On Fourier series with gaps
نویسندگان
چکیده
منابع مشابه
On lacunary series with random gaps
We prove Strassen’s law of the iterated logarithm for sums ∑N k=1 f(nkx), where f is a smooth periodic function on the real line and (nk)k≥1 is an increasing random sequence. Our results show that classical results of the theory of lacunary series remain valid for sequences with random gaps, even in the nonharmonic case and if the Hadamard gap condition fails.
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Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine exhibited some of the basic similarities and differences between the trigonometric orthonormal system and the Walsh system. He identified the Walsh functions with the full set of characters of the dyadic ...
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متن کاملOn the Absolute Convergence of Small Gaps Fourier Series of Functions
Let f be a 2π periodic function in L[0, 2π] and ∑∞ k=−∞ f̂(nk)e inkx be its Fourier series with ‘small’ gaps nk+1 − nk ≥ q ≥ 1. Here we have obtained sufficiency conditions for the absolute convergence of such series if f is of ∧ BV (p) locally. We have also obtained a beautiful interconnection between lacunary and non-lacunary Fourier series.
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ژورنال
عنوان ژورنال: Proceedings of the Japan Academy, Series A, Mathematical Sciences
سال: 1966
ISSN: 0386-2194
DOI: 10.3792/pja/1195522022