Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II

Jae Gil Choi; Sang Kil Shim

Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read On Wiener space, the generalized convolution of two generalized Fourier–Feynman transforms equals the generalized Fourier–Feynman transform of the pointwise product—the infinite-dimensional analogue of F(f)*F(g)=F(fg).

desk verdict A correct, clearly positioned, though incremental companion to the authors' first-relationship paper; Theorem 3.6 is new in the mixed-kernel case and the proof is a transparent composition of published results. read the letter →

arxiv 1908.01890 v3 pith:TD3LIT64 submitted 2019-08-05 math.FA

Sang Kil Shim , Jae Gil Choi This is my paper · ORCID

classification math.FA MSC 46G1228C2060G1560J65

keywords WienerspacegeneralizedFourier-FeynmantransformconvolutionproductBanachalgebraS(L2[0T])GaussianprocessPaley-Wiener-Zygmundstochasticintegralscale-invariantalmosteverywhereanalyticFeynman

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a second fundamental relationship between the generalized Fourier–Feynman transform and the generalized convolution product on Wiener space $C_0[0,T]$: convolving the transforms of two functionals gives the transform of their product. Concretely, for $F,G$ in the Banach algebra $S(L_2[0,T])$, nonzero kernels $h,k_1,k_2$ in $L_\infty[0,T]$ with $h^2=k_1k_2$ almost everywhere, all $p\in[1,2]$, and all nonzero real $q$, the identity $(T^{(p)}_{q,s(h,k_1)/\sqrt{2}}(F) \ast T^{(p)}_{q,s(h,k_2)/\sqrt{2}}(G))^{(k_1,k_2)}_{-q}(y) = T^{(p)}_{q,h}(F(\cdot/\sqrt{2})G(\cdot/\sqrt{2}))(y)$ holds for scale-invariant almost every path $y$. This is the infinite-dimensional analogue of the classical finite-dimensional identity $\mathcal{F}(f) \ast \mathcal{F}(g) = \mathcal{F}(fg)$, and it provides a computational shortcut: the convolution of transforms can be evaluated by taking one transform of a product instead.

What carries the argument

The machinery has three parts. The Banach algebra $S(L_2[0,T])$ consists of functionals $F(x)=\int_{L_2[0,T]}\exp\{i\langle u,x\rangle\}\,d\mu(u)$ with $\mu$ a countably additive complex Borel measure; it is the space on which both transforms exist and stay in the algebra. The generalized transform $T^{(p)}_{q,h}$ is the $L_p$ limit, as $\lambda\to -iq$, of the analytic Wiener integral built from the Gaussian process $Z_h(x,t)=\int_0^t h(s)\,d\tilde{x}(s)$ with the Paley–Wiener–Zygmund stochastic integral. The generalized convolution product $(F\ast G)^{(h_1,h_2)}_q$ integrates $F((y+Z_{h_1})/\sqrt{2})G((y-Z_{h_2})/\sqrt{2})$ with respect to the Feynman integral. The load-bearing object is the square-root convention $s(h_1,h_2)$, defined by $s(h_1,h_2)^2=h_1^2+h_2^2$ almost everywhere; together with the condition $h^2=k_1k_2$, it makes the quadratic forms in the two sides of the identity match, so Theorem 3.1 and Theorem 3.2 convert both sides into explicit exponential integrals over product measures.

What would settle it

Take $F(x)=\exp\{i\langle u_0,x\rangle\}$ and $G(x)=\exp\{i\langle v_0,x\rangle\}$, which belong to $S(L_2[0,T])$ as point-mass measures. Using the explicit measure formulas in Theorems 3.1 and 3.2, compute both sides of (3.4) directly for kernels $h,k_1,k_2$ satisfying $h^2=k_1k_2$ and for a fixed path $y$; the equality reduces to a concrete identity of exponentials. If any choice of $p\in[1,2]$, nonzero real $q$, or any admissible kernel triple yields different values, the theorem is false; checking agreement for all such choices would confirm it.

