Results 281 to 290 of about 373,199 (312)

Approximation by Rational Functions

Journal of the London Mathematical Society, 1977
This paper contains eight theorems on the rational approximation of \(e^{-x}\) . We cite one of them by way of an example: ''Let \(p(x)\) and \(q(x)\) be any polynomials of degress at most \(n-1\) where \(n\geq 2\). Then we have \[ \left\|e^{-x}-\frac{p(x)}{q(x)}\right\|_{l_{\infty}(N)}\geq\frac{(e-1)^ne^{-4n}2^{-7n}}{n(3+2\sqrt2)^{n-1}}.'', \] (\(N ...
Erdős, Paul, Newman, D. J., Reddy, A. R.   +2 more
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ON THE APPROXIMATIONS TO ANALYTIC FUNCTIONS

1994
Summary: We prove a theorem which shows that the uniqueness problem for entire functions of exponential type is equivalent to the approximation problem for analytic functions. This theorem is then combined with theorems on uniqueness to produce a number of results on the approximation of analytic functions.
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ON APPROXIMATION OF FUNCTIONS ON THE SPHERE

Russian Academy of Sciences. Izvestiya Mathematics, 1994
Let \(S^ n\) be the \(n\)-dimensional unit sphere and \(\| \cdot \|_ p\) the norm in the space \(L_ p (S^ n)\), \(1 \leq p \leq \infty\), where \(L_ \infty (S^ n) = C(S^ n)\). An \(r\)-th order modulus of smoothness is introduced for \(f \in L_ p (S^ n)\) by \(\omega_ r (f; \tau)_ p : = \sup \{\| (E - \text{sh}_ t)^{r/2} f \|_ p\), \(0 < t \leq \tau\}\)
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Approximately Midconvex Functions

2008
Let X be a vector space and let D ⊂ X be a nonempty convex set. We say that a function f is δ-midconvex if $$ f\left( {\frac{{x + y}} {2}} \right) \leqslant \frac{{f(x) + f(y)}} {2} + \delta\;\;for x,y \in D. $$ We find the smallest function C : [0, 1] ∩ ℚ → ℝ such that for every δ-midconvex function f : D → ℝ the following estimate holds $$
Misztal, Krzysztof, Tabor, Jacek, Tabor, Józef   +2 more
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Approximately Multiplicative Functionals

Journal of the London Mathematical Society, 1986
Let \({\mathfrak A}\) be a commutative Banach algebra with dual \({\mathfrak A}^*\). For \(\phi \in A^*\), define \({\breve \phi}\)(a,b)\(=\phi (ab)- \phi (a)\phi (b)\), and call \(\phi\delta\)-multiplicative iff \(\| {\breve \phi}\| \leq \delta\). \({\mathfrak A}\) is an algebra in which approximately multiplicative functionals are near multiplicative
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Classifiers that approximate functions

Natural Computing, 2002
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Approximations for nonlinear functions

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 1992
By applying a version of the Stone-Weierstrass theorem the author shows that a continuous real-valued function on a nonempty compact topological space can be uniformly approximated by a sum of the form \[ a_ 1 e^{\phi(x,p_ 1)}+\cdots+ a_ me^{\phi(x,p_ m)}. \] {}.
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Conservatively Approximable Functions

2013
In the study of computable functions on the Cantor space 2ℕ, it is well-known that the image of such a function is an effectively closed set, or \(\Pi^0_1\) class and in fact a decidable closed set. Here a closed subset Q of the Cantor space is decidable if the set of finite strings w which have an extension in Q is a computable set.
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Approximation by Unimodular Functions

Canadian Journal of Mathematics, 1971
The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous
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Approximation of the dilogarithm function

2007
Summary: In this short note we approximate the dilogarithm function defined by \(\text{dilog}(x)=\int_1^x\frac{\log t}{1-t}\,dt\). Letting \[ \mathcal{D}(x,n)=-\frac12\log^2x-\frac{\pi^2}6 +\sum_{k=1}^n\biggl(\frac1{k^2x^k}+\frac{\log x}{kx^k}\biggr), \] we show that for every \(x>1\) the inequalities \[ \mathcal{D}(x,N)
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