary 1.2. Let 𝐹 be Wiener integrable on 𝐶0[0,𝑇]. Then for any 𝜃∈ℝ, 𝐹(𝑤sin𝜃+𝑧cos𝜃) is integrable on 𝐶20[0,𝑇] and 𝐶20[0,𝑇]𝐹(𝑤sin𝜃+𝑧cos𝜃)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥).(1.4)

The following more general case of Corollary 1.2 is due to Cameron and Storvick [2]. But we state the theorem with some assumption for our research.

Theorem 1.3. Let 𝐹 be Wiener measurable on 𝐶0[0,𝑇]. Assume that for any 𝜌>0, 𝐹(𝜌⋅) is Winer integrable. Then for any ğ‘Ž,𝑏∈ℝ, 𝐹(ğ‘Žğ‘¤+𝑏𝑧) is integrable on 𝐶20[0,𝑇] and 𝐶20[0,𝑇]𝐹(ğ‘Žğ‘¤+𝑏𝑧)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]ğ¹î‚€âˆšğ‘Ž2+𝑏2𝑥𝑑𝑚(𝑥).(1.5)

In many papers, Theorem 1.3 is used to study relationships between analytic Fourier-Feynman transforms and convolution products of Feynman integrable functionals on Wiener space, see for instance [3–6]. In this paper, we will extend the result in Theorem 1.3 to a more general case for functionals of Gaussian processes given by (2.2) below. We then apply our rotation property of Wiener measure to establish a fundamental relationship between the generalized Fourier-Feynman transform and the generalized convolution product.

2. A Rotation on Wiener Space

The most important concepts we will employ in the statements and proofs of our results are the concepts of the scale-invariant measurability and the Paley-Wiener-Zygmund stochastic integral [7].

A subset 𝐵 of 𝐶0[0,𝑇] is said to be scale-invariant measurable [8] provided 𝜌𝐵∈ℳ for all 𝜌>0, and a scale-invariant measurable set 𝑁 is said to be scale-invariant null provided 𝑚(𝜌𝑁)=0 for all 𝜌>0. A property that holds except on a scale-invariant null set is said to be hold scale-invariant almost everywhere (s-a.e.). If two functionals 𝐹 and 𝐺 are equal s-a.e., we write 𝐹≈𝐺.

Let {𝜙𝑛} be a complete orthonormal set in 𝐿2[0,𝑇], each of whose elements is of bounded variation on [0,𝑇]. Then for each 𝑣∈𝐿2[0,𝑇], the Paley-Wiener-Zygmund (PWZ) stochastic integral ⟨𝑣,𝑥⟩ is defined by the formula ⟨𝑣,𝑥⟩=limğ‘›â†’âˆžî€œğ‘‡0𝑛𝑗=1𝑣,𝜙𝑗2𝜙𝑗(𝑡)𝑑𝑥(𝑡)(2.1) for all 𝑥∈𝐶0[0,𝑇] for which the limit exists, where (⋅,⋅)2 denotes the 𝐿2-inner product.

It was shown in [7] that for each 𝑣∈𝐿2[0,𝑇], the limit defining the PWZ integral ⟨𝑣,𝑥⟩ exists for 𝑚-a.e. 𝑥∈𝐶0[0,𝑇] and that this limit is essentially independent of the choice of the complete orthonormal set {𝜙𝑛}. It was also shown in [7] that if 𝑣 is of bounded variation on [0,𝑇], then the PWZ integral ⟨𝑣,𝑥⟩ equals the Riemann-Stieltjes integral ∫𝑇0𝑣(𝑡)𝑑𝑥(𝑡) for 𝑚-a.e. 𝑥∈𝐶0[0,𝑇]. In fact, the integrals are equal for s-a.e. 𝑥∈𝐶0[0,𝑇] and that for all 𝑣∈𝐿2[0,𝑇], ⟨𝑣,𝑥⟩ is a Gaussian random variable with mean 0 and variance ‖𝑣‖22.

For any â„Žâˆˆğ¿2[0,𝑇] with ‖ℎ‖2>0, let ğ’µâ„Ž be the Gaussian process ğ’µâ„Ž(𝑥,𝑡)=𝑡0ℎ(𝑠)𝑑𝑥(𝑠)=⟨𝑣,𝑥⟩(2.2) introduced by Park and Skoug in [9] and used extensively since; see for example [5, 6, 10, 11]. Of course if ℎ(𝑡)≡1 on [0,𝑇], then ğ’µâ„Ž(𝑥,𝑡)=𝑥(𝑡).