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Extended reading notes

Core claim

The central claim is Theorem 3.6. Let $k_1,k_2,F,G,h$ be as in Theorem 3.4: $F$ and $G$ are functionals in $S(L_2[0,T])$, $h,k_1,k_2$ are nonzero functions in $L_\infty[0,T]$, and $h^2=k_1k_2$ almost everywhere. Then for every $p\in[1,2]$ and every nonzero real $q$, $$\left($T^{{(p)}}$_{q,s(h,k_1)/\sqrt{2}}(F) \ast $T^{{(p)}}$_{q,s(h,k_2)/\sqrt{2}}(G)\right)^{(k_1,k_2)}_{-q}(y) = $T^{{(p)}}$_{q,h}\left(F(\cdot/\sqrt{2})G(\cdot/\sqrt{2})\right)(y)$$ for scale-invariant almost every $y\in C_0[0,T]$, where $s(h,k_j)$ is any function with $s(h,k_j)^2 = h^2 + k_j^2$ almost everywhere. The proof writes the left side as $T^{(p)}_{q,h}(T^{(p)}_{-q,h}(\text{left side}))$, applies the first fundamental relationship (Theorem 3.4) with $q$ replaced by $-q$, and then uses the inverse property $T^{(p)}_{-q,h}(T^{(p)}_{q,h}(F))\approx F$ to recover the product $F(\cdot/\sqrt{2})G(\cdot/\sqrt{2})$ under the transform.

Load-bearing premise

The identity rests on an earlier theorem about the same objects and on applying that theorem to the already-transformed functions with the sign of the parameter $q$ flipped; if that earlier theorem fails on those functions, or if the transforms leave the class of functions it covers, the new identity does not follow.

Editorial extensions

If this is right

Equation (3.4) gives a shortcut for computation: the generalized convolution of two generalized Fourier–Feynman transforms equals one generalized Fourier–Feynman transform of the product $F(\cdot/\sqrt{2})G(\cdot/\sqrt{2})$, with no convolution integral to evaluate.
The classical Fourier identity $\mathcal{F}(f)\ast\mathcal{F}(g)=\mathcal{F}(fg)$ now has a complete infinite-dimensional analogue on Wiener space, complementing the already-known analogue of $\mathcal{F}(f\ast g)=\mathcal{F}(f)\mathcal{F}(g)$.
Setting $h=k_1=k_2\equiv 1$ recovers the ordinary analytic Fourier–Feynman result of Corollary 3.8, and setting $h=k_1=k_2$ recovers the earlier generalized result of Corollary 3.9.
The iterated versions (Theorems 5.2 and 5.3) let nested transforms with kernels $h_1,\dots,h_n$ be collapsed into a single transform with composite kernel $s(H)$, so the same product-type formula holds for iterated transforms.
For transforms with different parameters $q_1,q_2$ (Lemma 6.2 and Theorem 6.3), the identity survives after rescaling each kernel by $\sqrt{q_j/(2q)}$, giving a mixed-parameter shortcut as well.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The condition $h^2=k_1k_2$ is the real structural content: it says the two Gaussian processes feeding the convolution combine, in squared norm, to reproduce the process of the target transform. A testable extension is whether the identity persists for kernels in $L_2[0,T]$ that are not in $L_\infty$, where the algebra-membership argument from Theorem 3.1 no longer applies.
Abstractly, the transform $T_{q,h}$ acts like a Fourier-type involution (its inverse is $T_{-q,h}$) and the generalized convolution behaves like a twisted product on $S(L_2[0,T])$; the identity suggests a bialgebra-like structure, and checking the standard Hopf-algebra axioms for this pair would be a natural next step.
The examples are generated entirely by classical trigonometric and hyperbolic identities that satisfy $h^2=k_1k_2$; this points to a recipe for producing new concrete identities on Wiener space from any algebraic relation among kernel functions, such as $\cosh^2 t = 1+\sinh^2 t$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Here's the quick read on Shim–Choi II. The paper proves the "second" fundamental relationship between their generalized Fourier–Feynman transform and generalized convolution product: the convolution of two transformed functionals equals the transform of their pointwise product, after the usual √2 scaling. That is the infinite-dimensional analogue of F(f)∗F(g)=F(fg), where the first paper had the homomorphism direction. The statement (3.4) for mixed kernels satisfying h²=k1k2 is new; the known results of Park–Skoug–Storvick and Chang et al. fall out as corollaries.

