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Springer Mathematische Hilfsmittel des Ingenieurs: 139 (Grundlehren der mathematischen Wissenschaften, 139)

black & white illustrations, bibliograph

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234 Originalvariable x nur ganzzahlige Werte annimmt, dann ist das Integral durch eine unendliche Summe zu ersetzen. Einige der im folgenden behandelten Transformationen gehOren zu diesen beiden Typen. Da wir nUr lineare Transformationen betrachten, wird spiiter die Eigenschaft der Linearitat nicht mehr eigens erwahnt. § 2. Der Hilbertsche Raum L2 Bei einer Integraitransformation HiBt man i. aUg. als Original funktionen aUe I (x) zu, fur die das Integral existiert. Manche Eigen schaften der Transformation lassen sich aber nUr dann exakt formu lieren und beweisen, wenn man die I (x) auf engere Raume beschrankt, die durch innere, von der Transformation unabhangige Eigenschaften charakterisiert sind. In dieser Beziehung ist der Raum der quadratisch l integrablen Funktionen am wichtigsten . Dieser laBt sich auffassen als Analogon zu dem Euklidischen Raum Rn von n Dimensionen, in dem sich die Variablen der gew6hnlichen Funktionen bewegen. Der Rn ist dadurch ausgezeichnet, daB in ihm die Distanz zweier Punkte Xl = (Xll' ••• , Xl II), X2 = (X21> ••. , X2 n) als die positive Wurzel aus n d (Xl, X2)2 = ~ (Xl v - X2v)2 . -1 definiert ist. Es liegt nahe, im Raum der in dem endlichen oder unend lichen IntervaU (a, b) definierten Funktionen die Distanz zweier Ele 2 mente 11, 12 durch den entsprechenden Ausdruck b d (11, 12)2 = jill (X) - 12 (X) 12 dx a zu definieren. Insbesondere ist die Distanz einer Funktion I (x) vom NuUpunkt, d. h.

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paperback
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Amazon UK
Release Date
12 February 2012
Listed Since
13 July 2012

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