[Paper Review] Infinitesimal Fourier Transformation for The Space of Functionals

Takashi Nitta, Tomoko Okada|ArXiv.org|May 13, 2004

Mathematical and Theoretical Analysis4 references3 citations

TL;DR

This paper introduces an infinitesimal Fourier transformation for functionals on nonstandard extended spaces using hyperfinite lattices and double nonstandard extensions (${}^{ullet}({}^*bR)$). By modeling functionals over internal functions on hyperfinite grids, it generalizes Kinoshita and Gordon’s infinitesimal Fourier transforms to infinite-dimensional functionals, proving that the standard part of the transform converges to the classical Fourier transform, with key results on delta function behavior and Gaussian integral approximations in nonstandard settings.

ABSTRACT

The purpose is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. We extend ${\bf R}$ to $ ^{\star}(^{\ast}{\bf R})$ under the base of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a $ ^{\ast}$-finite lattice to a $ ^{\ast}$-finite lattice with a double meaning. Considering a $ ^{\ast}$-finite lattice with a double meaning, we find how to treat the domain for a functional in our theory of Fourier transformation, and calculate two typical examples.

Motivation & Objective

• To develop a nonstandard analysis framework for Fourier transformation on the space of functionals, extending prior work on functions to functionals.
• To address the need for higher-order infinitesimals in Feynman path integrals by introducing a double extension $^{ullet}({}^*bR)$ of the real numbers.
• To define a rigorous infinitesimal Fourier transform for functionals over hyperfinite lattices, modeling infinite-dimensional function spaces via internal functions.
• To generalize Kinoshita and Gordon’s discrete Fourier transforms to the functional setting, preserving key properties like $F ilde{ ho} = 1$ for delta-like functionals.
• To demonstrate convergence of the nonstandard transform to the classical Fourier transform via standard part maps, particularly for Gaussian and delta-type functionals.

Proposed method

• Extends the real numbers to $^{ullet}({}^*bR)$ using nonstandard analysis to handle higher-order infinitesimals and infinities required for functional spaces.
• Models the domain of functionals as internal functions from a hyperfinite lattice $[-H/2, H/2)$ to another $[-H'/2, H'/2)$, enabling discrete approximation of functionals.
• Adopts a discrete Fourier transform formulation inspired by Kinoshita and Gordon: $(F ilde{ ho})(p) = rac{1}{H} ext{sum over } z ext{ in } [-H^2/2, H^2/2) ext{ of } ext{exp}(-2 ilde{ ho}ipz/H) ilde{ ho}(z/H)$.
• Applies the standard part map (st) to the nonstandard transform to recover classical Fourier transform behavior, showing convergence under conditions like $M o ilde{ ho}^{-1}$ and $M ilde{ ho} o ilde{ ho}^{-1}$.
• Uses hyperfinite sums and nonstandard analysis to approximate integrals over $bR$, such as $ ext{st}ig( ext{sum over } z ext{ of } ilde{ ho} ext{exp}(- ilde{ ho}^2 z^2)ig) = ext{int}_{-bR} ext{exp}(-x^2)dx = 1$.
• Analyzes error terms via bounds on exponential sums, showing that the difference between discrete sum and integral is infinitesimal in $^{ullet}({}^*bR)$, e.g., $| ext{sum} - ext{int} | o 0$ as $ ilde{ ho} o 0$.

Experimental results

Research questions

• RQ1 How can a Fourier transformation be rigorously defined for functionals on infinite-dimensional spaces using nonstandard analysis?
• RQ2 What role do higher-order infinitesimals (via $^{ullet}({}^*bR)$) play in enabling a consistent functional Fourier transform?
• RQ3 Can the standard part of the nonstandard Fourier transform of a functional converge to the classical Fourier transform?
• RQ4 How do delta-like functionals behave under the infinitesimal Fourier transform, and do they satisfy $F ilde{ ho} = 1$?
• RQ5 What conditions ensure that the discrete nonstandard transform approximates the continuous Fourier transform for functionals with Gaussian or compact support?

Key findings

• The standard part of the infinitesimal Fourier transform of a functional converges to the classical Fourier transform, as shown by $ ext{st}( ext{st}(C_2(b))) = 1$ for Gaussian-type functionals.
• The transform of the infinitesimal delta functional $ ilde{ ho}$ satisfies $F ilde{ ho} = 1$, extending the classical property to the nonstandard setting.
• The error between the discrete sum and the continuous integral is infinitesimal in $^{ullet}({}^*bR)$, bounded by terms like $ ext{exp}( ilde{ ho}^2) imes 2 ilde{ ho} imes ext{sum of exponentials}$, which vanish in the standard part.
• For Gaussian functionals, the nonstandard sum $ ext{sum}_z ilde{ ho} ext{exp}(- ilde{ ho}^2 z^2)$ has standard part equal to 1, matching $ ext{int}_{-bR} ext{exp}(-x^2)dx = 1$.
• The transform of a product of Gaussians and cosine terms converges to the classical integral $ ext{int} ext{exp}(- ilde{ ho}^2(x^2 - b^2)) ext{cos}(2 ilde{ ho} b x) dx$, with error bounded by an infinitesimal expression.
• The inverse transform behavior is preserved: $(1 + 1/(N ext{int} ext{exp}(- ilde{ ho}^2 x^2)dx))^{N} o ext{exp}(- ext{int} ext{exp}(- ilde{ ho}^2 x^2)dx)$, and its standard part is 1 when $N$ is infinite.