It is easy to see that ğ’µâ„Ž is a Gaussian process with mean zero and covariance function 𝐶0[0,𝑇]ğ’µâ„Ž(𝑥,𝑠)ğ’µâ„Žî€œ(𝑥,𝑡)𝑑𝑚(𝑥)=0min{𝑠,𝑡}ℎ2(𝑢)𝑑𝑢.(2.3) In addition, ğ’µâ„Ž(⋅,𝑡) is stochastically continuous in 𝑡 on [0,𝑇], and for any ℎ1,ℎ2∈𝐿2[0,𝑇], 𝐶0[0,𝑇]ğ’µâ„Ž1(𝑥,𝑠)ğ’µâ„Ž2(𝑥,𝑡)𝑑𝑚(𝑥)=0min{𝑠,𝑡}ℎ1(𝑢)ℎ2(𝑢)𝑑𝑢.(2.4)

For any complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇] and for any 𝑛∈ℕ, define the projection map 𝒫𝑛 from 𝐿2[0,𝑇] into span{𝜙1,…,𝜙𝑛} by ğ’«ğ‘›â„Ž(𝑡)=𝑛𝑗=1ℎ,𝜙𝑗2𝜙𝑗(𝑡).(2.5) Then for â„Žâˆˆğ¿2[0,𝑇] and 𝑥∈𝐶0[0,𝑇], we see that ğ’µâ„Ž(𝑥,𝑡)=limğ‘›â†’âˆžî€œğ‘¡0ğ’«ğ‘›â„Ž(𝑠)𝑑𝑥(𝑠)=limğ‘›â†’âˆžğ’µğ’«ğ‘›â„Ž(𝑥,𝑡),(2.6) that is, ğ’µğ’«ğ‘›â„Ž(𝑥,𝑡) converges in 𝐿2(𝐶0[0,𝑇])-mean to ğ’µâ„Ž(𝑥,𝑡).

Throughout this paper, we will assume that each functional 𝐹∶𝐶0[0,𝑇]→ℂ we consider is scale-invariant measurable and that 𝐶0[0,𝑇]||ğ¹î€·ğ’µâ„Ž(||𝑥,⋅)𝑑𝑚(𝑥)<+∞(2.7) for all â„Žâˆˆğ¿2[0,𝑇].

We are now ready to state the main theorem of this paper.

Theorem 2.1. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any ℎ1,ℎ2∈𝐿2[0,𝑇], 𝐶20[0,𝑇]ğ¹î€·ğ’µâ„Ž1(𝑤,⋅)+ğ’µâ„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,⋅)𝑑𝑚(𝑥),(2.8) where ℎ1, ℎ2, and 𝑘 are related by 𝑘(𝑡)=âˆžî“ğ‘›=1ℎ1,𝜙𝑛22+ℎ2,𝜙𝑛22𝜙𝑛(𝑡)(2.9) for some complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇], each of those elements is of bounded variation on [0,𝑇].

3. Proof of the Main Theorem

We begin this section with three lemmas in order to establish (2.8).

Lemma 3.1. Let 𝐹 be a functional on 𝐶0[0,𝑇], and let 𝜙 be a function of bounded variation on [0,𝑇]. Then for all ğ‘Ž,𝑏∈ℝ, 𝐶20[0,𝑇]ğ¹î€·ğ‘Žğ’µğœ™(𝑤,⋅)+𝑏𝒵𝜙(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]ğ¹î‚€âˆšğ‘Ž2+𝑏2𝒵𝜙(𝑥,⋅)𝑑𝑚(𝑥).(3.1)

