International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 578174 | 13 pages | https://doi.org/10.5402/2012/578174

A Rotation on Wiener Space with Applications

Academic Editor: M. Langthjem
Received25 Apr 2012
Accepted26 Jun 2012
Published05 Aug 2012

Abstract

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 ๐ถ0[0,๐‘‡] denote one-parameter Wiener space, that is, the space of all real-valued continuous functions ๐‘ฅ on [0,๐‘‡] with ๐‘ฅ(0)=0. Let โ„ณ denote the class of all Wiener measurable subsets of ๐ถ0[0,๐‘‡], and let ๐‘š denote Wiener measure. Then (๐ถ0[0,๐‘‡],โ„ณ,๐‘š) is a complete measure space, and we denote the Wiener integral of a Wiener integrable functional ๐น by ๎€œ๐ถ0[0,๐‘‡]๐น(๐‘ฅ)๐‘‘๐‘š(๐‘ฅ).(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 ๐บ(๐‘ค,๐‘ง) be an ๐‘šร—๐‘š-integrable functional on ๐ถ20[0,๐‘‡], the product of 2 copies of ๐ถ0[0,๐‘‡], and let ๐œƒ be a function of bounded variation on [0,๐‘‡]. Let ๐‘‡๐œƒโˆถ๐ถ20[0,๐‘‡]โ†’๐ถ20[0,๐‘‡] be the transformation defined by ๐‘‡๐œƒ(๐‘ค,๐‘ง)=(๐‘ค๎…ž,๐‘ง๎…ž) with ๐‘ค๎…ž(๎€œ๐‘ก)=๐‘ก0๎€œcos๐œƒ(๐‘ )๐‘‘๐‘ค(๐‘ )โˆ’๐‘ก0๐‘งsin๐œƒ(๐‘ )๐‘‘๐‘ง(๐‘ ),๎…ž๎€œ(๐‘ก)=๐‘ก0๎€œsin๐œƒ(๐‘ )๐‘‘๐‘ค(๐‘ )+๐‘ก0cos๐œƒ(๐‘ )๐‘‘๐‘ง(๐‘ ).(1.2) Then the transform ๐‘‡๐œƒ is measure preserving and ๎€œ๐ถ20[0,๐‘‡]๎€œ๐บ(๐‘ค,๐‘ง)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ20[0,๐‘‡]๐บ๎€ท๐‘‡๐œƒ(๎€ธ๐‘ค,๐‘ง)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง).(1.3)

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

Corollary 1.2. Let ๐น be Wiener integrable on ๐ถ0[0,๐‘‡]. Then for any ๐œƒโˆˆโ„, ๐น(๐‘คsin๐œƒ+๐‘งcos๐œƒ) is integrable on ๐ถ20[0,๐‘‡] and ๎€œ๐ถ20[0,๐‘‡]๎€œ๐น(๐‘คsin๐œƒ+๐‘งcos๐œƒ)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น(๐‘ฅ)๐‘‘๐‘š(๐‘ฅ).(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 ๐น be Wiener measurable on ๐ถ0[0,๐‘‡]. Assume that for any ๐œŒ>0, ๐น(๐œŒโ‹…) is Winer integrable. Then for any ๐‘Ž,๐‘โˆˆโ„, ๐น(๐‘Ž๐‘ค+๐‘๐‘ง) is integrable on ๐ถ20[0,๐‘‡] and ๎€œ๐ถ20[0,๐‘‡]๎€œ๐น(๐‘Ž๐‘ค+๐‘๐‘ง)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎‚€โˆš๐‘Ž2+๐‘2๐‘ฅ๎‚๐‘‘๐‘š(๐‘ฅ).(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 [3โ€“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 ๐ต of ๐ถ0[0,๐‘‡] is said to be scale-invariant measurable [8] provided ๐œŒ๐ตโˆˆโ„ณ for all ๐œŒ>0, and a scale-invariant measurable set ๐‘ is said to be scale-invariant null provided ๐‘š(๐œŒ๐‘)=0 for all ๐œŒ>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 ๐น and ๐บ are equal s-a.e., we write ๐นโ‰ˆ๐บ.

Let {๐œ™๐‘›} be a complete orthonormal set in ๐ฟ2[0,๐‘‡], each of whose elements is of bounded variation on [0,๐‘‡]. Then for each ๐‘ฃโˆˆ๐ฟ2[0,๐‘‡], the Paley-Wiener-Zygmund (PWZ) stochastic integral โŸจ๐‘ฃ,๐‘ฅโŸฉ is defined by the formula โŸจ๐‘ฃ,๐‘ฅโŸฉ=lim๐‘›โ†’โˆž๎€œ๐‘‡0๐‘›๎“๐‘—=1๎€ท๐‘ฃ,๐œ™๐‘—๎€ธ2๐œ™๐‘—(๐‘ก)๐‘‘๐‘ฅ(๐‘ก)(2.1) for all ๐‘ฅโˆˆ๐ถ0[0,๐‘‡] for which the limit exists, where (โ‹…,โ‹…)2 denotes the ๐ฟ2-inner product.

