Proof of Shift Theorem: e^{iwv} F(w) = f(x - v)
- Thread starter mario99
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Please show us what you have tried and exactly where you are stuck.
Please follow the rules of posting in this forum, as enunciated at:
READ BEFORE POSTING
Please share your work/thoughts about this problem
Please follow the rules of posting in this forum, as enunciated at:
READ BEFORE POSTING
Please share your work/thoughts about this problem
Dr.Peterson
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Start by writing out any relevant definitions. The problem as shown is lacking any information about, say, what F means. Writing out such things will not only help others see what you want to do, but also help you see what you need to do!
Then, look at what you wrote, and restate what has to be proved in terms of the definitions.
Look up example problems in your textbook - dealing with similar problems (Oh I forgot that you did not want to make our life easy by showing your work!!!)
topsquark
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Thank you Dr.Peterson, Steven G and topsquark for helping me.
Here he is already using a different symbol, [math]\nu[/math]. [math]e^{i\omega\nu}[/math]This is confusing.
Who said? Please show us what you have tried and exactly where you are stuck.
You know exactly where I am stuck. Where is your help?
F is the fourier transform of f. The definition uses [math]e^{-i\omega x}[/math]
Here he is already using a different symbol, [math]\nu[/math]. [math]e^{i\omega\nu}[/math]This is confusing.
You will be nicer if you contribute with something useful. Instead of waiting for my work and steal my ideas, show your work!
You are always like that. You kick students away by telling them to use google or textbook. Change your habits and you will be more useful. You are laughing at me because I don't show my work. What about you?
Who said? Please show us what you have tried and exactly where you are stuck.
You know exactly where I am stuck. Where is your help?
Of course it is not going to work if tutors like khansaheb, Steven G and stapel are watching and laughing. See how Dr.Peterson attacked my problem and you will know the difference.
Dr.Peterson
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Rather than complain about tutors trying to get you to think, you need to do the thinking, and show it to us. You haven't yet written out the definition you were given for the Fourier transform, and of the variables used in that definition. (Note that some authors use slightly different notation, such as different variables, so we need to see what you are working with. That's why I asked you for your definition, rather than giving you a link to one.)
You did not do what I asked, but you have at least now given us some idea of where your confusion lies. Once you show us your details, we will have something specific to talk about.
People can't help you get unstuck if they don't know where you are stuck. There is good reason for all the resistance to showing you anything until you have shown something. We are not mind readers. If I were tutoring you face to face, I would have asked to see your book, and found the definitions. Since I can't do that, you need to do it for us.
(By the way, note that I have changed your math expressions from "block" to "inline" to make it easier to read.)
topsquark
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No, we do not know exactly where you are stuck, we are not psychics. You didn't post anything. How could we possibly know?
mario, once again, it is you that is coming here for help. We already know how to do this. If you want someone to just give you an answer, please go somewhere (anywhere!) else.
None of this is original work and there is nothing to steal: you can look all of this up in a textbook. The "new" mario is just making up more excuses to not post his work and get us to do all of it for him. Same old, same old.
Please stop your whining and just start following the forum rules. We cannot help you (or anyone else) unless we know where you are taking a wrong turn, like you showed in post #8. Dr.Peterson was able to use that post to give you an assist. See how that works?
And, yet again, this thread is being contaminated by comments about you not following forum rules. If you would just start doing what you agreed to do when you signed up for your membership, these would be a lot easier to read and more informative.
-Dan
Dr.Peterson
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That is paraphrased slightly from here:
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Welcome to our tutoring boards! :) This page summarizes the main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally post immediate answers or step-by-step solutions. We don't do your homework. We prefer to...
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3. Show work that you've already done (even if you think it's wrong), or try to explain why you're stuck.
Simply posting an exercise statement without showing work or asking specific questions is not enough for us to help you quickly. As tutors, we need clues about parts you already understand versus what you find confusing, so that we can determine where to begin helping you. The sooner you show efforts or share what you've been thinking, the sooner we can get to the heart of the matter. If you cannot begin an exercise, then please tell us why (eg: unknown concept, confusing example, unfamiliar symbol, missing definition, unclear language).
That in turn is paraphrased from here:
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Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck. Don't worry that your work might be wrong; learning is a process of making mistakes. We'd like to check your efforts so far or read what you've been thinking or getting confused about (especially if you're not able to start). If you show no work at all, tutors may assume that you need lessons instead of online tutoring or they might think you're looking for somebody to do your homework. Even if you're asking about only the very end of the solution process, please include your intermediate steps. Errors may have occurred earlier than you realize; for example, correcting part (b) may clear up your confusion on part (d).