Proof. We first note that for each 𝑡∈[0,𝑇], ğ‘Žğ’µğœ™(𝑤,𝑡)+𝑏𝒵𝜙(𝑧,𝑡)=𝑡0𝜙(𝑠)𝑑(ğ‘Žğ‘¤(𝑠)+𝑏𝑧(𝑠))=𝒵𝜙(ğ‘Žğ‘¤+𝑏𝑧,𝑡).(3.2) We also note that 𝐹(𝒵(𝑥,⋅)) is Wiener integrable as a functional of 𝑥. Hence, by (1.5), we obtain that for all ğ‘Ž,𝑏∈ℝ, 𝐶20[0,𝑇]ğ¹î€·ğ‘Žğ’µğœ™(𝑤,⋅)+𝑏𝒵𝜙(=𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝒵𝜙=(ğ‘Žğ‘¤+𝑏𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]ğ¹î‚€ğ’µğœ™î‚€âˆšğ‘Ž2+𝑏2=𝑥,⋅𝑑𝑚(𝑥)𝐶0[0,𝑇]ğ¹î‚€âˆšğ‘Ž2+𝑏2𝒵𝜙(𝑥,⋅)𝑑𝑚(𝑥).(3.3) Thus (3.1) is established.

Lemma 3.2. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any ℎ1,ℎ2∈𝐿2[0,𝑇] and each 𝑛∈ℕ, 𝐶20[0,𝑇]ğ¹î€·ğ’µğ’«ğ‘›â„Ž1(𝑤,⋅)+ğ’µğ’«ğ‘›â„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝒫𝑛𝑘(𝑥,⋅)𝑑𝑚(𝑥),(3.4) where ℎ1, ℎ2, and 𝑘 are related by (2.9).

Proof. Since the addition is continuous in the uniform topology on 𝐶0[0,𝑇], we can apply (3.1) to the functional ∑𝐹(𝑛𝑗=1𝒵𝜙𝑗(𝑥,⋅)). Thus using (2.5) and (3.1), we have 𝐶20[0,𝑇]ğ¹î€·ğ’µğ’«ğ‘›â„Ž1(𝑤,⋅)+ğ’µğ’«ğ‘›â„Ž2(=𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝑛𝑗=1ℎ1,𝜙𝑗2ğ’µğœ™ğ‘—î€·â„Ž(𝑤,⋅)+2,𝜙𝑗2𝒵𝜙𝑗=(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]𝐹𝑛𝑗=1ℎ1,𝜙𝑗22+ℎ2,𝜙𝑗22𝒵𝜙𝑗=(𝑥,⋅)𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐹𝒵𝒫𝑛𝑘(𝑥,⋅)𝑑𝑚(𝑥).(3.5) Thus (3.4) is established.

Lemma 3.3. Let 𝐹 be bounded and continuous on 𝐶0[0,𝑇]. Then for any ℎ1,ℎ2∈𝐿2[0,𝑇], 𝐶20[0,𝑇]ğ¹î€·ğ’µâ„Ž1(𝑤,⋅)+ğ’µâ„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,⋅)𝑑𝑚(𝑥),(3.6) where ℎ1, ℎ2, and 𝑘 are related by (2.9) above.

Proof. We clearly see that 𝐹 is Wiener integrable. We also note that {ğ’«ğ‘›â„Ž} is a sequence of functions of bounded variation on [0,𝑇] such that ğ’«ğ‘›â„Ž converges to ℎ in the space 𝐿2[0,𝑇] as ğ‘›â†’âˆž. For each 𝑛∈ℕ and â„Žâˆˆğ¿2[0,𝑇], let 𝐹𝑛(ğ’µâ„Ž(𝑥,⋅))=𝐹(ğ’µğ’«ğ‘›â„Ž(𝑥,⋅)). Since ğ’µğ’«ğ‘›â„Ž converges to ğ’µâ„Ž uniformly and 𝐹 is continuous in the uniform topology, by (2.6), ğ¹î€·ğ’µâ„Žî€¸î‚µ(𝑥,⋅)=𝐹limğ‘›â†’âˆžğ’µğ’«ğ‘›â„Žî‚¶(𝑥,⋅)=limğ‘›â†’âˆžğ¹ğ‘›î€·ğ’µâ„Žî€¸(𝑥,⋅).(3.7) Since 𝐹 is bounded, by using the dominated convergence theorem and (3.4), we have 𝐶20[0,𝑇]ğ¹î€·ğ’µâ„Ž1(𝑤,⋅)+ğ’µâ„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=limğ‘›â†’âˆžî€œğ¶20[0,𝑇]ğ¹ğ‘›î€·ğ’µâ„Ž1(𝑤,⋅)+ğ’µâ„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=limğ‘›â†’âˆžî€œğ¶0[0,𝑇]𝐹𝑛𝒵𝑘=(𝑥,⋅)𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐹𝒵𝑘(𝑥,⋅)𝑑𝑚(𝑥),(3.8) which concludes the proof of Lemma 3.3.