It was shown in [7] that for each ๐‘ฃโˆˆ๐ฟ2[0,๐‘‡], the limit defining the PWZ integral โŸจ๐‘ฃ,๐‘ฅโŸฉ exists for ๐‘š-a.e. ๐‘ฅโˆˆ๐ถ0[0,๐‘‡] and that this limit is essentially independent of the choice of the complete orthonormal set {๐œ™๐‘›}. It was also shown in [7] that if ๐‘ฃ is of bounded variation on [0,๐‘‡], then the PWZ integral โŸจ๐‘ฃ,๐‘ฅโŸฉ equals the Riemann-Stieltjes integral โˆซ๐‘‡0๐‘ฃ(๐‘ก)๐‘‘๐‘ฅ(๐‘ก) for ๐‘š-a.e. ๐‘ฅโˆˆ๐ถ0[0,๐‘‡]. In fact, the integrals are equal for s-a.e. ๐‘ฅโˆˆ๐ถ0[0,๐‘‡] and that for all ๐‘ฃโˆˆ๐ฟ2[0,๐‘‡], โŸจ๐‘ฃ,๐‘ฅโŸฉ is a Gaussian random variable with mean 0 and variance โ€–๐‘ฃโ€–22.

For any โ„Žโˆˆ๐ฟ2[0,๐‘‡] with โ€–โ„Žโ€–2>0, let ๐’ตโ„Ž be the Gaussian process ๐’ตโ„Ž(๎€œ๐‘ฅ,๐‘ก)=๐‘ก0โ„Ž(๐‘ )๐‘‘๐‘ฅ(๐‘ )=โŸจ๐‘ฃ,๐‘ฅโŸฉ(2.2) introduced by Park and Skoug in [9] and used extensively since; see for example [5, 6, 10, 11]. Of course if โ„Ž(๐‘ก)โ‰ก1 on [0,๐‘‡], then ๐’ตโ„Ž(๐‘ฅ,๐‘ก)=๐‘ฅ(๐‘ก).

It is easy to see that ๐’ตโ„Ž is a Gaussian process with mean zero and covariance function ๎€œ๐ถ0[0,๐‘‡]๐’ตโ„Ž(๐‘ฅ,๐‘ )๐’ตโ„Ž๎€œ(๐‘ฅ,๐‘ก)๐‘‘๐‘š(๐‘ฅ)=0min{๐‘ ,๐‘ก}โ„Ž2(๐‘ข)๐‘‘๐‘ข.(2.3) In addition, ๐’ตโ„Ž(โ‹…,๐‘ก) is stochastically continuous in ๐‘ก on [0,๐‘‡], and for any โ„Ž1,โ„Ž2โˆˆ๐ฟ2[0,๐‘‡], ๎€œ๐ถ0[0,๐‘‡]๐’ตโ„Ž1(๐‘ฅ,๐‘ )๐’ตโ„Ž2๎€œ(๐‘ฅ,๐‘ก)๐‘‘๐‘š(๐‘ฅ)=0min{๐‘ ,๐‘ก}โ„Ž1(๐‘ข)โ„Ž2(๐‘ข)๐‘‘๐‘ข.(2.4)

For any complete orthonormal set {๐œ™๐‘›} in ๐ฟ2[0,๐‘‡] and for any ๐‘›โˆˆโ„•, define the projection map ๐’ซ๐‘› from ๐ฟ2[0,๐‘‡] into span{๐œ™1,โ€ฆ,๐œ™๐‘›} by ๐’ซ๐‘›โ„Ž(๐‘ก)=๐‘›๎“๐‘—=1๎€ทโ„Ž,๐œ™๐‘—๎€ธ2๐œ™๐‘—(๐‘ก).(2.5) Then for โ„Žโˆˆ๐ฟ2[0,๐‘‡] and ๐‘ฅโˆˆ๐ถ0[0,๐‘‡], we see that ๐’ตโ„Ž(๐‘ฅ,๐‘ก)=lim๐‘›โ†’โˆž๎€œ๐‘ก0๐’ซ๐‘›โ„Ž(๐‘ )๐‘‘๐‘ฅ(๐‘ )=lim๐‘›โ†’โˆž๐’ต๐’ซ๐‘›โ„Ž(๐‘ฅ,๐‘ก),(2.6) that is, ๐’ต๐’ซ๐‘›โ„Ž(๐‘ฅ,๐‘ก) converges in ๐ฟ2(๐ถ0[0,๐‘‡])-mean to ๐’ตโ„Ž(๐‘ฅ,๐‘ก).

Throughout this paper, we will assume that each functional ๐นโˆถ๐ถ0[0,๐‘‡]โ†’โ„‚ we consider is scale-invariant measurable and that ๎€œ๐ถ0[0,๐‘‡]||๐น๎€ท๐’ตโ„Ž(๎€ธ||๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)<+โˆž(2.7) for all โ„Žโˆˆ๐ฟ2[0,๐‘‡].

