Operator of Integrated Weighted Derivatives is Linear Mapping

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Theorem

Let $n \in \N$.

Let $I := \closedint a b$ be a closed real interval.

Let $\map {a_i} x : I \to \R$ be Riemann integrable functions.

Let $f, g \in \map {C^n} I$ be Riemann integrable real-valued functions of differentiability class $k$.

Let $L : \map {C^n} I \to \R$ be the operator of integrated weighted derivatives.


Then $L$ is a linear mapping.


Proof

Distributivity

\(\ds \map L {f + g}\) \(=\) \(\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren{\map f x + \map g x} \rd x\) Definition of Operator of Integrated Weighted Derivatives
\(\ds \) \(=\) \(\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \paren{\map {f^{\paren i} } x + \map {g^{\paren i} } x} \rd x\) Linear Combination of Derivatives
\(\ds \) \(=\) \(\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {f^{\paren i} } x \rd x + \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {g^{\paren i} } x \rd x\) Linear Combination of Definite Integrals
\(\ds \) \(=\) \(\ds \map L f + \map L g\) Definition of Operator of Integrated Weighted Derivatives

$\Box$


Positive homogenity

\(\ds \map L {\alpha f}\) \(=\) \(\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \dfrac {\d^i}{ {\d x}^i} \paren {\alpha \map f x} \rd x\) Definition of Riemann Integral Operator
\(\ds \) \(=\) \(\ds \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \alpha \map {f^{\paren i} } x \rd x\) Linear Combination of Derivatives
\(\ds \) \(=\) \(\ds \alpha \int_a^b \sum_{i \mathop = 0}^n \map {a_i} x \map {f^{\paren i} } x \rd x\) Primitive of Constant Multiple of Function
\(\ds \) \(=\) \(\ds \alpha \map L f\) Definition of Riemann Integral Operator

$\blacksquare$


Sources

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