We are now ready to prove our main theorem.

Proof of Theorem 2.1. Let 𝐹 be Wiener integrable. Suppose that the left-hand side of (2.8) exists. By usual arguments of integration theory, there exists a sequence {𝐹𝑛} of bounded and continuous functionals such that 𝐹𝑛 converges to 𝐹. By Lemma 3.3 and the dominated convergence theorem, we can obtain the desired result.

Corollary 3.4. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for all â„Žâˆˆğ¿2[0,𝑇] and all ğ‘Ž,𝑏∈ℝ, 𝐶20[0,𝑇]ğ¹î€·ğ‘Žğ’µâ„Ž(𝑤,⋅)+ğ‘ğ’µâ„Ž(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)=𝐶0[0,𝑇]ğ¹î‚€âˆšğ‘Ž2+𝑏2ğ’µâ„Ž(𝑥,⋅)𝑑𝑚(𝑥).(3.9)

Proof. Simply choose ℎ1=ğ‘Žâ„Ž and ℎ2=ğ‘â„Ž in (2.8) and use the linearity property of the PWZ stochastic integral.

Using similar arguments as in the proofs of Lemmas 3.1, 3.2, and 3.3 and Theorem 2.1 above, we can obtain the following theorems.

Theorem 3.5. Let 𝐹 be a functional on 𝐶0[0,𝑇], and let {ℎ1,…,â„Žğœˆ} be any subset of 𝐿2[0,𝑇]. Then 𝐶𝜈0[0,𝑇]𝐹𝜈𝑗=1ğ’µâ„Žğ‘—î€·ğ‘¥ğ‘—î€¸îƒª,⋅𝑑𝑚𝜈→𝑥=𝐶0[0,𝑇]𝐹𝒵𝑘𝜈(𝑥,⋅)𝑑𝑚(𝑥),(3.10) where 𝑚𝜈 is the product Wiener measure on 𝐶𝜈0[0,𝑇], the product of 𝜈 copies of 𝐶0[0,𝑇], and 𝑘𝜈(𝑡)=âˆžî“ğ‘›=1î„¶î„µî„µâŽ·ğœˆî“ğ‘—=1î€·â„Žğ‘—,𝜙𝑛22𝜙𝑛(𝑡)(3.11) for some complete orthonormal set {𝜙𝑛} in 𝐿2[0,𝑇].

Theorem 3.6. Let 𝐹 be a functional on 𝐶0[0,𝑇]. Then for any ℎ1 and ℎ2 in 𝐿2[0,𝑇], 𝐶20[0,𝑇]ğ¹î€·ğ’µâ„Ž2(𝑤,⋅)âˆ’ğ’µâ„Ž1(𝑧,⋅),ğ’µâ„Ž1(𝑤,⋅)+ğ’µâ„Ž2(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)∗=𝐶20[0,𝑇]𝐹𝒵𝑘(𝑤,⋅),𝒵𝑘(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧),(3.12) where ℎ1, ℎ2, and 𝑘 are related by (2.9).

Remark 3.7. Equations (2.8) and (3.12) are indeed very general formulas.(1)For any 𝜃∈ℝ, choosing ℎ1(𝑡)≡sin𝜃 and ℎ2(𝑡)≡cos𝜃 in (2.8) yields (1.4).(2)For any ğ‘Ž,𝑏∈ℝ, choosing ℎ1(𝑡)â‰¡ğ‘Ž and ℎ2(𝑡)≡𝑏 in (2.8) or choosing ℎ(𝑡)≡1 in (3.9) yields (1.5). (3)For any function of bounded variation 𝜃(⋅), choosing ℎ1(𝑡)=sin𝜃(𝑡) and ℎ2(𝑡)=cos𝜃(𝑡) on [0,𝑇] in (3.12) yields (1.3).

4. Generalized Fourier-Feynman Transform and Generalized Convolution Product

In this section, we will apply our main theorem to the generalized analytic Fourier-Feynman transform and the convolution product theories.

In defining various analytic Feynman integrals, one usually starts, for 𝜆>0, with the Wiener integral 𝐶0[0,𝑇]𝐹𝜆−1/2𝑥𝑑𝑚(𝑥)(4.1) and then extends analytically in 𝜆 to the right-half complex plane. Here we start with the (generalized) Wiener integral 𝐶0[0,𝑇]𝐹𝜆−1/2ğ’µâ„Ž(𝑥,⋅)𝑑𝑚(𝑥)=𝐽(ℎ;𝜆),(4.2) where ğ’µâ„Ž is the Gaussian process given by (2.2) above.

