Displaying similar documents to “General uniform approximation theory by multivariate singular integral operators”

On global smoothness preservation in complex approximation

George A. Anastassiou, Sorin G. Gal (2002)

Annales Polonici Mathematici

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By using the properties of convergence and global smoothness preservation of multivariate Weierstrass singular integrals, we establish multivariate complex Carleman type approximation results with rates. Here the approximants fulfill the global smoothness preservation property. Furthermore Mergelyan's theorem for the unit disc is strengthened by proving the global smoothness preservation property.

Tractability of multivariate problems for weighted spaces of functions

H. Woźniakowski (2006)

Banach Center Publications

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We survey recent results on tractability of multivariate problems. We mainly restrict ourselves to linear multivariate problems studied in the worst case setting. Typical examples include multivariate integration and function approximation for weighted spaces of smooth functions.

Professor Haruo Yanai and multivariate analysis

Yoshio Takane (2016)

Special Matrices

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The late Professor Yanai has contributed to many fields ranging from aptitude diagnostics, epidemiology, and nursing to psychometrics and statistics. This paper reviews some of his accomplishments in multivariate analysis through his collaborative work with the present author, along with some untold episodes for the inception of key ideas underlying the work. The various topics covered include constrained principal component analysis, extensions of Khatri’s lemma, theWedderburn-Guttman...

Variational inequalities for singular integral operators

Albert Mas (2012)

Journées Équations aux dérivées partielles

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In these notes we survey some new results concerning the ρ -variation for singular integral operators defined on Lipschitz graphs. Moreover, we investigate the relationship between variational inequalities for singular integrals on AD regular measures and geometric properties of these measures. An overview of the main results and applications, as well as some ideas of the proofs, are given.

Estimates for maximal singular integrals

Loukas Grafakos (2003)

Colloquium Mathematicae

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It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1,1) and L p -bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.

Estimates for singular integrals and extrapolation

Shuichi Sato (2009)

Studia Mathematica

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We study singular integrals with rough kernels, which belong to a class of singular Radon transforms. We prove certain estimates for the singular integrals that are useful in an extrapolation argument. As an application, we prove L p boundedness of the singular integrals under a certain sharp size condition on their kernels.

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

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If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then λ k ( | a b * | ) ≤ λ k ( 1 / p | a | p + 1 / q | b | q ) for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if | a | p = | b | q .

Boundedness of vector-valuedB-singular integral operators in Lebesgue spaces

Seyda Keles, Mehriban N. Omarova (2017)

Open Mathematics

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We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator △B=∑k=1n−1∂ 2∂x k 2+(∂2∂x n 2+2vxn∂∂x n),v>0. ▵ B = ∑ k = 1 n - 1 ∂ 2 ∂ x k 2 + ( ∂ 2 ∂ x n 2 + 2 v x n ∂ ∂ x n ) , v > 0 . We prove the boundedness of vector-valued B-singular integral operators A from [...] Lp,v(R+n,H1)toLp,v(R+n,H2), L p , v ( ℝ + n , H 1 ) to L p , v ( ℝ + n , H 2 ) , 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.

Asymptotically normal confidence intervals for a determinant in a generalized multivariate Gauss-Markoff model

Wiktor Oktaba (1995)

Applications of Mathematics

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By using three theorems (Oktaba and Kieloch [3]) and Theorem 2.2 (Srivastava and Khatri [4]) three results are given in formulas (2.1), (2.8) and (2.11). They present asymptotically normal confidence intervals for the determinant | σ 2 ∑ | in the MGM model ( U , X B , σ 2 ∑ ⊗ V ) , ∑ > 0 , scalar σ 2 > 0 , with a matrix V ≥ 0 . A known n × p random matrix U has the expected value E ( U ) = X B , where the n × d matrix X is a known matrix of an experimental design, B is an unknown d × p matrix of parameters and σ 2 ∑ ⊗ V is the covariance matrix of U , ⊗ being the symbol...

On the differentiability of certain saltus functions

Gerald Kuba (2011)

Colloquium Mathematicae

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We investigate several natural questions on the differentiability of certain strictly increasing singular functions. Furthermore, motivated by the observation that for each famous singular function f investigated in the past, f’(ξ) = 0 if f’(ξ) exists and is finite, we show how, for example, an increasing real function g can be constructed so that g ' ( x ) = 2 x for all rational numbers x and g’(x) = 0 for almost all irrational numbers x.

Singular solutions to systems of conservation laws and their algebraic aspects

V. M. Shelkovich* (2010)

Banach Center Publications

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We discuss the definitions of singular solutions (in the form of integral identities) to systems of conservation laws such as shocks, δ-, δ’-, and δ ( n ) -shocks (n = 2,3,...). Using these definitions, the Rankine-Hugoniot conditions for δ- and δ’-shocks are derived. The weak asymptotics method for the solution of the Cauchy problems admitting δ- and δ’-shocks is briefly described. The algebraic aspects of such singular solutions are studied. Namely, explicit formulas for flux-functions of...

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

Approximating real linear operators

Marko Huhtanen, Olavi Nevanlinna (2007)

Studia Mathematica

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A framework to extend the singular value decomposition of a matrix to a real linear operator ℳ : ℂ ⁿ → ℂ p is suggested. To this end real linear operators called operets are introduced, to have an appropriate generalization of rank-one matrices. Then, adopting the interpretation of the singular value decomposition of a matrix as providing its nearest small rank approximations, ℳ is approximated with a sum of operets.

Cauchy-Poisson transform and polynomial inequalities

Mirosław Baran (2009)

Annales Polonici Mathematici

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We apply the Cauchy-Poisson transform to prove some multivariate polynomial inequalities. In particular, we show that if the pluricomplex Green function of a fat compact set E in ℝ N is Hölder continuous then E admits a Szegö type inequality with weight function d i s t ( x , ∂ E ) - ( 1 - κ ) with a positive κ. This can be viewed as a (nontrivial) generalization of the classical result for the interval E = [-1,1] ⊂ ℝ.

Eigenvalues and subelliptic estimates for non-selfadjoint semiclassical operators with double characteristics

Michael Hitrik, Karel Pravda-Starov (2013)

Annales de l’institut Fourier

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For a class of non-selfadjoint h –pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description...

An extension of a boundedness result for singular integral operators

Deniz Karlı (2016)

Colloquium Mathematicae

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We study some operators originating from classical Littlewood-Paley theory. We consider their modification with respect to our discontinuous setup, where the underlying process is the product of a one-dimensional Brownian motion and a d-dimensional symmetric stable process. Two operators in focus are the G* and area functionals. Using the results obtained in our previous paper, we show that these operators are bounded on L p . Moreover, we generalize a classical multiplier theorem by weakening...