Some methods of constructing kernels in statistical learning

Tomasz Górecki; Maciej Łuczak

Displaying similar documents to “Some methods of constructing kernels in statistical learning”

Strict plurisubharmonicity of Bergman kernels on generalized annuli

Yanyan Wang (2014)

Annales Polonici Mathematici

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Let $A_{ζ} = Ω - \bar{ρ (ζ) \cdot Ω}$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{ζ} (z)$ of $A_{ζ}$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{ζ}$ is non-pseudoconvex when the dimension of $A_{ζ}$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ² l o g K_{ζ} / \partial ζ \partial ζ ̅$ , as well as its boundary behavior.

The method of rotation and Marcinkiewicz integrals on product domains

Jiecheng Chen, Dashan Fan, Yiming Ying (2002)

Studia Mathematica

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We give some rather weak sufficient condition for $L^{p}$ boundedness of the Marcinkiewicz integral operator $μ_{Ω}$ on the product spaces $ℝ ⁿ \times ℝ^{m}$ (1 < p < ∞), which improves and extends some known results.

Bounded evaluation operators from $H^{p}$ into $ℓ^{q}$

Martin Smith (2007)

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z, p} (f) = {(1 - | z ₙ | ²)}^{1 / p} f (z ₙ)$ . Necessary and sufficient conditions on zₙ are given such that $T_{z, p}$ maps the Hardy space $H^{p}$ boundedly into the sequence space $ℓ^{q}$ . A corresponding result for Bergman spaces is also stated.

Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

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Let 1 < p < ∞, q = p/(p-1) and for $f \in L^{p} (0, \infty)$ define $F (x) = (1 / x) ʃ_{0}^{x} f (t) d t$ , x > 0. Moser’s Inequality states that there is a constant $C_{p}$ such that $s u p_{a \leq 1} s u p_{f \in B_{p}} ʃ_{0}^{\infty} e x p [a x^{q} {| F (x) |}^{q} - x] d x = C_{p}$ where $B_{p}$ is the unit ball of $L^{p}$ . Moreover, the value a = 1 is sharp. We observe that $F = K_{1}$ f where the integral operator $K_{1}$ has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...

$L^{p} - L^{q}$ boundedness of analytic families of fractional integrals

Valentina Casarino, Silvia Secco (2008)

Studia Mathematica

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We consider a double analytic family of fractional integrals $S_{z}^{γ, α}$ along the curve ${t \mapsto | t |}^{α}$ , introduced for α = 2 by L. Grafakos in 1993 and defined by $(S_{z}^{γ, α} f) (x ₁, x ₂) : = 1 / Γ (z + 1 / 2) {\int \int | u - 1 |}^{z} ψ (u - 1) {f (x ₁ - t, x ₂ - u | t |}^{α} {) d u | t |}^{γ} d t / t$ , where ψ is a bump function on ℝ supported near the origin, $f \in_{c}^{\infty} (ℝ ²)$ , z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_{z}^{γ, α}$ maps $L^{p} (ℝ ²)$ to $L^{q} (ℝ ²)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{- 1 + i θ}^{i ϱ, α}$ is a product kernel on ℝ², adapted to the curve ${t \mapsto | t |}^{α}$ ; as a consequence, we show...

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $Δ_{A, B}$ with respect to the wider class of unitarily invariant...

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the $L^{p}$ boundedness of the Marcinkiewicz integral operators $μ_{Ω}$ on $ℝ^{n ₁} \times \dots \times ℝ^{n_{k}}$ under the condition that $Ω \in L {(l o g L)}^{k / 2} (^{n ₁ - 1} \times \dots \times^{n_{k} - 1})$ . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Classical boundary value problems for integrable temperatures in a $C^{1}$ domain

Anna Grimaldi Piro, Francesco Ragnedda (1991)

Annales Polonici Mathematici

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Abstract. We study a Neumann problem for the heat equation in a cylindrical domain with $C^{1}$ -base and data in $h_{c}^{1}$ , a subspace of $L$ 1. We derive our results, considering the action of an adjoint operator on $B_{T} M O C$ , a predual of $h_{c}^{1}$ , and using known properties of this last space.