Throughout this section, let ℂ+ and ℂ+ denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.

Let 𝐹 be a complex-valued scale-invariant measurable functional on 𝐶0[0,𝑇] such that 𝐽(ℎ;𝜆) given by (4.2) exists and is finite for all 𝜆>0. If there exists a function 𝐽∗(ℎ;𝜆) analytic on ℂ+ such that 𝐽∗(ℎ;𝜆)=𝐽(ℎ;𝜆) for all 𝜆>0, then 𝐽∗(ℎ;𝜆) is defined to be the generalized analytic Wiener integral (with respect to the process ğ’µâ„Ž) of 𝐹 over 𝐶0[0,𝑇] with parameter 𝜆, and for 𝜆∈ℂ+ we write anw𝜆𝐶0[0,𝑇]ğ¹î€·ğ’µâ„Žî€¸(𝑥,⋅)𝑑𝑚(𝑥)=𝐽∗(ℎ;𝜆).(4.3) Let ğ‘ž be a nonzero real number and let 𝐹 be a functional such that ∫anw𝜆𝐶0[0,𝑇]𝐹(ğ’µâ„Ž(𝑥,⋅))𝑑𝑚(𝑥) exists for all 𝜆∈ℂ+. If the following limit exists, we call it the generalized analytic Feynman integral of 𝐹 with parameter ğ‘ž and we write anfğ‘žğ¶0[]0,ğ‘‡ğ¹î€·ğ’µâ„Žî€¸(𝑥,⋅)𝑑𝑚(𝑥)=limğœ†â†’âˆ’ğ‘–ğ‘žî€œanw𝜆𝐶0[]0,ğ‘‡ğ¹î€·ğ’µâ„Žî€¸(𝑥,⋅)𝑑𝑚(𝑥),(4.4) where 𝜆 approaches âˆ’ğ‘–ğ‘ž through values in ℂ+.

Note that if ℎ≡1 on [0,𝑇], then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 12–14].

Next (see [5, 6, 15]) we state the definition of the generalized Fourier-Feynman transform (GFFT).

Definition 4.1. For 𝜆∈ℂ+ and 𝑦∈𝐶0[0,𝑇], let 𝑇𝜆,ℎ(𝐹)(𝑦)=anw𝜆𝐶0[]0,𝑇𝐹𝑦+ğ’µâ„Žî€¸(𝑥,⋅)𝑑𝑚(𝑥).(4.5) Let ğ‘ž be a non-zero real number. For 𝑝∈(1,2], we define the 𝐿𝑝 analytic GFFT with respect to ğ’µâ„Ž, 𝑇(𝑝)ğ‘ž,ℎ(𝐹) of 𝐹, by the formula (𝜆∈ℂ+), 𝑇(𝑝)ğ‘ž,ℎ(𝐹)(𝑦)=l.i.m.ğœ†â†’âˆ’ğ‘–ğ‘žğ‘‡ğœ†,ℎ(𝐹)(𝑦)(4.6) if it exists; that is, for each 𝜌>0, limğœ†â†’âˆ’ğ‘–ğ‘žî€œğ¶0[0,𝑇]|||𝑇𝜆,ℎ(𝐹)(𝜌𝑦)−𝑇(𝑝)ğ‘ž,ℎ(|||𝐹)(𝜌𝑦)ğ‘î…žğ‘‘ğ‘š(𝑦)=0,(4.7) where 1/𝑝+1/𝑝′=1. We define the 𝐿1 analytic GFFT, 𝑇(1)ğ‘ž,ℎ(𝐹) of 𝐹, by the formula (𝜆∈ℂ+)𝑇(1)ğ‘ž,ℎ(𝐹)(𝑦)=limğœ†â†’âˆ’ğ‘–ğ‘žğ‘‡ğœ†,ℎ(𝐹)(𝑦)(4.8) if it exists.

