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• W2052805565 abstract "A nonnegative function on the real line satisfies the condition <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A Subscript normal infinity> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{{mathbf {A}}_infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if, given <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there exists a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>delta > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an interval, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E subset-of upper I> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>E subset I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartAbsoluteValue upper E EndAbsoluteValue greater-than delta StartAbsoluteValue upper I EndAbsoluteValue> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>E</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>I</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>|E| > delta |I|</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=integral Underscript upper E Endscripts upper W less-than-or-equal-to epsilon integral Underscript upper I Endscripts upper W> <mml:semantics> <mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>E</mml:mi> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>W</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>I</mml:mi> </mml:msub> <mml:mi>W</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>int _E {W leq varepsilon int _I W }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A nonnegative function on the real line satisfies the condition <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathbf {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if for every interval <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I comma integral Underscript 2 upper I Endscripts upper W less-than-or-equal-to upper C integral Underscript upper I Endscripts upper W> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>2</mml:mn> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>W</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>C</mml:mi> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi>I</mml:mi> </mml:msub> <mml:mi>W</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>I,int _{2I} {W leq C} int _I W</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 upper I> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the interval with the same center as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and twice as long, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is independent of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding=application/x-tex>I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. An example is given of a function that satisfies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathbf {A}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> but not <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper A Subscript normal infinity> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>A</mml:mi> </mml:mrow> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{{mathbf {A}}_infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2052805565 date "1974-01-01" @default.
• W2052805565 modified "2023-10-14" @default.
• W2052805565 title "Two nonequivalent conditions for weight functions" @default.
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