Question

Is the vector space C∞[a,b] of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite inte

Transformations of functions
ANSWERED
asked 2021-06-23

Is the vector space \(C \infty[a,b]\) of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by \(D(f)(x)=f′(x)D(f)(x)=f′(x)\ and\ I(f)(x)=∫xaf(t)dt f(t)dt\). (a) Compute \((DI)(f)=D(I(f))(DI)(f)=D(I(f))\). (b) Compute \( (ID)(f)=I(D(f))(ID)(f)=I(D(f))\). (c) Do this transformations commute? That is to say, is it true that \((DI)(f)=(ID)(f)(DI)(f)=(ID)(f)\) for all vectors f in the space?

Answers (1)

2021-06-24

a) \((DI)(f)=f(x)\)
b) \((ID)(f)=f(x)-f(a)\)
c) \((DI)(f)=(ID)(f) \) only when \( f(a)=0\)

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