The Reconstruction of a function whose fourier transfrom is known by means of summability methods

ABSTRACT THE RECONSTRUCTION OF A FUNCTION WHOSE FOURIER TRANSFORM IS KNOWN BY MEANS OF SUMMABILITY METHODS Melih ERYİ?İT M.S. in Mathematics Advisor: Doç. Dr. İlham ALİYEV August, 1998, 33 pages The question " Given the Fourier transform of a Lebesgue integrable function /, how do we obtain / back again from its Fourier transform, similarly given the Fourier coefficients of a Lebesgue integrable function at [0, 27r], how do we obtain / back again from its Fourier coefficients ? " was an important problem of harmonic analysis and its applications, and it has drawn many famous mathematicans' atten tions, for example, Hardy and Littlewod, Cesaro, Poisson, Bochner, Riesz, Zygmund etc. The Fourier transform of a Lebesgue integrable function may not be integrable therefore we could not obtain / back again by inverse Fourier transform of /. In order to get rid of this difficulty, mathematicians reached different summability methods. The most well-known ones of these methods are Abel-Poisson, Gauss- Weierstrass and Riesz-Bochner. In this work we concentrate on the above mentioned summabilitiy methods. We have olso pointed out their significant role in harmonic analysis. In the last section, we study Riesz-Bochner means' convergence rate at some kind of smoothness point of the function in details. KEY WORDS : The Fourier transform, Poisson-Gauss Summability, Gauss- Weierstrass Summability, Riesz-Bochner Summability, //-Smoothness, Lebesgue point, Convergence rate. COMMITTEE : Doç. Dr. Ilham ALIYEV Prof. Dr. Halil İ. KARAKAŞ Prof. Dr. Doğan ÇÖKER ıı