Hilbert Transforms of Periodic Functions

1971 ◽  
pp. 334-354 ◽  
Author(s):  
Paul L. Butzer ◽  
Rolf J. Nessel
Keyword(s):  
Periodic Functions ◽  
Hilbert Transforms

Hilbert transforms and almost periodic functions

1960 ◽  
Vol 56 (4) ◽  
pp. 354-366 ◽  
Author(s):  
J. Cossar
Keyword(s):  
Hilbert Transform ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Hilbert Transforms ◽  
Cauchy Principal ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

The Hilbert transform, Hf, of a function f is defined by Hf = g, whereP denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists andIn the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

Hilbert transforms of periodic functions

1973 ◽  
Vol 20 (4) ◽  
pp. 454-454
Author(s):  
A. Papoulis
Keyword(s):  
Periodic Functions ◽  
Hilbert Transforms

(VC)(n)-ALMOST PERIODIC FUNCTIONS

1999 ◽  
Vol 32 (2) ◽  
Author(s):  
Stanislaw Stoinski
Keyword(s):  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

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