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Runge’S Approximation Theorem

Runge's Approximation Theorem ist ein bedeutendes Resultat in der Funktionalanalysis und der Approximationstheorie, das sich mit der Approximation von Funktionen durch rationale Funktionen beschäftigt. Der Kern des Theorems besagt, dass jede stetige Funktion auf einem kompakten Intervall durch rationale Funktionen beliebig genau approximiert werden kann, vorausgesetzt, dass die Approximation in einem kompakten Teilbereich des Intervalls erfolgt. Dies wird häufig durch die Verwendung von Runge-Polynomen erreicht, die eine spezielle Form von rationalen Funktionen sind.

Ein wichtiger Aspekt des Theorems ist die Identifikation von Rationalen Funktionen als eine geeignete Klasse von Funktionen, die eine breite Anwendbarkeit in der Approximationstheorie haben. Wenn beispielsweise fff eine stetige Funktion auf einem kompakten Intervall [a,b][a, b][a,b] ist, gibt es für jede positive Zahl ϵ\epsilonϵ eine rationale Funktion R(x)R(x)R(x), sodass:

∣f(x)−R(x)∣<ϵfu¨r alle x∈[a,b]|f(x) - R(x)| < \epsilon \quad \text{für alle } x \in [a, b]∣f(x)−R(x)∣<ϵfu¨r alle x∈[a,b]

Dies zeigt die Stärke von Runge's Theorem in der Approximationstheorie und seine Relevanz in verschiedenen Bereichen wie der Numerik und Signalverarbeitung.

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Nucleosome Positioning

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ΔS≥QT\Delta S \geq \frac{Q}{T}ΔS≥TQ​

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