The proof is short: apply the first relationship with parameter −q and functionals replaced by their transforms, then use the inverse property T_{−q,h}(T_{q,h}F)=F. The derivation checks out. The functionals produced by Theorem 3.1 remain in the Banach algebra S(L2[0,T]), so the inverse property applies. I did the Gaussian phase calculation, and the sign change to −q works because the kernel condition is exactly what makes the cross terms cancel. No circularity: (3.4) is derived from Theorem 3.4, which is proven in the earlier paper, not assumed here.

What the paper does well: it states the result cleanly, proves it transparently, and gives a nice collection of polynomial, trigonometric, and hyperbolic kernel examples satisfying h²=k1k2. The examples are not merely decorative; they show the condition is satisfiable in interesting ways. The iterated extensions in Sections 5 and 6 are straightforward but not vacuous.

Soft spots. The novelty is real but modest: the main step is a two-line composition of the authors' own first-relationship theorem and the inverse property. Someone might call this a "remark" rather than a "II," though the corollaries and examples give it enough body for a paper. The paper leans heavily on [3]; if that theorem had a subtle flaw, this one would collapse. I don't see such a flaw. There are minor notational slips: in Section 5, "the Hilbert space L∞[0,1]" should read L2[0,1], and in Example 5.5 the notation s(H)(t) ≡ s(h1,h2,h3)2(t) is a bit confusing. These are typesetting issues, not mathematical ones.

Who is this for? Specialists in analytic Feynman integrals who want a computational shortcut: (3.4) lets you compute a transform of a product via a convolution of transforms, or vice versa. For that audience it is useful. For anyone outside the subfield it is a minor entry in a long series. I would give it a fair referee slot: it is correct, clearly written, and honestly positioned relative to the prior literature. I wouldn't cite it unless I was already working on GFFTs.

Referee Report

0 major / 6 minor

Summary. The paper establishes a second fundamental relationship between the generalized Fourier–Feynman transform (GFFT) and the generalized convolution product (GCP) on Wiener space C0[0,T]. The main result, Theorem 3.6, states that for functionals F and G in the Banach algebra S(L2[0,T]) and kernels h,k1,k2 in L∞[0,T] satisfying h² = k1k2 almost everywhere, the GCP of the GFFTs of F and G with parameters s(h,k1)/√2 and s(h,k2)/√2 equals the GFFT of the pointwise product F(·/√2)G(·/√2). The proof is a short combination of the inverse property (Corollary 3.3) and the first fundamental relationship (Theorem 3.4) applied with q replaced by -q. The paper also derives iterated versions in Sections 5 and 6 and provides explicit kernel systems satisfying the hypotheses in Section 4.

Significance. The result is the infinite-dimensional analogue of the classical identity F(f)*F(g) = F(fg) for the Fourier transform on Euclidean space. It complements the first relationship established in the authors' earlier paper [3] and extends prior work on ordinary Wiener space. The proof of Theorem 3.6 is valid, and the paper is clearly written. The main theorem is, however, a short consequence of earlier results; the novelty is incremental but genuine, especially the iterated extensions in Theorems 5.2, 5.3, and 6.3 and the worked kernel examples. The paper is suitable for a specialized journal in measure theory and stochastic analysis.

minor comments (6)