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

Theorem 2.1. Let ๐น be a functional on ๐ถ0[0,๐‘‡]. Then for any โ„Ž1,โ„Ž2โˆˆ๐ฟ2[0,๐‘‡], ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ตโ„Ž1(๐‘ค,โ‹…)+๐’ตโ„Ž2(๎€ธ๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐‘˜(๎€ธ๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(2.8) where โ„Ž1, โ„Ž2, and ๐‘˜ are related by ๐‘˜(๐‘ก)=โˆž๎“๐‘›=1๎”๎€ทโ„Ž1,๐œ™๐‘›๎€ธ22+๎€ทโ„Ž2,๐œ™๐‘›๎€ธ22๐œ™๐‘›(๐‘ก)(2.9) for some complete orthonormal set {๐œ™๐‘›} in ๐ฟ2[0,๐‘‡], each of those elements is of bounded variation on [0,๐‘‡].

3. Proof of the Main Theorem

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

Lemma 3.1. Let ๐น be a functional on ๐ถ0[0,๐‘‡], and let ๐œ™ be a function of bounded variation on [0,๐‘‡]. Then for all ๐‘Ž,๐‘โˆˆโ„, ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐‘Ž๐’ต๐œ™(๐‘ค,โ‹…)+๐‘๐’ต๐œ™(๎€ธ๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎‚€โˆš๐‘Ž2+๐‘2๐’ต๐œ™(๎‚๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ).(3.1)

Proof. We first note that for each ๐‘กโˆˆ[0,๐‘‡], ๐‘Ž๐’ต๐œ™(๐‘ค,๐‘ก)+๐‘๐’ต๐œ™(๎€œ๐‘ง,๐‘ก)=๐‘ก0๐œ™(๐‘ )๐‘‘(๐‘Ž๐‘ค(๐‘ )+๐‘๐‘ง(๐‘ ))=๐’ต๐œ™(๐‘Ž๐‘ค+๐‘๐‘ง,๐‘ก).(3.2) We also note that ๐น(๐’ต(๐‘ฅ,โ‹…)) is Wiener integrable as a functional of ๐‘ฅ. Hence, by (1.5), we obtain that for all ๐‘Ž,๐‘โˆˆโ„, ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐‘Ž๐’ต๐œ™(๐‘ค,โ‹…)+๐‘๐’ต๐œ™(๎€ธ=๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ20[0,๐‘‡]๐น๎€ท๐’ต๐œ™๎€ธ=๎€œ(๐‘Ž๐‘ค+๐‘๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ0[0,๐‘‡]๐น๎‚€๐’ต๐œ™๎‚€โˆš๐‘Ž2+๐‘2=๎€œ๐‘ฅ,โ‹…๎‚๎‚๐‘‘๐‘š(๐‘ฅ)๐ถ0[0,๐‘‡]๐น๎‚€โˆš๐‘Ž2+๐‘2๐’ต๐œ™๎‚(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ).(3.3) Thus (3.1) is established.

Lemma 3.2. Let ๐น be a functional on ๐ถ0[0,๐‘‡]. Then for any โ„Ž1,โ„Ž2โˆˆ๐ฟ2[0,๐‘‡] and each ๐‘›โˆˆโ„•, ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ต๐’ซ๐‘›โ„Ž1(๐‘ค,โ‹…)+๐’ต๐’ซ๐‘›โ„Ž2(๎€ธ๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐’ซ๐‘›๐‘˜(๎€ธ๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(3.4) where โ„Ž1, โ„Ž2, and ๐‘˜ are related by (2.9).

Proof. Since the addition is continuous in the uniform topology on ๐ถ0[0,๐‘‡], we can apply (3.1) to the functional โˆ‘๐น(๐‘›๐‘—=1๐’ต๐œ™๐‘—(๐‘ฅ,โ‹…)). Thus using (2.5) and (3.1), we have ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ต๐’ซ๐‘›โ„Ž1(๐‘ค,โ‹…)+๐’ต๐’ซ๐‘›โ„Ž2(๎€ธ=๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ20[0,๐‘‡]๐น๎ƒฉ๐‘›๎“๐‘—=1๎‚ƒ๎€ทโ„Ž1,๐œ™๐‘—๎€ธ2๐’ต๐œ™๐‘—๎€ทโ„Ž(๐‘ค,โ‹…)+2,๐œ™๐‘—๎€ธ2๐’ต๐œ™๐‘—๎‚„๎ƒช=๎€œ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ0[0,๐‘‡]๐น๎ƒฉ๐‘›๎“๐‘—=1๎”๎€ทโ„Ž1,๐œ™๐‘—๎€ธ22+๎€ทโ„Ž2,๐œ™๐‘—๎€ธ22๐’ต๐œ™๐‘—๎ƒช=๎€œ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐’ซ๐‘›๐‘˜๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ).(3.5) Thus (3.4) is established.

Lemma 3.3. Let ๐น be bounded and continuous on ๐ถ0[0,๐‘‡]. Then for any โ„Ž1,โ„Ž2โˆˆ๐ฟ2[0,๐‘‡], ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ตโ„Ž1(๐‘ค,โ‹…)+๐’ตโ„Ž2(๎€ธ๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐‘˜(๎€ธ๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(3.6) where โ„Ž1, โ„Ž2, and ๐‘˜ are related by (2.9) above.

