| Summary: | The Banach Gelfand Triple <inline-formula> <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msup> <mrow> <mi mathvariant="bold">L</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <msubsup> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> consists of <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msub> <mrow> <mo>∥</mo> <mrow> <mspace width="0.166667em"></mspace> <mo>·</mo> <mspace width="0.166667em"></mspace> </mrow> <mo>∥</mo> </mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> </msub> </mfenced> </semantics> </math> </inline-formula>, a very specific <i>Segal algebra</i> as algebra of test functions, the Hilbert space <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mrow> <msup> <mrow> <mi mathvariant="bold">L</mi> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mspace width="0.166667em"></mspace> <msub> <mrow> <mo>∥</mo> <mrow> <mspace width="0.166667em"></mspace> <mo>·</mo> <mspace width="0.166667em"></mspace> </mrow> <mo>∥</mo> </mrow> <mn>2</mn> </msub> </mfenced> </semantics> </math> </inline-formula> and the dual space <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, whose elements are also called <i>“mild distributions”</i>. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of <i>Gabor Analysis</i>, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msub> <mrow> <mo>∥</mo> <mrow> <mspace width="0.166667em"></mspace> <mo>·</mo> <mspace width="0.166667em"></mspace> </mrow> <mo>∥</mo> </mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> </msub> </mfenced> </semantics> </math> </inline-formula> and hence <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mrow> <msubsup> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <mo>∥</mo> <mrow> <mspace width="0.166667em"></mspace> <mo>·</mo> <mspace width="0.166667em"></mspace> </mrow> <msub> <mo>∥</mo> <msubsup> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </msub> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called <i>mild Cauchy sequences</i> (in short <b>ECmiCS</b>). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra <inline-formula> <math display="inline"> <semantics> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>)</mo> </mrow> </mrow> <mo>,</mo> <msub> <mrow> <mo>∥</mo> <mrow> <mspace width="0.166667em"></mspace> <mo>·</mo> <mspace width="0.166667em"></mspace> </mrow> <mo>∥</mo> </mrow> <msub> <mi mathvariant="bold">S</mi> <mrow> <mspace width="-1.66672pt"></mspace> <mn>0</mn> </mrow> </msub> </msub> </mfenced> </semantics> </math> </inline-formula> can be used to establish this natural identification.
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