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• W3136361983 abstract "We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis bold-italic upper L squared left-parenthesis double-struck upper R right-parenthesis comma double-vertical-bar dot double-vertical-bar Subscript 2 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold-italic>L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mspace width=thinmathspace /> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mspace width=thinmathspace /> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding=application/x-tex>big ( {{{boldsymbol {L}}^2}(mathbb {R})}, , |,cdot ,|_{2} big )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold-script upper S prime left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic-bold mathvariant=bold-script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{{{boldsymbol {mathcal {S}}}’}(mathbb {R}^d)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>d geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis bold-italic upper Q Subscript s Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis comma double-vertical-bar dot double-vertical-bar Subscript bold-italic upper Q Sub Subscript s Subscript Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold-italic>Q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=thinmathspace /> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mspace width=thinmathspace /> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mspace width=thinmathspace /> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold-italic>Q</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> </mml:msub> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding=application/x-tex>big ( {{{boldsymbol {Q}}_s}({{mathbb {R}^{d}}})}, , |,cdot ,|_{{{boldsymbol {Q}}_s}} big )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, showing that they are special cases of Katsnelson’s setting (only) for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s greater-than-or-equal-to 0> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W3136361983 title "Completeness of shifted dilates in invariant Banach spaces of tempered distributions" @default.
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