This is how we work, and it is based on standard principles of tutoring. We do the same at the face-to-face tutoring center where I work.
This is also standard procedure for asking anyone for help. If you don't want to cooperate with them, then you politely go elsewhere.
Sorry. I don't get block to inline thing!
This is the definition [math]f(x) = \int_{-\infty}^{\infty}F(\omega)e^{-i\omega x} \ d\omega[/math]
Also I want to correct something. In my first post, I should have written, the inverse fourier transform of [math]e^{i\omega\nu}F(\omega)[/math]is
[math]f(x - \nu)[/math]
And I should prove that as that result is known as the shift theorem for fourier transform.
I am so shocked that you still say Mario did not change. You have known I was a hard working student. In MHF forum where I was blocked, I have posted tons of problems in a daily basis and I have worked at them at the same time. In that same day I had a lot of assignments and exams but I did not ignore anything. I worked in all of them at once.
Now, I post only one or a few problems not even in a daily basis but I don't get a single help. When I post a problem and I don't show my work it means I can't show anything. I am fully stuck and I need any push to help me. At least one line of the solution or Hint. And you will be so appreciated if you post the whole solution. Who said that will not help me? I will stick that solution inside my brain for future similar problems. It is a big benefit.
I apologize for the seniors. Not showing my work doesn't always mean I am hiding my thoughts from burglars. It sometimes mean I can't or I don't have anything to show.
Dr.Peterson
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If you're using the "f(x)" button on the toolbar, it gives the option of "block" or "inline" at the top. If you're typing "[ math ]" manually, change it to "[ imath ]", which is what I did to correct your work. If you're doing something else, read
LaTeX Math Formatting is Available
A system named LaTeX is available in the forum, to generate math symbols and formatting. Using LaTeX requires learning programming codes and syntax. Tutorials and examples are available on the Internet, and some references and links appear below. (As an alternative to LaTeX, the purplemath site...
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It may help if you show us images of the definition and of the problem, so we can be sure of the exact formulation (especially any conditions given for the theorem). Every word matters, so a paraphrase of either a definition or a theorem or a problem may make it incorrect, or at least interfere in the work. Does the problem talk about the Fourier transform, or the inverse Fourier transform? Please show the whole thing.
Before, you said you were confused by the "different symbol" [imath]\nu[/imath]. It is just a constant; it doesn't matter what it is called.
Just put what you were given (either the [imath]f(x - \nu)[/imath] part, or the [imath]e^{i\omega\nu}F(\omega)[/imath] part) into the definition you were given, and show that to us; then think about what you might do to transform one side into the other.
What you need to do, when you think you have nothing to show, is to say something like "I have no idea how to start, but ...", followed perhaps by a definition or an example you were given, or something to show what you do know. You are starting to do that, as I demand it of you; you should do it without having to be prompted.
My face to face students quickly learn to show me what they have tried before they do anything else; and they also thank me for not doing too much for them, so that they get the chance to practice thinking themselves. They never complain that I am stealing their ideas ...
topsquark
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You are not that stupid. Neither am I. You persist in not following the forum rules; this is what has not changed.
Now, please follow Dr.Peterson's excellent advice.
-Dan
I got you now. I should write imath. This is something new. I will do it in my future posts.
Both. You will see now.
Images of the problem will not help you much as you will not be able to read my HandWriting. I will post the exact problem with the new imath.
If [imath]F(\omega)[/imath] is the fourier transform of [imath]f(x)[/imath], show that the inverse Fourier transform of [imath]e^{i\omega \nu} F(\omega)[/imath] is [imath]f(x - \nu)[/imath]. This result is known as the shift theorem for Fourier transforms.
The exact definition is written with symbols I don't know how to write them in Latex. Let me see if google can help.
After searching for the symbols.....this is the exact definition.
[imath]\mathcal{F}^{-1}[\widehat{f}](x) = \int_{-\infty}^{\infty} \widehat{f}(\omega) e^{-i\omega x} \ d\omega[/imath]
But in class we don't use [imath]\widehat{f}(\omega)[/imath]. We use [imath]F(\omega)[/imath].
I will talk more about my confusion. For the inverse, we always use [imath]F(\omega) e^{-i\omega x}[/imath] inside the integral, not [imath]F(\omega) e^{i\omega \nu}[/imath].
You said [imath]\nu[/imath] is a constant, but in the definition [imath]x[/imath] is a variable. This increases confusion.
I am so tired of thinking in this problem. Can you please solve the problem for me and I promise in future problems I will work hard to do them by myself.
