On the L{sup p}{sub {mu}}-strong property of orthonormal systems

Journal Article · Fri Oct 31 04:00:00 EST 2003 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM2003V194N10ABEH000774; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21208347

Grigorian, M G ^[1]

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1. Yerevan State University, Yerevan (Armenia)

Let {l_brace}{phi}{sub n}(x){r_brace} be a system of bounded functions complete and orthonormal in L{sup 2}{sub [0,1]} and assume that ||{phi}||{sub p{sub 0}}{<=}const, n{>=}1, for some p{sub 0}>2. Then the elements of the system can be rearranged so that the resulting system has the L{sup p}{sub {mu}}-strong property: for each {epsilon}>0 there exists a (measurable) subset E subset of [0,1] of measure |E|>1-{epsilon} and a measurable function {mu}(x), 0<{mu}(x){<=}1, {mu}(x)=1 on E such that for all p>2 and f(x) element of L{sup p}{sub {mu}}[0,1] one can find a function g(x) element of L{sup 1}{sub [0,1]} coinciding with f(x) on E such that its Fourier series in the system ({phi}{sub {sigma}}{sub (k)}(x)) converges to g(x) in the L{sup p}{sub {mu}}[0,1]-norm and the sequence of Fourier coefficients of this function belongs to all spaces l{sup q}, q>2.

• OSTI ID: 21208347
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 10 Vol. 194; ISSN 1064-5616
• Country of Publication: United States
• Language: English

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