[Section 3, proof of Theorem 3.6] The proof should explicitly state that the functionals A = T^{(p)}_{q, s(h,k1)/√2}(F) and B = T^{(p)}_{q, s(h,k2)/√2}(G) belong to S(L2[0,T]) by Theorem 3.1, so that Theorem 3.4 may be applied with parameter -q.
[Section 3, Theorem 3.6] The right-hand side of (3.4) involves the GFFT of the product F(·/√2)G(·/√2); the authors should justify that this product belongs to S(L2[0,T]). This follows from the closure of S under pointwise multiplication and under the scaling x ↦ x/√2, but a brief remark would make the proof self-contained.
[Section 2 and Section 3] The notation s(h1,h2) is defined as an equivalence class in Section 2, yet Theorems 3.4 and 3.6 treat s(h,kj)/√2 as a specific function in L∞[0,T]. It should be remarked that the GFFT and GCP depend only on the square of the kernel function, so the choice of representative does not affect the identities.
[Section 6, equation (6.1)] The scaling property T^{(p)}_{βq,h}(F) ≈ T^{(p)}_{q,h/√β}(F) is asserted without proof or reference; a short derivation from Theorem 3.1 would clarify the arguments in Lemma 6.2 and Theorem 6.3.
[Example 5.5] The notation s(H)(t) ≡ s(h1,h2,h3)2(t) is ambiguous; it should be written as s(H)(t)² = s(h1,h2,h3)(t)² to distinguish the function from its square.
[General] The manuscript contains several typographical issues that appear to be artefacts of the PDF conversion, such as 'F ourier' in the title and misplaced parentheses in Theorem 5.2; the authors should ensure the final published version is free of these artefacts.

Circularity Check

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No significant circularity: Theorem 3.6 follows by applying the prior first-relationship theorem to transformed functionals and cancelling via the inverse transform.

full rationale

The main identity (3.4) is derived, not assumed. The proof starts from the left-hand side of (3.4), inserts T_{-q,h}(T_{q,h}(·)) via the inverse property (3.2), applies Theorem 3.4 with q replaced by -q and with F and G replaced by T_{q,s(h,k1)/√2}(F) and T_{q,s(h,k2)/√2}(G), and then cancels the inverse transforms using (3.2) again. Equation (3.4) never appears as a hypothesis: it is the conclusion. The only imported external result, Theorem 3.4, is the earlier 'first relationship' from [3]; it requires exactly the same kernel condition h^2 = k1k2 and does not contain the new identity. The substituted functionals are in S(L2[0,T]) by Theorem 3.1, so the hypotheses of Theorem 3.4 are satisfied; the final functional F(·/√2)G(·/√2) is also in S by the same measure-pushforward reasoning used in Theorem 3.2. Although [3] shares an author with the present paper, citing a previously proved theorem with stated hypotheses is ordinary mathematical practice and does not smuggle the target result into the proof. The paper recovers known special cases as corollaries rather than renaming them. I therefore find no step in which the claimed derivation reduces by construction to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem is a corollary of two cited results (Theorems 3.1 and 3.4) plus the inverse property (3.2), none of which are proved here. There are no fitted parameters and no invented entities; the s-functions are an equivalence-class notation inherited from [3]. The essential hypotheses are the kernel condition h² = k1k2 and membership of F, G in S(L2[0,T]).

assumptions (5)

