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Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) Distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

b) Within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice do at many points and this is significant.

c) Most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of distributions which depend on a parameter--distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) The mathematician's dream then comes true--one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the classical one. This explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in French but also some in Portuguese. He was working on an encyclopedic monograph of his ideas and their applications but his untimely passing prevented its completion.

Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) Distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational (even meromophic) function can be regarded as a distribution. The details of all this are in the article I cited.

b) Within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice do at many points and this is significant (despite some comments to the contrary on this post)

c) Most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of distributions which depend on a parameter--distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) The mathematician's dream then comes true--one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the classical one. This explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in French but also some in Portuguese. He was working on an encyclopedic monograph of his ideas and their applications but his untimely passing prevented its completion.

Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

b) within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice DO at many points and this is significant.

c) most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of disributions which depend on a parameter---distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) the mathematician's dream then comes true--- one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the clasicaloneThis explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in french but also some in portuguese. He was working on an encyclopediac monograph of his ideas and their applications but his untimely passing prevented its completion.

Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) Distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

b) Within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice do at many points and this is significant.

c) Most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of distributions which depend on a parameter--distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) The mathematician's dream then comes true--one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the classical one. This explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in French but also some in Portuguese. He was working on an encyclopedic monograph of his ideas and their applications but his untimely passing prevented its completion.

Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

b) within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice DO at many points and this is significant.

c) most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of disributions which depend on a parameter---distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) the mathematician's dream then comes true--- one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the clasicaloneThis explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in french but also some in portuguese. He was working on an encyclopediac monograph of his ideas and their applications but his untimely passing prevented its completion.

Let me add the comment that in the article of Sebastião e Silva I quoted in a recent post, some of the questions you pose are addressed.

a) distributions are not introduced as functionals on spaces of test functions but as derivatives in a generalised sense of continuous functions. Thus the delta distribution is the second derivative of $\frac 12 |x|$. Any locally integrable function is a distribution--take the distributional derivative of its primitive. Hence, $\log |x|$ is a distribution and so are its distributional derivatives--this gives the negative powers of $x$. In a similar manner, any rational function can be regarded as a distribution. The details of all this is in the article I cited.

b) within this framework the concept of limits of distributions at finite or infinite points, values of distributions at a point and integrals of distributions are developed. Although a general distribution doesn't have a value at a point, most ones which occur in practice DO at many points and this is significant.

c) most significantly for your question, Sebastião e Silva developed the concept of a parametrised integral, i.e., the integral of a family of disributions which depend on a parameter---distributionally. This is particularly important for Fourier transforms which are defined exactly as classically, not via duality. The resulting integrals converge in the above sense, even when they diverge in the usual sense,

d) the mathematician's dream then comes true--- one has Fubini and can differentiate under the integral in parametrised integrals in situations where this would forbidden in the clasicaloneThis explains why many formal computations give the correct answer and puts them firmly in a rigorous framework.

Sebastião e Silva's article is completely elementary and he works through several examples very much in the style of your question. You can also find more sophisticated material at the site I pointed out, some in french but also some in portuguese. He was working on an encyclopediac monograph of his ideas and their applications but his untimely passing prevented its completion.