Proof. We clearly see that ๐น is Wiener integrable. We also note that {๐’ซ๐‘›โ„Ž} is a sequence of functions of bounded variation on [0,๐‘‡] such that ๐’ซ๐‘›โ„Ž converges to โ„Ž in the space ๐ฟ2[0,๐‘‡] as ๐‘›โ†’โˆž. For each ๐‘›โˆˆโ„• and โ„Žโˆˆ๐ฟ2[0,๐‘‡], let ๐น๐‘›(๐’ตโ„Ž(๐‘ฅ,โ‹…))=๐น(๐’ต๐’ซ๐‘›โ„Ž(๐‘ฅ,โ‹…)). Since ๐’ต๐’ซ๐‘›โ„Ž converges to ๐’ตโ„Ž uniformly and ๐น is continuous in the uniform topology, by (2.6), ๐น๎€ท๐’ตโ„Ž๎€ธ๎‚ต(๐‘ฅ,โ‹…)=๐นlim๐‘›โ†’โˆž๐’ต๐’ซ๐‘›โ„Ž๎‚ถ(๐‘ฅ,โ‹…)=lim๐‘›โ†’โˆž๐น๐‘›๎€ท๐’ตโ„Ž๎€ธ(๐‘ฅ,โ‹…).(3.7) Since ๐น is bounded, by using the dominated convergence theorem and (3.4), we have ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ตโ„Ž1(๐‘ค,โ‹…)+๐’ตโ„Ž2(๎€ธ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=lim๐‘›โ†’โˆž๎€œ๐ถ20[0,๐‘‡]๐น๐‘›๎€ท๐’ตโ„Ž1(๐‘ค,โ‹…)+๐’ตโ„Ž2๎€ธ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=lim๐‘›โ†’โˆž๎€œ๐ถ0[0,๐‘‡]๐น๐‘›๎€ท๐’ต๐‘˜๎€ธ=๎€œ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐‘˜๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(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 ๐น be Wiener integrable. Suppose that the left-hand side of (2.8) exists. By usual arguments of integration theory, there exists a sequence {๐น๐‘›} of bounded and continuous functionals such that ๐น๐‘› converges to ๐น. By Lemma 3.3 and the dominated convergence theorem, we can obtain the desired result.

Corollary 3.4. Let ๐น be a functional on ๐ถ0[0,๐‘‡]. Then for all โ„Žโˆˆ๐ฟ2[0,๐‘‡] and all ๐‘Ž,๐‘โˆˆโ„, ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐‘Ž๐’ตโ„Ž(๐‘ค,โ‹…)+๐‘๐’ตโ„Ž(๎€ธ๎€œ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=๐ถ0[0,๐‘‡]๐น๎‚€โˆš๐‘Ž2+๐‘2๐’ตโ„Ž(๎‚๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ).(3.9)

Proof. Simply choose โ„Ž1=๐‘Žโ„Ž and โ„Ž2=๐‘โ„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 ๐น be a functional on ๐ถ0[0,๐‘‡], and let {โ„Ž1,โ€ฆ,โ„Ž๐œˆ} be any subset of ๐ฟ2[0,๐‘‡]. Then ๎€œ๐ถ๐œˆ0[0,๐‘‡]๐น๎ƒฉ๐œˆ๎“๐‘—=1๐’ตโ„Ž๐‘—๎€ท๐‘ฅ๐‘—๎€ธ๎ƒช,โ‹…๐‘‘๐‘š๐œˆ๎‚ตโ†’๐‘ฅ๎‚ถ=๎€œ๐ถ0[0,๐‘‡]๐น๎€ท๐’ต๐‘˜๐œˆ๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(3.10) where ๐‘š๐œˆ is the product Wiener measure on ๐ถ๐œˆ0[0,๐‘‡], the product of ๐œˆ copies of ๐ถ0[0,๐‘‡], and ๐‘˜๐œˆ(๐‘ก)=โˆž๎“๐‘›=1๎„ถ๎„ต๎„ตโŽท๐œˆ๎“๐‘—=1๎€ทโ„Ž๐‘—,๐œ™๐‘›๎€ธ22๐œ™๐‘›(๐‘ก)(3.11) for some complete orthonormal set {๐œ™๐‘›} in ๐ฟ2[0,๐‘‡].

Theorem 3.6. Let ๐น be a functional on ๐ถ0[0,๐‘‡]. Then for any โ„Ž1 and โ„Ž2 in ๐ฟ2[0,๐‘‡], ๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ตโ„Ž2(๐‘ค,โ‹…)โˆ’๐’ตโ„Ž1(๐‘ง,โ‹…),๐’ตโ„Ž1(๐‘ค,โ‹…)+๐’ตโ„Ž2(๎€ธ๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)โˆ—=๎€œ๐ถ20[0,๐‘‡]๐น๎€ท๐’ต๐‘˜(๐‘ค,โ‹…),๐’ต๐‘˜๎€ธ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง),(3.12) where โ„Ž1, โ„Ž2, and ๐‘˜ are related by (2.9).