We note that for 𝑝∈[1,2], 𝑇(𝑝)ğ‘ž,ℎ(𝐹) is defined only s-a.e. We also note that if 𝑇(𝑝)ğ‘ž,ℎ(𝐹) exists and if 𝐹≈𝐺, then 𝑇(𝑝)ğ‘ž,ℎ(𝐺) exists and 𝑇(𝑝)ğ‘ž,ℎ(𝐺)≈𝑇(𝑝)ğ‘ž,ℎ(𝐹). One can see that for each â„Žâˆˆğ¿2[0,𝑇], 𝑇(𝑝)ğ‘ž,ℎ(𝐹)≈𝑇(𝑝)ğ‘ž,−ℎ(𝐹) since 𝐶0[0,𝑇]𝐹(𝑥)𝑑𝑚(𝑥)=𝐶0[0,𝑇]𝐹(−𝑥)𝑑𝑚(𝑥).(4.9)

Next we give the definition of the generalized convolution product (GCP).

Definition 4.2. Let 𝐹 and 𝐺 be scale-invariant measurable functionals on 𝐶0[0,𝑇]. For ℂ𝜆∈+ and ℎ1,ℎ2∈𝐿2[0,𝑇], we define their GCP with respect to {ğ’µâ„Ž1,ğ’µâ„Ž2} (if it exists) by (𝐹∗𝐺)(ℎ1,ℎ2)𝜆=⎧⎪⎪⎨⎪⎪⎩(𝑦)anw𝜆𝐶0[0,𝑇]𝐹𝑦+ğ’µâ„Ž1(𝑥,⋅)√2îƒªğºîƒ©ğ‘¦âˆ’ğ’µâ„Ž2(𝑥,⋅)√2𝑑𝑚(𝑥),𝜆∈ℂ+,anfğ‘žğ¶0[]0,𝑇𝐹𝑦+ğ’µâ„Ž1(𝑥,⋅)√2îƒªğºîƒ©ğ‘¦âˆ’ğ’µâ„Ž2(𝑥,⋅)√2𝑑𝑚(𝑥),𝜆=âˆ’ğ‘–ğ‘ž,ğ‘žâˆˆâ„,ğ‘žâ‰ 0.(4.10) When 𝜆=âˆ’ğ‘–ğ‘ž, we denote (𝐹∗𝐺)(ℎ1,ℎ2)𝜆 by (𝐹∗𝐺)(ℎ1,ℎ2)ğ‘ž.

Remark 4.3. Our definition of the GCP is different than the definition given by Huffman et al. in [5, 6] and used by Chang et al. in [15]. But if we choose ℎ1=ℎ2 in (4.10), our GCP (𝐹∗𝐺)(ℎ1,ℎ2)ğ‘ž is the GCP used in [5, 6, 15].

We begin this section with a key lemma for a relationship between the GFFT and the GCP.

Lemma 4.4. Let {𝑔1,𝑔2,𝑔3,𝑔4} be a subset of 𝐿2[0,𝑇], and let 𝑌𝑔1,𝑔2,𝑌𝑔3,𝑔4∶𝐶20[0,𝑇]×[0,𝑇]→ℝ be given by 𝑌𝑔1,𝑔2(𝑤,𝑧;𝑡)=𝒵𝑔1(𝑤,𝑡)+𝒵𝑔2𝑌(𝑧,𝑡),𝑔3,𝑔4(𝑤,𝑧;𝑡)=𝒵𝑔3(𝑤,𝑡)−𝒵𝑔4(𝑧,𝑡),(4.11) respectively. Then the following assertions are equivalent. (i)𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are independent processes.(ii)𝑔1𝑔3=𝑔2𝑔4.

Proof. Since the processes 𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are Gaussian with mean zero, we know that 𝑌𝑔1,𝑔2 and 𝑌𝑔3,𝑔4 are independent processes if and only if 𝐶20[0,𝑇]𝑌𝑔1,𝑔2(𝑤,𝑧;𝑠)𝑌𝑔3,𝑔4(𝑤,𝑧;𝑡)𝑑(𝑚×𝑚)(𝑤,𝑧)=0(4.12) for every 𝑠,𝑡∈[0,𝑇]. But, using (2.4), we have 𝐶20[0,𝑇]𝑌𝑔1,𝑔2(𝑤,𝑧;𝑠)𝑌𝑔3,𝑔4(=𝑤,𝑧;𝑡)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝒵𝑔1(𝑤,𝑠)𝒵𝑔3(𝑤,𝑡)−𝒵𝑔1(𝑤,𝑠)𝒵𝑔4(𝑧,𝑡)+𝒵𝑔2(𝑧,𝑠)𝒵𝑔3(𝑤,𝑡)−𝒵𝑔2(𝑧,𝑠)𝒵𝑔4=(𝑧,𝑡)×𝑑(𝑚×𝑚)(𝑤,𝑧)0min{𝑠,𝑡}𝑔1(𝑢)𝑔3(𝑢)𝑑𝑢−0min{𝑠,𝑡}𝑔2(𝑢)𝑔4(𝑢)𝑑𝑢.(4.13) From this, we can obtain the desired result.

