the-space-is-c02-pi-with-the-inner-product-6-find-the-third-order-fourier-approximation-to-ftt-1

Question

The space is $C[0,2 \pi]$ with the inner product $(6)$.
Find the third-order Fourier approximation to $f(t)=t-1$

Angelo Rendina · Accepted Answer

from example for we know that the four year approximation off T is given by buying minus twice. Scientist E minus. Sign to teen Minus all the way to mind a study. Very bad, man. Sign off. It&#x27;s a mistake. This is Sign off, aunty. Okay, so we&#x27;re looking for the third or the approximation of team minus one and of course, is just by minus one for the constant term. There we have minus twice. Scientist E minus, sign to T and minus two, divided by three sign three t.

Angelo Rendina · Answer

that exist are consigned to application formula. That says Consign off three team is equal to four times because I Ncube&#x27;s off T minus three times Co sign Off t and now the exercises over immediately. Because we are so late. Consign cube off t on that three divided by four course 90 both one over four co sign off three t And of course, since therefore they&#x27;re serious, is unique. These must be the fully expansion off consign cube of tea and in particularities is correct to the third.

Angelo Rendina · Answer

So we&#x27;re giving the function f, which is equal to one from zero to buy and minus one from by 22 by, and we want to find its for the expansion. So let&#x27;s start with computed integral from 0 to 2 pi off F 15 Consign aunty. And of course, we do this because we were completed the full coefficients. So because about half is defined that is the integral from zero to buy off consign aunty miners integral from pi to to buy off course an anti the tea he&#x27;s in. The guy&#x27;s immediate is signing TD valuable and evaluated from zero to by minors Sign indeed divided by an evaluated from pi to two Pi is equal to well, one in the denominator and that we have signed bye and minus zero minus zero minus son my end and therefore our first put a confusion a n is given by one divided by by these integral they were just computed and in this case is exactly zero. So all the a Ensor zeros now, Stephen, only we compute integral from here to buy off ft Sign Aunty and again was played these into two Integral Sze and these 20 girls are again immediate. So this is the the minus consigning t divided by and have a little from zero to pi minus the same function valued it from buy to toe By now this is minus consign by n plus one minus minus one plus sign by n and therefore are for the coefficients B N r. One divided by pi Dizzy, integral and these entire Elise to minors to co sign by n divided by N and of course, co sign off by n We have a simple expression for it. It is twice one over and by they&#x27;re multiplies one minus minus one to the power of an Okay, Now we can proceed to write the foot expansion. We we need one last thing, which is a constant term. The constant term A zero is given by one over pi identical from zero to buy off the function itself. He&#x27;s one over pi ng that they are multiplies integral from zero to pie off the T minus integral from party to buy up the teen of course disease one over pi that multiplies Pyman spy. So the constant room is zero so we can finally find our expansion. So I 15 is given by constant constant terror much zero. Then we have the first or their terms, which are well, there&#x27;s no consign because a n y zero But we have the end. So we have 40 valuable pie Sinan t Well, time want e There is no degree to terms and the degree ther three term is getting by $4 3 pi sign off three t.

Angelo Rendina · Answer

we&#x27;re given the function G, which is our first order for the approximation over the given function f plus the term A to consign to t. And he said, H is exactly the order to fully approximation off recalled that on the space off continues functions from zero to buy. We have a norm we just given by the route off the product X against axe. But of course in a park is the integral off X times X over zero to buy and by construction, the distance between F and the order and for the approximation goes down to zero as an increases. So essentially the higher the harder the better. Approximation is so absurd that G is almost, um, order true for the approximation because it lacks the B two signed to T term, which is that age has. And since H is constructed as or as projection onto north diagonal bases along the way to a degree too, were ordered to, we know that it isn&#x27;t f to age. It is more or less between all the other or they&#x27;re too approximations in particular. This is routine F an H is muller, or eventually equal to the since between F N G, and therefore age is are more accurate. Description of F