Remark 3.7. Equations (2.8) and (3.12) are indeed very general formulas.(1)For any ๐œƒโˆˆโ„, choosing โ„Ž1(๐‘ก)โ‰กsin๐œƒ and โ„Ž2(๐‘ก)โ‰กcos๐œƒ in (2.8) yields (1.4).(2)For any ๐‘Ž,๐‘โˆˆโ„, choosing โ„Ž1(๐‘ก)โ‰ก๐‘Ž and โ„Ž2(๐‘ก)โ‰ก๐‘ in (2.8) or choosing โ„Ž(๐‘ก)โ‰ก1 in (3.9) yields (1.5). (3)For any function of bounded variation ๐œƒ(โ‹…), choosing โ„Ž1(๐‘ก)=sin๐œƒ(๐‘ก) and โ„Ž2(๐‘ก)=cos๐œƒ(๐‘ก) on [0,๐‘‡] 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 ๐œ†>0, with the Wiener integral ๎€œ๐ถ0[0,๐‘‡]๐น๎€ท๐œ†โˆ’1/2๐‘ฅ๎€ธ๐‘‘๐‘š(๐‘ฅ)(4.1) and then extends analytically in ๐œ† to the right-half complex plane. Here we start with the (generalized) Wiener integral ๎€œ๐ถ0[0,๐‘‡]๐น๎€ท๐œ†โˆ’1/2๐’ตโ„Ž(๎€ธ๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)=๐ฝ(โ„Ž;๐œ†),(4.2) where ๐’ตโ„Ž is the Gaussian process given by (2.2) above.

Throughout this section, let โ„‚+ and ๎‚โ„‚+ denote the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part, respectively.

Let ๐น be a complex-valued scale-invariant measurable functional on ๐ถ0[0,๐‘‡] such that ๐ฝ(โ„Ž;๐œ†) given by (4.2) exists and is finite for all ๐œ†>0. If there exists a function ๐ฝโˆ—(โ„Ž;๐œ†) analytic on โ„‚+ such that ๐ฝโˆ—(โ„Ž;๐œ†)=๐ฝ(โ„Ž;๐œ†) for all ๐œ†>0, then ๐ฝโˆ—(โ„Ž;๐œ†) is defined to be the generalized analytic Wiener integral (with respect to the process ๐’ตโ„Ž) of ๐น over ๐ถ0[0,๐‘‡] with parameter ๐œ†, and for ๐œ†โˆˆโ„‚+ we write ๎€œanw๐œ†๐ถ0[0,๐‘‡]๐น๎€ท๐’ตโ„Ž๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)=๐ฝโˆ—(โ„Ž;๐œ†).(4.3) Let ๐‘ž be a nonzero real number and let ๐น be a functional such that โˆซanw๐œ†๐ถ0[0,๐‘‡]๐น(๐’ตโ„Ž(๐‘ฅ,โ‹…))๐‘‘๐‘š(๐‘ฅ) exists for all ๐œ†โˆˆโ„‚+. If the following limit exists, we call it the generalized analytic Feynman integral of ๐น with parameter ๐‘ž and we write ๎€œanf๐‘ž๐ถ0[]0,๐‘‡๐น๎€ท๐’ตโ„Ž๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ)=lim๐œ†โ†’โˆ’๐‘–๐‘ž๎€œanw๐œ†๐ถ0[]0,๐‘‡๐น๎€ท๐’ตโ„Ž๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ),(4.4) where ๐œ† approaches โˆ’๐‘–๐‘ž through values in โ„‚+.

Note that if โ„Žโ‰ก1 on [0,๐‘‡], then these definitions agree with the previous definitions of the analytic Wiener integral and the analytic Feynman integral [3, 4, 8, 12โ€“14].

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

Definition 4.1. For ๐œ†โˆˆโ„‚+ and ๐‘ฆโˆˆ๐ถ0[0,๐‘‡], let ๐‘‡๐œ†,โ„Ž๎€œ(๐น)(๐‘ฆ)=anw๐œ†๐ถ0[]0,๐‘‡๐น๎€ท๐‘ฆ+๐’ตโ„Ž๎€ธ(๐‘ฅ,โ‹…)๐‘‘๐‘š(๐‘ฅ).(4.5) Let ๐‘ž be a non-zero real number. For ๐‘โˆˆ(1,2], we define the ๐ฟ๐‘ analytic GFFT with respect to ๐’ตโ„Ž, ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น) of ๐น, by the formula (๐œ†โˆˆโ„‚+), ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น)(๐‘ฆ)=l.i.m.๐œ†โ†’โˆ’๐‘–๐‘ž๐‘‡๐œ†,โ„Ž(๐น)(๐‘ฆ)(4.6) if it exists; that is, for each ๐œŒ>0, lim๐œ†โ†’โˆ’๐‘–๐‘ž๎€œ๐ถ0[0,๐‘‡]|||๐‘‡๐œ†,โ„Ž(๐น)(๐œŒ๐‘ฆ)โˆ’๐‘‡(๐‘)๐‘ž,โ„Ž(|||๐น)(๐œŒ๐‘ฆ)๐‘๎…ž๐‘‘๐‘š(๐‘ฆ)=0,(4.7) where 1/๐‘+1/๐‘โ€ฒ=1. We define the ๐ฟ1 analytic GFFT, ๐‘‡(1)๐‘ž,โ„Ž(๐น) of ๐น, by the formula (๐œ†โˆˆโ„‚+)๐‘‡(1)๐‘ž,โ„Ž(๐น)(๐‘ฆ)=lim๐œ†โ†’โˆ’๐‘–๐‘ž๐‘‡๐œ†,โ„Ž(๐น)(๐‘ฆ)(4.8) if it exists.

