On the convolution algebra of Beurling

Sunouchi, Gen-ichirô

doi:10.2748/tmj/1178243279

Cited by 10 publications

(5 citation statements)

References 9 publications

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“…Now it is not difficult to see that the two series within the square brackets are covergent if and only if (5) and 2 (3) follows; and hence the proof.…”

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confidence: 81%

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Contractions of functions and their Fourier series

Yadav¹

1969

Pacific J. Math.

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The object of the present paper is to define a new type of contraction, called ' Shrivel', of a function and to prove a theorem on the absolute convergence of its Fourier series. Our theorem is similar to a theorem of M. Kinukawa, but as it is shown in the end the two results are essentially different. The original results in this direction are due to A. Beurling and R. P. Boas. According to Beurling [1] a function g is said to be a contraction of function / if | g(x)-g(y) | ^ A \f(x)-f(y) |, for all x, y, where A is an absolute constant. We shall assume throughout that the functions / and g are each L-integrable in (π, π) and periodic with the period 2π. We shall further let

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“…Now it is not difficult to see that the two series within the square brackets are covergent if and only if (5) and 2 (3) follows; and hence the proof.…”

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confidence: 81%

“…However, it has been shown by Leindler [4] and Sunouchi [5] 1 that the conditions (1) and 2 The object of this paper is to prove Theorem 2 which is similar to Theorem 1; but as we shall see in the end the two theorems are independent of each other. Before we can state our result precisely, we need introduce a couple of notations and a definition.…”

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confidence: 86%

Contractions of functions and their Fourier series

Yadav¹

1969

Pacific J. Math.

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“…(i) The subject of Theorems 1 and 2 was initiated by Beurling [2], Boas [3], Sunouchi [27]. Results related to Theorem 1 are found in IIardy-Littlewood [6], Leindler [17], [19], [20], Kinukawa [12] and Totik-Vincze [28].…”

Section: W 1 the General Bernstein Theoremmentioning

confidence: 99%

Inequalities related to smoothness conditions and Fourier series

Kinukawa

1994

Period Math Hung

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“…However, our proof of Theorems 3 and 4 adopted in this paper is quite elementary. That is, we shall discuss the problem along the line set by Beurling [1], Boas [2], Sunouchi [6], and Kinukawa [5].…”

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confidence: 99%

“…(Cf. Sunouchi[6]) Note that a BUF) = κ\~rH"( I F(x) fd Let 0 < p < a and 0 < a < j, then a C p , s , a (F)^K a B P , a (F). PROOF.…”

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confidence: 99%

Contractions of Fourier transforms in $R_{k}$

Kinukawa¹

1972

Tohoku Math. J. (2)

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Introduction.In the present paper, we shall characterize some functions, those which satisfy a Lipschitz condition, as Fourier transforms of a certain sub-class of L p (R k ), and we shall give a contraction theorem of L p -Fourier transforms.A complex valued function f(x ly x 2 , , x k ) on R k , the k-άim. Euclidean space, is denoted by f(x).When / has the following property (i) or (ii), we say / is (p)normalized: for any finiteJi+y interval I, where 1/p + 1/p' = 1; (ii) if p = 1, then / is continuous and lim^,.^ f(x) = 0. We denote the j-th difference of /(a?), with respect to heR k , by Δl(f(x)), that is, Δί(f(x)) -Σ(-l) i+m ( 3 )f(x + mh) . m=0 \m)

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On the convolution algebra of Beurling

Cited by 10 publications

References 9 publications

Contractions of functions and their Fourier series

Contractions of functions and their Fourier series

Inequalities related to smoothness conditions and Fourier series

Contractions of Fourier transforms in $R_{k}$

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