In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows: The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow.
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| - Lemme du soleil levant (fr)
- Rising sun lemma (en)
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| - Le lemme du soleil levant est un lemme d'analyse réelle dû à Frigyes Riesz, utilisé dans une preuve du théorème maximal de Hardy-Littlewood. Ce lemme a été un précurseur en dimension 1 du lemme de Calderón-Zygmund. Son nom imagé vient du fait qu'il concerne les points du graphe d'une fonction, vu comme un paysage, qui sont dans l'ombre lorsque ce paysage est éclairé horizontalement par la droite. (fr)
- In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows: The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow. (en)
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| - Le lemme du soleil levant est un lemme d'analyse réelle dû à Frigyes Riesz, utilisé dans une preuve du théorème maximal de Hardy-Littlewood. Ce lemme a été un précurseur en dimension 1 du lemme de Calderón-Zygmund. Son nom imagé vient du fait qu'il concerne les points du graphe d'une fonction, vu comme un paysage, qui sont dans l'ombre lorsque ce paysage est éclairé horizontalement par la droite. (fr)
- In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. The lemma is stated as follows: Suppose g is a real-valued continuous function on the interval [a,b] and S is the set of x in [a,b] such that there exists a y∈(x,b] with g(y) > g(x). (Note that b cannot be in S, though a may be.) Define E = S ∩ (a,b).Then E is an open set, and it may be written as a countable union of disjoint intervalssuch that g(ak) = g(bk), unless ak = a ∈ S for some k, in which case g(a) < g(bk) for that one k. Furthermore, if x ∈ (ak,bk), then g(x) < g(bk). The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow. (en)
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