$L^{p}$ type mapping estimates for oscillatory integrals in higher dimensions

G. Sampson (2006)

Studia Mathematica

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We show in two dimensions that if $K f = \int_{ℝ ₊ ²} k (x, y) f (y) d y$ , $k (x, y) = (e^{i x^{a} \cdot y^{b}}) {/ (| x - y |}^{η})$ , p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_{p} (y) = y^{(p / p^{'}) (1 ̅ - b / a)}$ , then ${| | K f | |}_{p} {\leq C | | f | |}_{p, v_{p}}$ if η + α₁ + α₂ < 2, $α_{j} = 1 - b_{j} / a_{j}$ , j = 1,2. Our methods apply in all dimensions and also for more general kernels.

Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric

Ngaiming Mok (2012)

Journal of the European Mathematical Society

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We study the extension problem for germs of holomorphic isometries $f : (D; x_{0}) \to (Ω; f (x_{0}))$ up to normalizing constants between bounded domains in Euclidean spaces equipped with Bergman metrics $d s_{D}^{2}$ on $D$ and $d s_{Ω}^{2}$ on $Ω$ . Our main focus is on boundary extension for pairs of bounded domains $(D, Ω)$ such that the Bergman kernel $K_{D} (z, w)$ extends meromorphically in $(z, \bar{w})$ to a neighborhood of $\bar{D} \times D$ , and such that the analogous statement holds true for the Bergman kernel $K_{Ω} (ς, ξ)$ on $Ω$ . Assuming that $(D; d s_{D}^{2})$ and $(Ω; d s_{Ω}^{2})$ are complete Kähler manifolds, we prove that...

A-Statistical Convergence of Subsequence of Double Sequences

Harry I. Miller (2007)

Bollettino dell'Unione Matematica Italiana

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The concept of statistical convergence of a sequence was first introduced by H. Fast [7] in 1951. Recently, in the literature, the concept of statistical convergence of double sequences has been studied. The main result in this paper is a theorem that gives meaning to the statement: $s={s_{ij}}$ converges statistically $A$ to $L$ if and only if "most" of the "subsequences" of $s$ converge to $L$ in the ordinary sense. The results presented here are analogue of theorems in [12], [13] and [6] and are concerned...

Endomorphism kernel property for finite groups

Heghine Ghumashyan, Jaroslav Guričan (2022)

Mathematica Bohemica

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A group $G$ has the endomorphism kernel property (EKP) if every congruence relation $θ$ on $G$ is the kernel of an endomorphism on $G$ . In this note we show that all finite abelian groups have EKP and we show infinite series of finite non-abelian groups which have EKP.

One-sided Gorenstein subcategories

Weiling Song, Tiwei Zhao, Zhaoyong Huang (2020)

Czechoslovak Mathematical Journal

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We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $𝒞$ of an abelian category $𝒜$ , and prove that the right Gorenstein subcategory $r 𝒢 (𝒞)$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $𝒞$ is self-orthogonal, we give a characterization for objects in $r 𝒢 (𝒞)$ , and prove that any object in $𝒜$ with finite $r 𝒢 (𝒞)$ -projective dimension is isomorphic to a kernel (or a cokernel) of a morphism from an object in $𝒜$ with finite $𝒞$ -projective...

About $w c s$ -covers and $w c s^{*}$ -networks on the Vietoris hyperspace $ℱ (X)$

Luong Quoc Tuyen, Ong V. Tuyen, Phan D. Tuan, Nguzen X. Truc (2023)

Commentationes Mathematicae Universitatis Carolinae

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We study some generalized metric properties on the hyperspace $ℱ (X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $w c s$ -covers if and only if so does $ℱ (X)$ . Moreover, $X$ has a sequence of $w c s$ -covers with property $(P)$ which is a point-star network if and only if so does $ℱ (X)$ , where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable....

The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

A generalization to the Hardy-Sobolev spaces $H^{k, p}$ of an $L^{p}$ - $L^{1}$ logarithmic type estimate

Imed Feki, Ameni Massoudi, Houda Nfata (2018)

Czechoslovak Mathematical Journal

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The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k, p} (G)$ ; $k \in ℕ^{*}$ , $1 \leq p \leq \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$ .

Embedding in $T_{3}$ - $F_{σ}$ spaces

D. W. Hajek, I. Irizarry (1981)

Matematički Vesnik

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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