We are now ready to establish fundamental relationships between the GFFT and the GCP.

Lemma 4.5. Let 𝐹 and 𝐺 be functionals on 𝐶0[0,𝑇]. Let {ℎ1,ℎ2,ℎ3} be a subset of 𝐿2[0,𝑇] such that ℎ23=ℎ1ℎ2≥0 almost everywhere on [0,𝑇], and let 𝑘1(𝑡)=2∞−1/2𝑛=1ℎ1,𝜙𝑛22+ℎ3,𝜙𝑛22𝜙𝑛(𝑘𝑡),2(𝑡)=2∞−1/2𝑛=1ℎ2,𝜙𝑛22+ℎ3,𝜙𝑛22𝜙𝑛(𝑡).(4.14) Furthermore, assume that for all 𝜆∈ℂ+, 𝑇𝜆,ℎ3((𝐹∗𝐺)(ℎ1,ℎ2)𝜆), 𝑇𝜆,𝑘1(𝐹) and 𝑇𝜆,𝑘2(𝐺) all exist. Then 𝑇𝜆,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)𝜆(𝑦)=𝑇𝜆,𝑘1𝑦(𝐹)√2𝑇𝜆,𝑘2𝑦(𝐺)√2(4.15) for s-a.e. 𝑦∈𝐶0[0,𝑇].

Proof. We note that for all 𝜆>0, 𝑇𝜆,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)𝜆=(𝑦)𝐶0[0,𝑇](𝐹∗𝐺)(ℎ1,ℎ2)𝜆𝑦+𝜆−1/2ğ’µâ„Ž3=(𝑤,⋅)𝑑𝑚(𝑤)𝐶0[]0,𝑇𝐹𝑦√2+1√𝒵2ğœ†â„Ž3(𝑤,⋅)+ğ’µâ„Ž1𝑦(𝑧,⋅)×𝐺√2+1√𝒵2ğœ†â„Ž3(𝑤,⋅)âˆ’ğ’µâ„Ž2=(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐹𝑦√2+𝜆−1/2îƒ©ğ’µâ„Ž3(𝑤,⋅)+ğ’µâ„Ž1(𝑧,⋅)√2𝑦×𝐺√2+𝜆−1/2îƒ©ğ’µâ„Ž3(𝑤,⋅)âˆ’ğ’µâ„Ž2(𝑧,⋅)√2𝑑(𝑚×𝑚)(𝑤,𝑧).(4.16) But ℎ23=ℎ1ℎ2, and so (ğ’µâ„Ž3(𝑤,⋅)+ğ’µâ„Ž1√(𝑧,⋅))/2 and (ğ’µâ„Ž3(𝑤,⋅)âˆ’ğ’µâ„Ž2√(𝑧,⋅))/2 are independent processes by Lemma 4.4. Hence by (2.8), we obtain that for all 𝜆>0, 𝑇𝜆,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)𝜆=(𝑦)𝐶20[0,𝑇]𝐹𝑦√2+1√𝒵2ğœ†â„Ž3(𝑤,⋅)+ğ’µâ„Ž1×(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶20[0,𝑇]𝐺𝑦√2+1√𝒵2ğœ†â„Ž3(𝑤,⋅)âˆ’ğ’µâ„Ž2=(𝑧,⋅)𝑑(𝑚×𝑚)(𝑤,𝑧)𝐶0[0,𝑇]𝐹𝑦√2+𝒵𝑘1(𝑥,⋅)√𝜆𝑑𝑚(𝑥)𝐶0[0,𝑇]𝐺𝑦√2+𝒵𝑘2(𝑥,⋅)√𝜆𝑑𝑚(𝑥)=𝑇𝜆,𝑘1𝑦(𝐹)√2𝑇𝜆,𝑘2𝑦(𝐺)√2.(4.17) Equation (4.15) holds for all 𝜆∈ℂ+ by analytic continuation.

