The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient f (n) is of o(1) as n → ∞ for any integrable function f on Vilenkin groups. However, it is known that the Vilenkin-Fourier coefficients of integrable functions can tend to zero as slowly as we wish. The definitive result is due to B. L. Ghodadra for functions of certain classes of generalized bounded fluctuations. We prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between 'localness' of the hypothesis and type of lacunarity and allow us to interpolate the results.
Abstract. In this paper the β -absolute convergence ( 0 < β 2 ) of Vilenkin-Fourier series for the functions of various classes of functions of generalized bounded fluctuation is studied. In proving our main results we use famous Hölder's inequality and Jensen's inequality for integrals.
Abstract. For a Lebesgue integrable complex-valued function f defined on R , letf be its Fourier transform. The Riemann-Lebesgue lemma says thatf (t) → 0 as |t| → ∞ . But in general, there is no definite rate at which the Fourier transform tends to zero. In fact, the Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of L 1 (R) there is a definite rate at which the Fourier transform tends to zero. In this paper, we determine this rate for functions of bounded variation on R . We also determine such rate of Fourier transform for functions of bounded variation in the sense of Vitali defined on R N (N ∈ N) . Mathematics subject classification (2010): 42A38, 42B10, 26A12, 26A45, 26B30, 26D15. Keywords and phrases: Fourier transform, function of bounded variation over R , function of bounded variation over R 2 , function of bounded variation over R N , order of magnitude.
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