We note that for ๐‘โˆˆ[1,2], ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น) is defined only s-a.e. We also note that if ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น) exists and if ๐นโ‰ˆ๐บ, then ๐‘‡(๐‘)๐‘ž,โ„Ž(๐บ) exists and ๐‘‡(๐‘)๐‘ž,โ„Ž(๐บ)โ‰ˆ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น). One can see that for each โ„Žโˆˆ๐ฟ2[0,๐‘‡], ๐‘‡(๐‘)๐‘ž,โ„Ž(๐น)โ‰ˆ๐‘‡(๐‘)๐‘ž,โˆ’โ„Ž(๐น) since ๎€œ๐ถ0[0,๐‘‡]๎€œ๐น(๐‘ฅ)๐‘‘๐‘š(๐‘ฅ)=๐ถ0[0,๐‘‡]๐น(โˆ’๐‘ฅ)๐‘‘๐‘š(๐‘ฅ).(4.9)

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

Definition 4.2. Let ๐น and ๐บ be scale-invariant measurable functionals on ๐ถ0[0,๐‘‡]. For ๎‚โ„‚๐œ†โˆˆ+ and โ„Ž1,โ„Ž2โˆˆ๐ฟ2[0,๐‘‡], we define their GCP with respect to {๐’ตโ„Ž1,๐’ตโ„Ž2} (if it exists) by (๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๎€œ(๐‘ฆ)anw๐œ†๐ถ0[0,๐‘‡]๐น๎ƒฉ๐‘ฆ+๐’ตโ„Ž1(๐‘ฅ,โ‹…)โˆš2๎ƒช๐บ๎ƒฉ๐‘ฆโˆ’๐’ตโ„Ž2(๐‘ฅ,โ‹…)โˆš2๎ƒช๐‘‘๐‘š(๐‘ฅ),๐œ†โˆˆโ„‚+,๎€œanf๐‘ž๐ถ0[]0,๐‘‡๐น๎ƒฉ๐‘ฆ+๐’ตโ„Ž1(๐‘ฅ,โ‹…)โˆš2๎ƒช๐บ๎ƒฉ๐‘ฆโˆ’๐’ตโ„Ž2(๐‘ฅ,โ‹…)โˆš2๎ƒช๐‘‘๐‘š(๐‘ฅ),๐œ†=โˆ’๐‘–๐‘ž,๐‘žโˆˆโ„,๐‘žโ‰ 0.(4.10) When ๐œ†=โˆ’๐‘–๐‘ž, we denote (๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ† by (๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘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 โ„Ž1=โ„Ž2 in (4.10), our GCP (๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘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 {๐‘”1,๐‘”2,๐‘”3,๐‘”4} be a subset of ๐ฟ2[0,๐‘‡], and let ๐‘Œ๐‘”1,๐‘”2,๐‘Œ๐‘”3,๐‘”4โˆถ๐ถ20[0,๐‘‡]ร—[0,๐‘‡]โ†’โ„ be given by ๐‘Œ๐‘”1,๐‘”2(๐‘ค,๐‘ง;๐‘ก)=๐’ต๐‘”1(๐‘ค,๐‘ก)+๐’ต๐‘”2๐‘Œ(๐‘ง,๐‘ก),๐‘”3,๐‘”4(๐‘ค,๐‘ง;๐‘ก)=๐’ต๐‘”3(๐‘ค,๐‘ก)โˆ’๐’ต๐‘”4(๐‘ง,๐‘ก),(4.11) respectively. Then the following assertions are equivalent. (i)๐‘Œ๐‘”1,๐‘”2 and ๐‘Œ๐‘”3,๐‘”4 are independent processes.(ii)๐‘”1๐‘”3=๐‘”2๐‘”4.

