FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W1934490935> ?p ?o ?g. }

• W1934490935 endingPage "4347" @default.
• W1934490935 startingPage "4297" @default.
• W1934490935 abstract "For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 greater-than beta greater-than 4> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>></mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>2>beta >4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we analyze the behavior, near the rational points <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x equals p pi slash q> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>x=ppi /q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts n Superscript negative beta Baseline exp left-parenthesis i x n cubed right-parenthesis> <mml:semantics> <mml:mrow> <mml:munderover> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:munderover> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mi>β<!-- β --></mml:mi> </mml:mrow> </mml:msup> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>i</mml:mi> <mml:mi>x</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>sum ^infty _{n=1}n^{-beta }exp (ixn^{3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, considered as a function of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding=application/x-tex>x</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We expand this series into a constant term, a term on the order of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis x minus p pi slash q right-parenthesis Superscript left-parenthesis beta minus 1 right-parenthesis slash 3> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>(x-ppi /q)^{(beta -1)/3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a term linear in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x minus p pi slash q> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>x-ppi /q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a “chirp term on the order of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis x minus p pi slash q right-parenthesis Superscript left-parenthesis 2 beta minus 1 right-parenthesis slash 4> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>β<!-- β --></mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>(x-ppi /q)^{(2beta -1)/4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and an error term on the order of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis x minus p pi slash q right-parenthesis Superscript beta slash 2> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mi>π<!-- π --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>β<!-- β --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>(x-ppi /q)^{beta /2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=beta less-than-or-equal-to left-parenthesis StartRoot 97 EndRoot minus 1 right-parenthesis slash 4 equals 2.212 ellipsis> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:msqrt> <mml:mn>97</mml:mn> </mml:msqrt> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>4</mml:mn> <mml:mo>=</mml:mo> <mml:mn>2.212</mml:mn> <mml:mo>…<!-- … --></mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>beta le (sqrt {97}-1)/4=2.212dots</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, both the real and imaginary parts of the cubic series are differentiable almost nowhere." @default.
• W1934490935 created "2016-06-24" @default.
• W1934490935 creator A5033354252 @default.
• W1934490935 date "2003-07-02" @default.
• W1934490935 modified "2023-10-17" @default.
• W1934490935 title "On cubic lacunary Fourier series" @default.
• W1934490935 cites W1965816151 @default.
• W1934490935 cites W2003880880 @default.
• W1934490935 cites W2021421272 @default.
• W1934490935 cites W2025576058 @default.
• W1934490935 cites W2044023594 @default.
• W1934490935 cites W2060556643 @default.
• W1934490935 cites W2079081125 @default.
• W1934490935 cites W2316976966 @default.
• W1934490935 cites W2325716611 @default.
• W1934490935 cites W4237726272 @default.
• W1934490935 cites W4246973793 @default.
• W1934490935 doi "https://doi.org/10.1090/s0002-9947-03-03149-0" @default.
• W1934490935 hasPublicationYear "2003" @default.
• W1934490935 type Work @default.
• W1934490935 sameAs 1934490935 @default.
• W1934490935 citedByCount "10" @default.
• W1934490935 countsByYear W19344909352014 @default.
• W1934490935 countsByYear W19344909352015 @default.
• W1934490935 countsByYear W19344909352017 @default.
• W1934490935 countsByYear W19344909352023 @default.
• W1934490935 crossrefType "journal-article" @default.
• W1934490935 hasAuthorship W1934490935A5033354252 @default.
• W1934490935 hasBestOaLocation W19344909351 @default.
• W1934490935 hasConcept C11413529 @default.
• W1934490935 hasConcept C154945302 @default.
• W1934490935 hasConcept C41008148 @default.
• W1934490935 hasConceptScore W1934490935C11413529 @default.
• W1934490935 hasConceptScore W1934490935C154945302 @default.
• W1934490935 hasConceptScore W1934490935C41008148 @default.
• W1934490935 hasIssue "11" @default.
• W1934490935 hasLocation W19344909351 @default.
• W1934490935 hasOpenAccess W1934490935 @default.
• W1934490935 hasPrimaryLocation W19344909351 @default.
• W1934490935 hasRelatedWork W2051487156 @default.
• W1934490935 hasRelatedWork W2073681303 @default.
• W1934490935 hasRelatedWork W2351491280 @default.
• W1934490935 hasRelatedWork W2371447506 @default.
• W1934490935 hasRelatedWork W2386767533 @default.
• W1934490935 hasRelatedWork W2748952813 @default.
• W1934490935 hasRelatedWork W2899084033 @default.
• W1934490935 hasRelatedWork W303980170 @default.
• W1934490935 hasRelatedWork W3141679561 @default.
• W1934490935 hasRelatedWork W4225307033 @default.
• W1934490935 hasVolume "355" @default.
• W1934490935 isParatext "false" @default.
• W1934490935 isRetracted "false" @default.
• W1934490935 magId "1934490935" @default.
• W1934490935 workType "article" @default.