By the way, thank you very much Dr. for letting me know "imath". It is a lot better than "math". As Dr.Peterson once said, everyday, you learn something new.
As you say Sir topsquark.?
Dr.Peterson
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I didn't ask for your handwriting. I asked for an image of the actual problem, in print, exactly as given to you. Why would I want handwriting???
And an image of the original would make it unnecessary for you to figure out the symbols in Latex. You are making things hard for yourself. (Is it that you have nothing in print or on a website, only handwritten notes? I know nothing about your context.)
Why do you show the definition using notation not used in your class? Show it exactly as you were taught it (as an image, ideally).
Take your definition of the inverse transform, and replace [imath]F(\omega)[/imath] with [imath]e^{i\omega \nu} F(\omega)[/imath]. Then see if you can turn what you write into [imath]f(x - \nu)[/imath]. (A u-substitution might be a good idea.)
Alternatively, you might put [imath]f(x - \nu)[/imath] into the definition you were given for the Fourier transform, in place of [imath]f(x)[/imath], and see if you can change the integral to look like the desired goal.
Let me add that if your goal is simply to be given the answer, you could search for the proof of the shift theorem, which you can find in many places. Why bother us?
Mario had declared before ...... I like to make tutors suffer!
Steven G
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Are you saying that this fact has not been proven yet? That is the only way I can think of how to steal your work.
You need to understand that solving the problem with assistance is better for you in the end versus someone here just giving you the solution.
You think that if the solution was given to you that would benefit you. You may be correct, but I assure you that if you are the one who ultimately solves the problem you would benefit more.
Here is a story that I posted many times that I will never forget.
In graduate school my algebra teacher gave the class a set of 10 theorems to prove.
I did 9 of them. Some took a very short time, while others took days for me to prove.
The 10th problem I looked out I did not immediately see how to prove it but did recall seeing this problem along with its proof in another algebra book I had. I looked up the proof and was amazed that I didn't see the proof on my own as it was basically a trivial proof.
The day the assignment was due I left my work at home. I thought that was not a big deal as I will just quickly write up the 10 proofs. I did the first 9 very quickly, including those proofs that took me days to figure out. That 10th theorem I could not see the proof! I had to go home and retrieve my assignment before class started. After seeing that 10th proof I again couldn't believe that I couldn't see/remember the proof on my own.
Why was I able to recall the 9 proofs and not the trivial 10th proofs? The answer is very simple--I suffered (maybe not for too long) coming up for the proof, but that 10th proof was just given to me. When you suffer you really learn something. When something is give to you, even if it is trivial, you never thought about it on your own. Thinking about something on your own really helps you remember/learn something.
I apologize for the lateness. I was studying.
Do you mean this?
[imath]\int_{-\infty}^{\infty}e^{i\omega \nu} F(\omega) \ d\omega[/imath]
Or this?
[imath]\int_{-\infty}^{\infty}e^{i\omega \nu} F(\omega) e^{-i\omega x} \ d\omega[/imath]
In either case, I can't answer anyone of them because they don't follow the definition notation.
If reverse the process, how am I supposed to know what should I write inside the integral?
[imath]f(x - \nu) = \int_{-\infty}^{\infty} \ ?[/imath]
Your story reminds me by an old saying.
Don't give me the fish. Teach me how to fish.
I only have handwriting.
It was written in class exactly like that, but when we solve problems the teacher just writes [imath]F(\omega)[/imath]. Next time I will ask him why you don't use [imath]\hat{f}(\omega)[/imath] when you wrote the definition with that notation.
Are you telling me to violate the definition and force it to include [imath]e^{i\omega \nu} F(\omega)[/imath]?
Do you mean this?
[imath]\int_{-\infty}^{\infty}e^{i\omega \nu} F(\omega) \ d\omega[/imath]
Or this?
[imath]\int_{-\infty}^{\infty}e^{i\omega \nu} F(\omega) e^{-i\omega x} \ d\omega[/imath]
In either case, I can't answer anyone of them because they don't follow the definition notation.
If reverse the process, how am I supposed to know what should I write inside the integral?
[imath]f(x - \nu) = \int_{-\infty}^{\infty} \ ?[/imath]
I am not telling you to give me the answer for no effort. I have already tried my best but failed. By giving me the answer, I will be able to solve any problem with shifting. If you don't want to help me, just say it. It seems that you are right, I should have not bothered you and searched in somewhere else.
Thank you very much Steven G for telling me this story. It really means something. I don't know if you added some Drama or it is fully real, but it has affected on me.
Your story reminds me by an old saying.
Don't give me the fish. Teach me how to fish.
It seems that you have taken it personally!