Proof. Since the processes ๐‘Œ๐‘”1,๐‘”2 and ๐‘Œ๐‘”3,๐‘”4 are Gaussian with mean zero, we know that ๐‘Œ๐‘”1,๐‘”2 and ๐‘Œ๐‘”3,๐‘”4 are independent processes if and only if ๎€œ๐ถ20[0,๐‘‡]๐‘Œ๐‘”1,๐‘”2(๐‘ค,๐‘ง;๐‘ )๐‘Œ๐‘”3,๐‘”4(๐‘ค,๐‘ง;๐‘ก)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)=0(4.12) for every ๐‘ ,๐‘กโˆˆ[0,๐‘‡]. But, using (2.4), we have ๎€œ๐ถ20[0,๐‘‡]๐‘Œ๐‘”1,๐‘”2(๐‘ค,๐‘ง;๐‘ )๐‘Œ๐‘”3,๐‘”4(=๎€œ๐‘ค,๐‘ง;๐‘ก)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ20[0,๐‘‡]๎€ฝ๐’ต๐‘”1(๐‘ค,๐‘ )๐’ต๐‘”3(๐‘ค,๐‘ก)โˆ’๐’ต๐‘”1(๐‘ค,๐‘ )๐’ต๐‘”4(๐‘ง,๐‘ก)+๐’ต๐‘”2(๐‘ง,๐‘ )๐’ต๐‘”3(๐‘ค,๐‘ก)โˆ’๐’ต๐‘”2(๐‘ง,๐‘ )๐’ต๐‘”4๎€พ=๎€œ(๐‘ง,๐‘ก)ร—๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)0min{๐‘ ,๐‘ก}๐‘”1(๐‘ข)๐‘”3๎€œ(๐‘ข)๐‘‘๐‘ขโˆ’0min{๐‘ ,๐‘ก}๐‘”2(๐‘ข)๐‘”4(๐‘ข)๐‘‘๐‘ข.(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 ๐น and ๐บ be functionals on ๐ถ0[0,๐‘‡]. Let {โ„Ž1,โ„Ž2,โ„Ž3} be a subset of ๐ฟ2[0,๐‘‡] such that โ„Ž23=โ„Ž1โ„Ž2โ‰ฅ0 almost everywhere on [0,๐‘‡], and let ๐‘˜1(๐‘ก)=2โˆžโˆ’1/2๎“๐‘›=1๎”๎€ทโ„Ž1,๐œ™๐‘›๎€ธ22+๎€ทโ„Ž3,๐œ™๐‘›๎€ธ22๐œ™๐‘›(๐‘˜๐‘ก),2(๐‘ก)=2โˆžโˆ’1/2๎“๐‘›=1๎”๎€ทโ„Ž2,๐œ™๐‘›๎€ธ22+๎€ทโ„Ž3,๐œ™๐‘›๎€ธ22๐œ™๐‘›(๐‘ก).(4.14) Furthermore, assume that for all ๐œ†โˆˆโ„‚+, ๐‘‡๐œ†,โ„Ž3((๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†), ๐‘‡๐œ†,๐‘˜1(๐น) and ๐‘‡๐œ†,๐‘˜2(๐บ) all exist. Then ๐‘‡๐œ†,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎‚(๐‘ฆ)=๐‘‡๐œ†,๐‘˜1๎ƒฉ๐‘ฆ(๐น)โˆš2๎ƒช๐‘‡๐œ†,๐‘˜2๎ƒฉ๐‘ฆ(๐บ)โˆš2๎ƒช(4.15) for s-a.e. ๐‘ฆโˆˆ๐ถ0[0,๐‘‡].

Proof. We note that for all ๐œ†>0, ๐‘‡๐œ†,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎‚=๎€œ(๐‘ฆ)๐ถ0[0,๐‘‡](๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎€ท๐‘ฆ+๐œ†โˆ’1/2๐’ตโ„Ž3๎€ธ=๎€(๐‘ค,โ‹…)๐‘‘๐‘š(๐‘ค)๐ถ0[]0,๐‘‡๐น๎ƒฉ๐‘ฆโˆš2+1โˆš๎€ท๐’ต2๐œ†โ„Ž3(๐‘ค,โ‹…)+๐’ตโ„Ž1๎€ธ๎ƒช๎ƒฉ๐‘ฆ(๐‘ง,โ‹…)ร—๐บโˆš2+1โˆš๎€ท๐’ต2๐œ†โ„Ž3(๐‘ค,โ‹…)โˆ’๐’ตโ„Ž2๎€ธ๎ƒช=๎€œ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ20[0,๐‘‡]๐น๎ƒฉ๐‘ฆโˆš2+๐œ†โˆ’1/2๎ƒฉ๐’ตโ„Ž3(๐‘ค,โ‹…)+๐’ตโ„Ž1(๐‘ง,โ‹…)โˆš2๎ƒฉ๐‘ฆ๎ƒช๎ƒชร—๐บโˆš2+๐œ†โˆ’1/2๎ƒฉ๐’ตโ„Ž3(๐‘ค,โ‹…)โˆ’๐’ตโ„Ž2(๐‘ง,โ‹…)โˆš2๎ƒช๎ƒช๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง).(4.16) But โ„Ž23=โ„Ž1โ„Ž2, and so (๐’ตโ„Ž3(๐‘ค,โ‹…)+๐’ตโ„Ž1โˆš(๐‘ง,โ‹…))/2 and (๐’ตโ„Ž3(๐‘ค,โ‹…)โˆ’๐’ตโ„Ž2โˆš(๐‘ง,โ‹…))/2 are independent processes by Lemma 4.4. Hence by (2.8), we obtain that for all ๐œ†>0, ๐‘‡๐œ†,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎‚=๎€œ(๐‘ฆ)๐ถ20[0,๐‘‡]๐น๎ƒฉ๐‘ฆโˆš2+1โˆš๎€ท๐’ต2๐œ†โ„Ž3(๐‘ค,โ‹…)+๐’ตโ„Ž1๎€ธ๎ƒชร—๎€œ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ20[0,๐‘‡]๐บ๎ƒฉ๐‘ฆโˆš2+1โˆš๎€ท๐’ต2๐œ†โ„Ž3(๐‘ค,โ‹…)โˆ’๐’ตโ„Ž2๎€ธ๎ƒช=๎€œ(๐‘ง,โ‹…)๐‘‘(๐‘šร—๐‘š)(๐‘ค,๐‘ง)๐ถ0[0,๐‘‡]๐น๎ƒฉ๐‘ฆโˆš2+๐’ต๐‘˜1(๐‘ฅ,โ‹…)โˆš๐œ†๎ƒช๎€œ๐‘‘๐‘š(๐‘ฅ)๐ถ0[0,๐‘‡]๐บ๎ƒฉ๐‘ฆโˆš2+๐’ต๐‘˜2(๐‘ฅ,โ‹…)โˆš๐œ†๎ƒช๐‘‘๐‘š(๐‘ฅ)=๐‘‡๐œ†,๐‘˜1๎ƒฉ๐‘ฆ(๐น)โˆš2๎ƒช๐‘‡๐œ†,๐‘˜2๎ƒฉ๐‘ฆ(๐บ)โˆš2๎ƒช.(4.17) Equation (4.15) holds for all ๐œ†โˆˆโ„‚+ by analytic continuation.

