Äärellisten Borel-mittojen Fourier-muunnoksista euklidisissa avaruuksissa

6533b822fe1ef96bd127da8c

RESEARCH PRODUCT

Äärellisten Borel-mittojen Fourier-muunnoksista euklidisissa avaruuksissa

Joonas Niinikoski

subject

Gaussin kuvaus kapasitiivinen dimensio Fourier-muunnos polynomiaalinen väheneminen Frostmanin lemma Ruellen siirto-operaattori symbolinen dynamiikka äärellinen Borel-mitta Rieszin energia Gibbsin mitta Rajchman-mitta

description

Tutkielmassa esitellään euklidisen avaruuden äärellisille Borel-mitoille Fourier-muunnokset kompleksiarvoisina kuvauksina ja tutkitaan niiden vähenemistä mentäessä ääarettömän kauas origosta. Keskeisenä kysymyksenä on, millä reunaehdoilla annetun kompkatikantajaisen ja äärellisen Borel-mitan Fourier muunnos vähenee polynomiaalisesti (eli sitä voidaan dominoida jollain euklidisen normin negatiivisella potenssilla) kaikkialla riittävän kaukana tai ainakin ”keskimääräisesti”. Mikäli tällainen Borel-mitta on absoluuttisesti jatkuva Lebesguen mitan suhteen kompaktikantajaisella ja sileällä tiheysfunktiolla, niin sen Fourier-muunnos vähenee aina polynomiaalisesti kaikkialla. Ongelmaa tarkastellaan keskeisesti potentiaaliteorian avulla. Jokaiselle äärelliselle ja kompaktikantajaiselle Borel-mitalle asetetaan Rieszin energia. Tällöin Rieszin energian äärellisyys implikoi mitan Fourier-muunnoksen polynomiaalisen vähenemisen keskimäärin. Rieszin energian avulla määritellään myös ns. kapasitiivinen dimensio jokaiselle euklidisen avaruuden Borel-joukolle. Työssä käydään läpi myös klassinen Frostmanin lemma euklidisen avaruuden F_σ-joukoille, jonka perusteella tällaisten joukkojen kapasitiiviset dimensiot yhtyvät niiden Hausdorff-dimensioihin. Toinen näkökulma ongelmaan otetaan tarkastelemalla yksikkövälin irrationaalilukujen Gaussin kuvauksen dynaamista systeemiä ja siinä olevia Gibbsin mittoja. Sopivalla reunaehdolla tällaiset mitat vähenevät aina polynomiaalisesti. Tarkastelu edellyttää kuitenkin symbolista dynamiikkaa, joka on myös työssä yksi käsitelty aihe.

year	journal	country	edition	language
2016-01-01

http://urn.fi/URN:NBN:fi:jyu-201609274216

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