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### Session S05 - Advances on Spaces of Non-absolutely Integrable Functions and Related Applications

Monday, July 12, 14:20 ~ 14:50 UTC-3

## A Constructive Definition of the Fourier Transform on a Separable Banach Space

### Timothy Myers

#### Howard University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakfbb9c0d7d44fe4c3753b702181ae9924').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyfbb9c0d7d44fe4c3753b702181ae9924 = 'tmy&#101;rs' + '&#64;'; addyfbb9c0d7d44fe4c3753b702181ae9924 = addyfbb9c0d7d44fe4c3753b702181ae9924 + 'h&#111;w&#97;rd' + '&#46;' + '&#101;d&#117;'; var addy_textfbb9c0d7d44fe4c3753b702181ae9924 = 'tmy&#101;rs' + '&#64;' + 'h&#111;w&#97;rd' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakfbb9c0d7d44fe4c3753b702181ae9924').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyfbb9c0d7d44fe4c3753b702181ae9924 + '\'>'+addy_textfbb9c0d7d44fe4c3753b702181ae9924+'<\/a>';

Gill and Myers proved that every separable Banach space, denoted $\mathcal{B}$, has an isomorphic, isometric embedding in $\mathbb{R}^{\infty}=\mathbb{R}\times\mathbb{R}\times\cdots$ . They used this result and a method due to Yamasaki to construct a sigma-finite Lebesgue measure $\lambda_{\mathcal{B}}$ for $\mathcal{B}$\ and defined the associated integral $\int_{\mathcal{B}}\cdot d\lambda_{\mathcal{B}}$ in a way that equals a limit of finite-dimensional Lebesgue integrals.

The objective of this talk is to develop a constructive definition of the Fourier transform on $L^1[\mathcal{B}]$. Our approach is constructive in the sense that this Fourier transform is defined as an integral on $\mathcal{B}$, which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension $n\to\infty$. Because this transform has all of the familiar properties, we will use these to derive the solution to the diffuse equation on $\mathcal{B}$.