Summability Of Fourier Series Research Articles

On the Poisson summability of Fourier series

On the Poisson summability of Fourier series

Remarks on the summability of Fourier series

Remarks on the summability of Fourier series

Remarks on the Cesàro Summability of Fourier Series and its Conjugate Fourier Series

Remarks on the Cesàro Summability of Fourier Series and its Conjugate Fourier Series

On the Summability of Fourier Series. Fifth Note

On the Summability of Fourier Series. Fifth Note

On the summation of Fourier series

On the summation of Fourier series

On the Cesáro Summation of Fourier Series and Allied Series

On the Cesáro Summation of Fourier Series and Allied Series

On the strong summability of Fourier series

On the strong summability of Fourier series

On the summability of fourier series. III

On the summability of fourier series. III

The Absolute Summability of Fourier Series

The problem under discussion is the relation between the behaviour of a function ɸ (t) near a particular point t = x and the behaviour of the partial sums sn of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x, then the Fourier series at that point is summable in the Cesàro sense, and conversely. The exact order of the Cesàro means in each case is unspecified.

On a Method of Summation of Fourier Series

On a Method of Summation of Fourier Series

Addition to the Paper "On the Summability of Fourier Series II"

Addition to the Paper "On the Summability of Fourier Series II"

On the Summability of Fourier Series. II

On the Summability of Fourier Series. II

On the Summability of Fourier Series and the Bounded Variation of Power Series

On the Summability of Fourier Series and the Bounded Variation of Power Series

On the summability of Fourier series by Riesz's logarithmic means

On the summability of Fourier series by Riesz's logarithmic means

On the Summability of Fourier Series. I

On the Summability of Fourier Series. I

On a Method of Summation of Fourier Series

On a Method of Summation of Fourier Series

Summation of Fourier series

Summation of Fourier series

On the Summability of Fourier Series

In Theorem 4, that ϕ(t) = ½{f(x+t)+f(x−t)−2s}→0 (C,β) as t→0 whenever the Denjoy-Fourier series of f(x) is summable (C, α) to s, where η>α+1 and a ⩾−1, my proof for the case −1⩽α<0,β<1, was incomplete*. The proof may be completed by the following lemma, valid also for 0⩽a⩽1. LEMMA. If Σan cos nt is the Denjoy-Fourier series of ϕ(t) and an =0(na), then ϕβ(t)=βΣanγβ(nt), for t≠0, where β > a+1 and − 1 ⩽ a < 0. Proof. Suppose that 0 < β < 2. Then, writing γ β , ɛ ( x ) = ∫ 0 1 - ɛ ( 1 - u ) β - 1 cosxy du, we have†, since (1−u)β−1 is of bounded variation in (0, 1 − ɛ), ∫ 0 1 - ɛ ( 1 - u ) β - 1 φ ( t u ) d u = Σanγβ, ɛ(nt. Moreover, for a given t≠0, it is easily verified that γβ, ɛ(nt), and hence ∑ a n { γ β ( n t ) - γ β , ɛ ( n t ) } = ∑ n ⩽ 1 / ɛ o ( n α ) O ( ɛ β ) + ∑ n > 1 / ɛ o ( n α - β ) = o ( ɛ β - α - 1 The result follows by letting ɛ → 0.

On the summability of Fourier series. I

On the summability of Fourier series. I

On the Summability of Fourier Series. Fourth Note

On the Summability of Fourier Series. Fourth Note