hi, i have a problem with the following question: if f(x)=u(x)*exp{-alpha*x} where u(x) is the step function ie 1 when x>=0 and 0 when x<0 then does f prime (ie 1st derivative of f wrt x) belong to the lebesgue space with p=2 on the real line.
The derivative is a linear combination of a piecewise continuous function and a dirac delta functional. Note the derivative of the step function is the dirac delta function). Now i know that the dirac delta function is a distribution, in fact it is a tempered distribution by wikipedia. However i don't know why this means f prime does not belong to lebesgue space with p=2. i'm close to the answer as i think it definitely has something to do with the definition of a distribution. Any help would be terrific. thanks
The derivative is a linear combination of a piecewise continuous function and a dirac delta functional. Note the derivative of the step function is the dirac delta function). Now i know that the dirac delta function is a distribution, in fact it is a tempered distribution by wikipedia. However i don't know why this means f prime does not belong to lebesgue space with p=2. i'm close to the answer as i think it definitely has something to do with the definition of a distribution. Any help would be terrific. thanks