A finite set S in the extended complex plane
is called a unique range set counting multiplicities (URSCM) (unique range set ignoring multiplicities, URSIM) for meromorphic functions on
if any two ...nonconstant meromorphic functions f and g in
satisfying
counting multiplicities (CM) (ignoring multiplicities IM), then f = g. The question 'What is the smallest cardinality for such a set?' is still a challenging and open one. In this article, for meromorphic functions on
we exhibit a URSCM and a URSIM of 13 and 19 elements, respectively. We also characterize a unique range set for entire functions of order <1 on
. Some conjectures are proposed for further studies.
§Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.
The authors define axiomatically a large class of function (or distribution) spaces on $N$-dimensional Euclidean space. The crucial property postulated is the validity of a vector-valued maximal ...inequality of Fefferman-Stein type. The scales of Besov spaces ($B$-spaces) and Lizorkin-Triebel spaces ($F$-spaces), and as a consequence also Sobolev spaces, and Bessel potential spaces, are included as special cases. The main results of Chapter 1 characterize our spaces by means of local approximations, higher differences, and atomic representations. In Chapters 2 and 3 these results are applied to prove pointwise differentiability outside exceptional sets of zero capacity, an approximation property known as spectral synthesis, a generalization of Whitney's ideal theorem, and approximation theorems of Luzin (Lusin) type.
We use the Implicit Function Theorem to establish a result of non-existence of limit to a certain class of functions of several variables. We consider functions given by quotients such that both the ...numerator and denominator functions are null at the limit point. We show that the non-existence of the limit of such function is related with the gradient vectors of the numerator and denominator functions. We prove the limit does not exist if the dimension of the vector subspace spanned by the gradient vectors is ≥1.
Introduces geometric measure theory through the notion of currents. This book provides background for the student and discusses techniques that are applicable to complex geometry, partial ...differential equations, harmonic analysis, differential geometry, and many other parts of mathematics.
The main goal of this paper is to prove that the classical theorem of local inversion for functions extends in finite dimension to everywhere differentiable functions. As usual, a theorem of implicit ...functions can be deduced from this “Local Inversion Theorem”. The deepest part of the local inversion theorem consists of showing that a differentiable function with non-vanishing Jacobian determinant is locally one-to-one. In turn, this fact allows one to extend the Darboux property of derivative functions on ℝ (the range of the derivative is an interval) to the Jacobian function Df of a differentiable function, under the condition that this Jacobian function does not vanish. It is also proved that these results are no longer true in infinite dimension. These results should be known in whole or part, but references to a complete proof could not be found.
Let ohm ⊂ R^sup n^ be a domain. The result of J. Kauhanen, P. Koskela and J. Maly 4 states that a function f : ohm arrow right R with a derivative in the Lorentz space L^sup n,1^ (ohm, R^sup n^) is ...n-absolutely continuous in the sense of 5. We give an example of an absolutely continuous function of two variables, whose derivative is not in L^sup 2,1^. The boundary behavior of n-absolutely continuous functions is also studied. PUBLICATION ABSTRACT
In the article, using Taylor's formula for functions of several variables, the author establishes some inequalities for the weighted multiple integral of a function defined on an m-dimensional ...rectangle, if its partial derivatives of (n+1)th order remain between bounds. Using this result, Iyengar's inequality is generalized and related results in references could be deduced.
We study analytic functions of several variables in the Korányi class such that, when applied to a contractive tuple of doubly commuting matrices, a positive definite matrix results. A criterion is ...given for a finite set to have the property that every measure supported on the set yields, via an integral formula, a function as above.
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