The special atom space and Haar wavelets in higher dimensions

• Autorzy: Kwessi Eddy , Souza de Geraldo , Djitte Ngalla , Ndiaye Mariama

Identyfikatory

DOI

10.1515/dema-2020-0011

• Języki publikacji: EN

Abstrakty

EN

In this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spacessuch as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higherdimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

Słowa kluczowe

EN

analytic function; special atom; Haar wavelets; high dimension

PL

funkcja analityczna; falka Haara

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2020
• Tom: Vol. 53, nr 1
• Strony: 131--151
• Opis fizyczny: Bibliogr. 9 poz., rys.

Twórcy

autor

Kwessi Eddy

• Department of Mathematics, Trinity University, San Antonio, TX 78212, USA,

autor

Souza de Geraldo

• Department of Mathematics, Auburn University, Auburn, AL 36849, USA

autor

Djitte Ngalla

• Department of Mathematics, Gaston Berger University, St. Louis, Senegal

autor

Ndiaye Mariama

• Department of Mathematics, Gaston Berger University, St. Louis, Senegal

Bibliografia

• [1] Ronald Coifman, A real variable characterization of hp, Studia Math. 51(1974), 269-274.
• [2] Geraldo De Souza, Spaces formed by special atoms, PhD thesis, SUNY at Albany, 1980.
• [3] Geraldo De Souza and Gary Sampson, Function in the Dirichlet space such that its Fourier series diverges almost everywhere, Proc. Am. Math. Soc. 120(1994) no. 3, 723-726.
• [4] Eddy Kwessi, Paul Alfonso, Geraldo De Souza, and Asheber Abebe, A note on multiplication and composition operators in Lorentz spaces, J. Funct. Spaces Appl. 2012(2012), 1-10, DOI: 10.1155/2012/293613.
• [5] Stephen Bloom and Geraldo De Souza, Atomic decomposition of generalized Lipschitz spaces, Illinois J. Math. 33(1989), no. 2, 181-209.
• [6] Geraldo De Souza, The atomic decomposition of Bergman-Besov-Lipschitz spaces, Proc. Am. Math. Soc. 14(1985), 682-686.
• [7] Steven Krantz, Handbook of Complex Variables, Birkhäuser Boston Inc., 1999.
• [8] Eddy Kwessi, Geraldo De Souza, Asheber Abebe, and Rauno Aulaskari, Characterization of Lacunary functions in Bergma-Besov-Lipschitz spaces, Complex Var. Elliptic Equ. 58(2013), no. 2, 157-162.
• [9] Antoni Zygmund, Trigonometric Series, vol. I and II, Cambridge Mathematical Library, 2002

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).