Generalized localization for the double trigonometric Fourier series and the Walsh-Fourier series of functions in L log {sup +}L log {sup +} log {sup +}L

Journal Article · Tue Jun 30 04:00:00 EDT 1998 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM1998V189N05ABEH000318; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21202779

Bloshanskaya, S K ^[1]; Bloshanskii, I L ^[2]; Roslova, T Y ^[3]

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1. Moscow Engineering Physics Institute (State University), Moscow (Russian Federation)
2. Moscow State Pedagogical University, Moscow (Russian Federation)
3. Moscow Pedagogical University, Moscow (Russian Federation)

For an arbitrary open set {omega} subset of I{sup 2}=[0,1){sup 2} and an arbitrary function f element of L log {sup +}L log {sup +} log {sup +}L(I{sup 2}) such that f=0 on {omega} the double Fourier series of f with respect to the trigonometric system {psi}=E and the Walsh-Paley system {psi}=W is shown to converge to zero (over rectangles) almost everywhere on {omega}. Thus, it is proved that generalized localization almost everywhere holds on arbitrary open subsets of the square I{sup 2} for the double trigonometric Fourier series and the Walsh-Fourier series of functions in the class L log {sup +}L log {sup +} log {sup +}L (in the case of summation over rectangles). It is also established that such localization breaks down on arbitrary sets that are not dense in I{sup 2}, in the classes {phi}{sub {psi}}(L)(I{sup 2}) for the orthonormal system {psi}=E and an arbitrary function such that {phi}{sub E}(u)=o(u log {sup +} log {sup +}u) as u{yields}{infinity} or for {phi}{sub W}(u)=u( log {sup +} log {sup +}u){sup 1-{epsilon}}, 0<{epsilon}<1.

• OSTI ID: 21202779
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 5 Vol. 189; ISSN 1064-5616
• Country of Publication: United States
• Language: English

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