On the Fourier series of Stepanov almost periodic functions

Tornehave, Hans

doi:10.7146/math.scand.a-10410

Cited by 4 publications

(7 citation statements)

References 3 publications

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“…1. The same argument holds for 0 < x < 2k, and then we get (12) for almost every x. This means that ei"t and e"I't (I X -V > 1) are orthogonal in s-measure.…”

mentioning

confidence: 64%

“…There are Nr different ordered r-tuples that can be formed from the integers 1, *--, N. We denote such an ordered r-tuple by {a1, I, pl}, or more briefly by {paj if also [ 6. We will next verify, as previously stated, that for the special (11) we have relation (12 N k Therefore (12) holds for 0 < x ? 1.…”

mentioning

confidence: 89%

“…We note that condition (9) cannot be omitted. We shall verify that for the absolutely convergent Fourier series (11) ~~~g(x) = " 1 Cos x (n=2 n(log n)1+ 2n for any s > 0, we have (12) (log N)4 BN 1 g(2kx)…”

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confidence: 98%

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Strong Law of Large Numbers for Sequences of Almost Periodic Functions

Bochner¹,

Izumi²

1959

The Annals of Mathematics

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“…1. The same argument holds for 0 < x < 2k, and then we get (12) for almost every x. This means that ei"t and e"I't (I X -V > 1) are orthogonal in s-measure.…”

mentioning

confidence: 64%

mentioning

confidence: 89%

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Strong Law of Large Numbers for Sequences of Almost Periodic Functions

Bochner¹,

Izumi²

1959

The Annals of Mathematics

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“…More precisely, the sequence of partial sums converges in S 2 to a limit f ∈ S 2 . If a n > 0 for every n, the converse is also true, see Tornehave [25].…”

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confidence: 98%

Pointwise convergence of almost periodic Fourier series and associated series of dilates

Cuny

Weber²

2018

Pacific J. Math.

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“…The sufficiency was proved by Wiener [27,Theorem 22,p. 583] and in a different way by Tornehave [26], who also proved the necessity. Moreover, by section 22 below, a trigonometrical series with non-negative coefficients is S 2 convergent if and only if it is the Fourier series of an S 2 almost periodic function.…”

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confidence: 99%

24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

Cossar

1976

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisThe series considered are of the form , where Σ | cn |2 is convergent and the real numbers λn (the exponents) are distinct. It is known that if the exponents are integers, the series is the Fourier series of a periodic function of locally integrable square (the Riesz-Fischer theorem); and more generally that if the exponents are not necessarily integers but are such that the difference between any pair exceeds a fixed positive number, the series is the Fourier series of a function of the Stepanov class, S2, of almost periodic functions.We consider in this paper cases where the exponents are subject to less stringent conditions (depending on the coefficients cn). Some of the theorems included here are known but had been proved by other methods. A fuller account of the contents of the paper is given in Sections 1-5.

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On the Fourier series of Stepanov almost periodic functions

Cited by 4 publications

References 3 publications

Strong Law of Large Numbers for Sequences of Almost Periodic Functions

Strong Law of Large Numbers for Sequences of Almost Periodic Functions

Pointwise convergence of almost periodic Fourier series and associated series of dilates

24.—Mean-square Convergence of Non-harmonic Trigonometrical Series

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