• Title/Summary/Keyword: F norm Function

Search Result 39, Processing Time 0.021 seconds

FOURIER TRANSFORM OF ANISOTROPIC MIXED-NORM HARDY SPACES WITH APPLICATIONS TO HARDY-LITTLEWOOD INEQUALITIES

  • Liu, Jun;Lu, Yaqian;Zhang, Mingdong
    • Journal of the Korean Mathematical Society
    • /
    • v.59 no.5
    • /
    • pp.927-944
    • /
    • 2022
  • Let $\vec{p}{\in}(0,\;1]^n$ be an n-dimensional vector and A a dilation. Let $H^{\vec{p}}_A(\mathbb{R}^n)$ denote the anisotropic mixed-norm Hardy space defined via the radial maximal function. Using the known atomic characterization of $H^{\vec{p}}_A(\mathbb{R}^n)$ and establishing a uniform estimate for corresponding atoms, the authors prove that the Fourier transform of $f{\in}H^{\vec{p}}_A(\mathbb{R}^n)$ coincides with a continuous function F on ℝn in the sense of tempered distributions. Moreover, the function F can be controlled pointwisely by the product of the Hardy space norm of f and a step function with respect to the transpose matrix of A. As applications, the authors obtain a higher order of convergence for the function F at the origin, and an analogue of Hardy-Littlewood inequalities in the present setting of $H^{\vec{p}}_A(\mathbb{R}^n)$.

GROWTH NORM ESTIMATES FOR ¯∂ ON CONVEX DOMAINS

  • Cho, Hong-Rae;Kwon, Ern-Gun
    • Communications of the Korean Mathematical Society
    • /
    • v.22 no.1
    • /
    • pp.111-119
    • /
    • 2007
  • We consider the growth norm of a measurable function f defined by defined by $${\parallel}f{\parallel}-\sigma=ess\;sup\{\delta_D(z)^\sigma{\mid}f(z)\mid:z{\in}D\}$$, where $\delta_D(z)$ denote the distance from z to ${\partial}D$. We prove some kind of optimal growth norm estimates for a on convex domains.

LIPSCHITZ TYPE CHARACTERIZATION OF FOCK TYPE SPACES

  • Hong Rae, Cho;Jeong Min, Ha
    • Bulletin of the Korean Mathematical Society
    • /
    • v.59 no.6
    • /
    • pp.1371-1385
    • /
    • 2022
  • For setting a general weight function on n dimensional complex space ℂn, we expand the classical Fock space. We define Fock type space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ of entire functions with a mixed norm, where 0 < p, q < ∞ and t ∈ ℝ and prove that the mixed norm of an entire function is equivalent to the mixed norm of its radial derivative on $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$. As a result of this application, the space $F^{p,q}_{{\phi},t}({\mathbb{C}}^n)$ is especially characterized by a Lipschitz type condition.

EQUIVALENT NORMS IN A BANACH FUNCTION SPACE AND THE SUBSEQUENCE PROPERTY

  • Calabuig, Jose M.;Fernandez-Unzueta, Maite;Galaz-Fontes, Fernando;Sanchez-Perez, Enrique A.
    • Journal of the Korean Mathematical Society
    • /
    • v.56 no.5
    • /
    • pp.1387-1401
    • /
    • 2019
  • Consider a finite measure space (${\Omega}$, ${\Sigma}$, ${\mu}$) and a Banach space $X({\mu})$ consisting of (equivalence classes of) real measurable functions defined on ${\Omega}$ such that $f{\chi}_A{\in}X({\mu})$ and ${\parallel}f{\chi}_A{\parallel}{\leq}{\parallel}f{\parallel}$, ${\forall}f{\in}({\mu})$, $A{\in}{\Sigma}$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.

