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- W3043995685 abstract "Abstract For a sequence of continuous functions $$f_1,f_2,ldots :I rightarrow mathbb {R}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>:</mml:mo> <mml:mi>I</mml:mi> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> ( I is a fixed closed interval) with $$f_1le f_2 le cdots $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≤</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:math> define a set $$begin{aligned} I_f:=big {x in I : lim _{n rightarrow infty } f_n(x)=+infty big }. end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> <mml:mo>:</mml:mo> <mml:munder> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mrow> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> We study the properties of the family of all admissible $$I_f$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> -s and the family of all admissible $$I_f$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:math> -s under the additional assumption $$begin{aligned} lim _{n rightarrow infty } int _x^y f_n(t),dt=+infty quad text { for all }x,y in Itext { with }x<y. end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:munder> <mml:mo>lim</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:munder> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mi>x</mml:mi> <mml:mi>y</mml:mi> </mml:msubsup> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mo>+</mml:mo> <mml:mi>∞</mml:mi> <mml:mspace /> <mml:mspace /> <mml:mtext>for all</mml:mtext> <mml:mspace /> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> <mml:mspace /> <mml:mtext>with</mml:mtext> <mml:mspace /> <mml:mi>x</mml:mi> <mml:mo><</mml:mo> <mml:mi>y</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> The origin of this problem is the limit behaviour of quasiarithmetic means." @default.
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- W3043995685 date "2022-11-04" @default.
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- W3043995685 title "On the $$L^1$$ and pointwise divergence of continuous functions" @default.
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