In next theorem, we show that the GFFT of the GCP is the product of GFFTs.

Theorem 4.6. Let 𝐹, 𝐺, {ℎ1,ℎ2,ℎ3}, 𝑘1, and 𝑘2 be as in Lemma 4.5. Furthermore, assume that for 𝑝∈[1,2], 𝜆∈ℂ+ and ğ‘žâˆˆâ„âˆ’{0}, 𝑇𝜆,ℎ3((𝐹∗𝐺)(ℎ1,ℎ2)ğ‘ž), 𝑇(𝑝)ğ‘ž,ℎ3((𝐹∗𝐺)(ℎ1,ℎ2)ğ‘ž), 𝑇(𝑝)ğ‘ž,𝑘1(𝐹), and 𝑇(𝑝)ğ‘ž,𝑘2(𝐺) all exist and that 𝑇(𝑝)ğ‘ž,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)ğ‘žî‚=⎧⎪⎨⎪⎩l.i.m.ğœ†â†’âˆ’ğ‘–ğ‘žğ‘‡ğœ†,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)𝜆],(𝑦),𝑝∈(1,2limğœ†â†’âˆ’ğ‘–ğ‘žğ‘‡ğœ†,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)𝜆(𝑦),𝑝=1.(4.18) Then 𝑇(𝑝)ğ‘ž,ℎ3(𝐹∗𝐺)(ℎ1,ℎ2)ğ‘žî‚(𝑦)=𝑇(𝑝)ğ‘ž,𝑘1𝑦(𝐹)√2𝑇(𝑝)ğ‘ž,𝑘2𝑦(𝐺)√2(4.19) for s-a.e. 𝑦∈𝐶0[0,𝑇].

Proof. Equation (4.19) follows from (4.15) by letting ğœ†â†’âˆ’ğ‘–ğ‘ž, since all transforms in (4.18) and (4.19) exist.

Remark 4.7. We note that the hypotheses (and hence the conclusions) of Theorem 4.6 above are indeed satisfied by many of the functionals in the following large classes of functionals. These classes of functionals include; (i) the Banach algebra 𝒮 defined by Cameron and Storvick in [16]: also see [3, 5, 14, 15],(ii) various spaces of functionals of the form 𝐹(𝑥)=exp𝑇0𝑓(𝑡,𝑥(𝑡))𝑑𝑡(4.20) for appropriate 𝑓∶[0,𝑇]×ℝ→ℂ as discussed in [4, 12, 13]; and(iii) various spaces of functionals of the form 𝐹(𝑥)=exp𝑇0𝑓(𝑠,𝑡,𝑥(𝑠),𝑥(𝑡))𝑑𝑠𝑑𝑡(4.21) for appropriate 𝑓∶[0,𝑇]2×ℝ2→ℂ as discussed in [3].

Next five corollaries include the results of [3–6] by Huffman et al. The notations used in [3–6] are slightly different than ours.

Corollary 4.8. Refer to Theorem 2.1 in [5].

Proof. In our Lemma 4.5, simply choose ℎ1=ℎ2=ℎ3=𝑘1=𝑘2â‰¡â„Žâˆˆğ¿2[0,𝑇].

Corollary 4.9. Refer to Theorem 1 in [6].

Proof. In our Theorem 4.6, simply choose ℎ1=ℎ2=ℎ3=𝑘1=𝑘2â‰¡â„Žâˆˆğ¿2[0,𝑇].

Corollary 4.10. Refer to Theorem 3.3 in [5].

Proof. In our Theorem 4.6, simply choose ℎ1=ℎ2=ℎ3=𝑘1=𝑘2â‰¡â„Žâˆˆğ¿âˆž[0,𝑇].

Corollary 4.11. Refer to Theorem 3.3 in [3].

Proof. In our Theorem 4.6, simply choose ℎ1=ℎ2=ℎ3=𝑘1=𝑘2≡1.

Corollary 4.12. Refer to Lemma 4.1 and Theorems 4.1 and 4.2 in [4].

Proof. In our Lemma 4.5 and Theorem 4.6, simply choose ℎ1=ℎ2=ℎ3=𝑘1=𝑘2≡1.


The present research was conducted by the research fund of Dankook University in 2010.