A New Wavelet Watermarking Based on Linear Bit Expansion (선형계수확장 기반의 새로운 웨이블릿 워터마킹)

  • Piao, Yong-Ri;Kim, Seok-Tae
    • The Journal of Korean Institute of Communications and Information Sciences
    • /
    • v.32 no.1C
    • /
    • pp.16-22
    • /
    • 2007
  • This study proposes a new wavelet watermark technique based on the Linear Bit Expansion. To ensure the security of the watermark, enlarged watermark by applying linear bit expansion is inserted in a given intensity to a low frequency subband of the image which is wavelet transformed after the Arnold Transformation. When detecting the presence of watermart F norm function is applied unlike the existing methods. The experiment results verify that the proposed watermarking technique has outstanding quality in regards to fidelity and robustness.

STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION IN QUASI NORMED SPACES

  • Mirmostafaee, Alireza Kamel
    • Bulletin of the Korean Mathematical Society
    • /
    • v.47 no.4
    • /
    • pp.777-785
    • /
    • 2010
  • Let X be a linear space and Y be a complete quasi p-norm space. We will show that for each function f : X $\rightarrow$ Y, which satisfies the inequality ${\parallel}{\Delta}_x^nf(y)\;-\;n!f(x){\parallel}\;{\leq}\;\varphi(x,y)$ for suitable control function $\varphi$, there is a unique monomial function M of degree n which is a good approximation for f in such a way that the continuity of $t\;{\mapsto}\;f(tx)$ and $t\;{\mapsto}\;\varphi(tx,\;ty)$ imply the continuity of $t\;{\mapsto}\;M(tx)$.

BOUNDED FUNCTION ON WHICH INFINITE ITERATIONS OF WEIGHTED BEREZIN TRANSFORM EXIST

  • Jaesung Lee
    • Korean Journal of Mathematics
    • /
    • v.31 no.3
    • /
    • pp.305-311
    • /
    • 2023
  • We exhibit some properties of the weighted Berezin transform Tαf on L(Bn) and on L1(Bn). As the main result, we prove that if f ∈ L(Bn) with limk→∞ Tkαf exists, then there exist unique M-harmonic function g and $h{\in}{\bar{(I-T_{\alpha})L^{\infty}(B_n)}}$ such that f = g + h. We also show that of the norm of weighted Berezin operator Tα on L1(Bn, ν) converges to 1 as α tends to infinity, where ν is an ordinary Lebesgue measure.

A REMARK ON SOME INEQUALITIES FOR THE SCHATTEN p-NORM

  • HEDAYATIAN, K.;BAHMANI, F.
    • Honam Mathematical Journal
    • /
    • v.24 no.1
    • /
    • pp.9-23
    • /
    • 2002
  • For a closed densely defined linear operator T on a Hilbert space H, let ${\prod}$ denote the function which corresponds to T, the orthogonal projection from $H{\oplus}H$ onto the graph of T. We extend some ordinary norm ineqralites comparing ${\parallel}{\Pi}(A)-{\Pi}(B){\parallel}$ and ${\parallel}A-B{\parallel}$ to the Schatten p-norm where A and B are bounded operators on H.

  • PDF

NOTES ON EXTENDED NEURAL NETWORK APPROXIMATION

  • Hahm, Nahm-Woo;Hong, Bum-Il;Choi, Sung-Hee
    • Journal of applied mathematics & informatics
    • /
    • v.5 no.3
    • /
    • pp.867-875
    • /
    • 1998
  • In this paper we prove that any continuous function on a bounded closed interval of can be approximated by the superposition of a bounded sigmoidal function with a fixed weight. In addition we show that any continuous function over $\mathbb{R}$ which vanishes at infinity can be approximated by the superposition f a bounded sigmoidal function with a weighted norm. Our proof is constructive.

WEIGHTED LEBESGUE NORM INEQUALITIES FOR CERTAIN CLASSES OF OPERATORS

  • Song, Hi Ja
    • Korean Journal of Mathematics
    • /
    • v.14 no.2
    • /
    • pp.137-160
    • /
    • 2006
  • We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.

  • PDF