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

Theorem 4.6. Let ๐น, ๐บ, {โ„Ž1,โ„Ž2,โ„Ž3}, ๐‘˜1, and ๐‘˜2 be as in Lemma 4.5. Furthermore, assume that for ๐‘โˆˆ[1,2], ๐œ†โˆˆโ„‚+ and ๐‘žโˆˆโ„โˆ’{0}, ๐‘‡๐œ†,โ„Ž3((๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘ž), ๐‘‡(๐‘)๐‘ž,โ„Ž3((๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘ž), ๐‘‡(๐‘)๐‘ž,๐‘˜1(๐น), and ๐‘‡(๐‘)๐‘ž,๐‘˜2(๐บ) all exist and that ๐‘‡(๐‘)๐‘ž,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘ž๎‚=โŽงโŽชโŽจโŽชโŽฉl.i.m.๐œ†โ†’โˆ’๐‘–๐‘ž๐‘‡๐œ†,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎‚],(๐‘ฆ),๐‘โˆˆ(1,2lim๐œ†โ†’โˆ’๐‘–๐‘ž๐‘‡๐œ†,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐œ†๎‚(๐‘ฆ),๐‘=1.(4.18) Then ๐‘‡(๐‘)๐‘ž,โ„Ž3๎‚€(๐นโˆ—๐บ)(โ„Ž1,โ„Ž2)๐‘ž๎‚(๐‘ฆ)=๐‘‡(๐‘)๐‘ž,๐‘˜1๎ƒฉ๐‘ฆ(๐น)โˆš2๎ƒช๐‘‡(๐‘)๐‘ž,๐‘˜2๎ƒฉ๐‘ฆ(๐บ)โˆš2๎ƒช(4.19) for s-a.e. ๐‘ฆโˆˆ๐ถ0[0,๐‘‡].

Proof. Equation (4.19) follows from (4.15) by letting ๐œ†โ†’โˆ’๐‘–๐‘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 ๐’ฎ defined by Cameron and Storvick in [16]: also see [3, 5, 14, 15],(ii) various spaces of functionals of the form ๎‚ป๎€œ๐น(๐‘ฅ)=exp๐‘‡0๎‚ผ๐‘“(๐‘ก,๐‘ฅ(๐‘ก))๐‘‘๐‘ก(4.20) for appropriate ๐‘“โˆถ[0,๐‘‡]ร—โ„โ†’โ„‚ as discussed in [4, 12, 13]; and(iii) various spaces of functionals of the form ๎‚ป๎€๐น(๐‘ฅ)=exp๐‘‡0๎‚ผ๐‘“(๐‘ ,๐‘ก,๐‘ฅ(๐‘ ),๐‘ฅ(๐‘ก))๐‘‘๐‘ ๐‘‘๐‘ก(4.21) for appropriate ๐‘“โˆถ[0,๐‘‡]2ร—โ„2โ†’โ„‚ as discussed in [3].

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

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

Proof. In our Lemma 4.5, simply choose โ„Ž1=โ„Ž2=โ„Ž3=๐‘˜1=๐‘˜2โ‰กโ„Žโˆˆ๐ฟ2[0,๐‘‡].

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

Proof. In our Theorem 4.6, simply choose โ„Ž1=โ„Ž2=โ„Ž3=๐‘˜1=๐‘˜2โ‰กโ„Žโˆˆ๐ฟ2[0,๐‘‡].

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

Proof. In our Theorem 4.6, simply choose โ„Ž1=โ„Ž2=โ„Ž3=๐‘˜1=๐‘˜2โ‰กโ„Žโˆˆ๐ฟโˆž[0,๐‘‡].

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

Proof. In our Theorem 4.6, simply choose โ„Ž1=โ„Ž2=โ„Ž3=๐‘˜1=๐‘˜2โ‰ก1.

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 โ„Ž1=โ„Ž2=โ„Ž3=๐‘˜1=๐‘˜2โ‰ก1.

Acknowledgment

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

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Copyright © 2012 Jae Gil Choi and Seung Jun Chang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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