domain assumption Theorem 3.1 (Huffman-Park-Skoug [8]): for F in S(L2[0,T]) with Borel measure f, the GFFT T^{(p)}_{q,h}(F) exists and its associated measure is exp(-i/(2q)||uh||²) d f(u).
Invoked at the start of the proof of Corollary 3.3 and used to guarantee transforms of F and G lie in S(L2[0,T]) when applying Theorem 3.4.
domain assumption Theorem 3.4 (Chang-Chung-Choi [3]): the first relationship T^{(p)}_{q,h}((F*G)^{(k1,k2)}_q)(y) = T^{(p)}_{q,s(h,k1)/√2}(F)(y/√2) T^{(p)}_{q,s(h,k2)/√2}(G)(y/√2) under h² = k1k2.
The load-bearing premise of the proof of Theorem 3.6, used at the second equality with q replaced by -q.
domain assumption Corollary 3.3: inverse property T^{(p)}_{-q,h}(T^{(p)}_{q,h}(F)) ≈ F.
Used at the first and third equalities of the proof of Theorem 3.6 to insert and remove an inverse transform pair.
domain assumption The kernel functions h, k1, k2 are nonzero in L∞[0,T] with h² = k1k2 mL-a.e.; the equivalence-class functions s(h,kj) satisfy s(h,kj)² = h² + kj² and are taken in L∞[0,T].
Stated in Theorem 3.4 and in the conventions of Section 2; required for the GFFT and GCP to be defined with L∞ kernels.
domain assumption Theorem 3.2 (Chang-Chung-Choi [3]): the GCP (F*G)^{(k1,k2)}_q exists for F, G in S(L2[0,T]) and belongs to S(L2[0,T]).
Ensures the convolution on the left-hand side of (3.4) exists when the inputs are the transformed functionals.

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Pith. "Pith review of Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II." pith.science (2026). https://pith.science/paper/TD3LIT64

@misc{pith2026190801890,
  author       = {Pith},
  title        = {Pith review of: Generalized Fourier--Feynman transforms and generalized convolution products on Wiener space II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TD3LIT64}},
  note         = {Machine review of arXiv:1908.01890}
}

read the original abstract

The purpose of this article is to present the second type fundamental relationship between the generalized Fourier--Feynman transform and the generalized convolution product on Wiener space. The relationships in this article are also natural extensions (to the case on an infinite dimensional Banach space) of the structure which exists between the Fourier transform and the convolution of functions on Euclidean spaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [3]

Chang, S.J., Chung, H.S., Choi, J.G.: Generalized Fourie r–Feynman transforms and generalized convolution products on Wiener space. Indag. M ath. 28, 566–579 (2017)

work page 2017

[2] [1]

Analytic Functions, Kozubnik 1979 (Proc

Cameron, R.H., Storvick, D.A.: Some Banach algebras of an alytic Feynman integrable functionals. Analytic Functions, Kozubnik 1979 (Proc. Sev enth Conf., Kozubnik, 1979), pp. 18–67, Lecture Notes in Math., 798, Springer, Berlin (1980)

work page 1980

[3] [2]

Integral Transforms Spec

Chang, K.S., Cho, D.H., Kim, B.S., Song, T.S., Yoo, I.: Rel ationships involving gener- alized Fourier–Feynman transform, convolution and ﬁrst va riation. Integral Transforms Spec. Funct. 16, 391–405 (2005)

work page 2005

[4] [4]

Michigan Math

Chung, D.M, Park, C., Skoug, D.: Generalized Feynman inte grals via conditional Feyn- man integrals. Michigan Math. J. 40, 377–391 (1993)

work page 1993

[5] [5]

Huﬀman, T., Park, C., Skoug, D.: Analytic Fourier–Feynma n transforms and convolu- tion. Trans. Amer. Math. Soc. 347, 661–673 (1995)

work page 1995

[6] [6]

Michigan Math

Huﬀman, T., Park, C., Skoug, D.: Convolutions and Fourier –Feynman transforms of functionals involving multiple integrals. Michigan Math. J. 43, 247–261 (1996)

work page 1996

[7] [7]

Rocky Mountain J

Huﬀman, T., Park, C., Skoug, D.: Convolution and Fourier– Feynman transforms. Rocky Mountain J. Math. 27, 827–841 (1997)

work page 1997

[8] [8]

Huﬀman, T., Park, C., Skoug, D.: Generalized transforms a nd convolutions. Int. J. Math. Math. Sci. 20, 19–32 (1997)

work page 1997