Angelo Rendina · Answer

So let&#x27;s call I m n the quantity integral from 0 to 2. Pi off co sign Empty Sign Anti DT, where m and n are positive integers. And of course, our goal is to show that I am any zero. So we do these integral by parts. So we see these as the integral from zero to buy off the derivative of Sign Mt. Divided by M And of course, he&#x27;s just coast on Mt. They&#x27;re multiply sign and the T d t and now by parts diseases sign MTV Alabam that multiply 17 evaluated from zero to Dubai minus the integral from 0 to 2. Pi it off sign MT Divided by AM that multiplies the derivative of their function So and co sign and d d t. Now these offers Brockett zero. They were minus. I need to go from zero to buy. Let&#x27;s eyesight I n divided by am And then we have sign empty co signing team. Let&#x27;s rev right in the other side. Let the other way so co sign anti sign, empty DT and we recognized as decent. A girl is nothing else but Minus and Avery by a m i n m observed that I am an is equal to something I n m So we&#x27;re switching the indices, but we can do the same integral. So I am in by parts by seeing the other function as a derivative. So this isn&#x27;t a girl from zero to buy off course in Mt. And then sign and TV seed as the derivative off miners co sign anti divided by N dt. Now again by parts disease consigning t that multiplies minus consigning TD Balaban have a letter from zero to buy minus So the derivative of the first function, which is minus m sign MT that multiplies Minako san and T divided van duty and again the old Rockets zero and they were minus the integral from zero to buy the two minuses constantly each are other, so we have m divided by end co sign anti sign, empty and again we recognize his expression to be just minus m a divided by N I and em. So we have two different expressions for I am n Let&#x27;s they&#x27;re not the 1st 1 they obtain up there in right by one and these other quality down here in green as to and we&#x27;re going to use them right away. So minus and divided by a m i n m is by the first equality in red I am and which is by a quality to in green equal to minus m d valuable and I n m And it is true for every n and m great recall than one in party Are we we right? I an m as by moving the Andy Very bam on the right inside m square that divides and square I and m And this expression in blue is a number which depends on them and then in any em and end are different Then the oil quantity in blue is different than one and therefore we see that i n m must be zero if n and m are different. So our last case they we need to check Is it an NMR Rico? So we need to compute i n n is of course, the integral from zero to the pie off sign anti sign anti the teeth by now. This is easy because you can just do a simple substitution. Aunty is equal to X therefore the axes and DT and the fourth is becomes integral from zero two to buy n call, Sign of X sign of X, then DT is one divided by and the ex. Okay. And I were there because you can simply compute is a well known fact. But you can compute yourself. That didn&#x27;t occur from 0 to 2 pi off course I necks Cenex the X zero and the function cause an X sign X is periodic would be to buy and therefore we see that is all integral is just equal to zero.

Carlos Pinilla · Answer

So we have a precision that is even by deploying expression seeks signed to t along the either action You are six goes well to be from the J Direction two d Jianye direction Plus why d no que aviation about the compute wasti your normal The one discover curvature on what he&#x27;s, uh the unit normal. So let&#x27;s compute first. What is V So in friendship these So there is little sign signs of because the the baby travel Get out to them 60 well crusade thing I from here you have a minus sign Good tea Now then by the channel your inner six times 2 12 one day and there&#x27;s five d just five over the cave. Original bodies over the J direction. So here again but cattle there so co sign off t I. I signed 30 j lost five so that it&#x27;s no people do the square root off these squared Sapele Square will sign I want to d squared Was the principal components squared plus well squared Sign off to t squared Was that wanted squared Those five squared Teoh here we conducted a poll I will sign up to d squared 12 times 12 square damn school side of the square because I&#x27;m a duty squared Waas sang off 30 squared. That&#x27;s fine. So something verification here is one. So be well the square. It&#x27;s fired square. Um, so this fire was quarries. Do you live and filled the square one for you? More. So this would be Discworld or, um 1 44 I lost 25 nine of being for so see one six mile, which seems to be 40 three because so all these number varies. Oh, squared. Uh, where we, uh So these, uh, Morgan live with us squirt of warm 16th of disease. Eagle do 16. Ah, that&#x27;s them. Waas three squared from the You can just be like that. So square the 1 69 So, uh, so that t will be equal to, um, these vector divided by disclosed 1 69 So Well, who Sino two. D. All right, I my no sign. Oh, to t j not divided by a squared. Oh, 1 69 Plus, um, my okay. R squared with one 69. So this is he I&#x27;m that is gonna be done. Is gonna be e e t. The very one. We&#x27;re gonna build a lost huge. There are relative disease, so we have to compute What is empty? Yeah. So, uh, well, the key, uh, will be the compactor. 12 for screwed of 1 69 So it&#x27;s great. The one sees nine from being really harsh. These so? Ah, two times minus sign to D I That being two guys co sign my bicycle sign school dams minds were saying to here j and then for day these we have really at their resume. So just be just that when we come factor do there she would be 24 doctoring You do? So I&#x27;m still bye bye squared of 169 one better. We have a minus. Minus sign two d I plus co sign T. J um so bodies the TT And then what is It&#x27;s norm also very would have squared of diet both this girl So 24 squared 1 69 The times sign duty squared plus rules to do square. So all of this part will be one. So it would be for 24. He&#x27;s embraced more over screen one 69. Well, so that, uh well these couple will be 24 divided by a squared with 1 69 onda. Uh, the exclude them a normal fees Screws of 1 69 So to that Well, like these to the four over one 16 night so that he&#x27;s, uh that is a curvature Kate on the no. The normal for people to t u t divided by its Not so. Is this these vector so Well, huh? Minus 10 4 over square you go. 169 Uh, yeah. Times sign of I Sino to t I Law school Sign off to t J from the Nazi divide. But ah, the norm is this. But why did he or were? I want 69 so that we see that these canceled giving us one on one So it is normal should be equal. Teoh Mine Sign of pity. All right, Miners co sign off to t Jay. Really? The well you can check Berna some simplifications. Well, so So this deal. Oh, yes. So as you can check here Well, is this components plus that from planet? But you can check that you didn&#x27;t the product off t you then Hewitt an The two components her are flu was it happens. Ah is the product should be should be zero, which it is so

Problem

The space is $C[0,2 \pi]$ with the inner product …

Problem 8 Easy Difficulty

The space is $C[0,2 \pi]$ with the inner product $(6)$.
Find the third-order Fourier approximation to $f(t)=t-1$

Answer

$-1+\pi-2 \sin t-\sin 2 t-\frac{2}{3} \sin 3 t$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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$-1+\pi-2 \sin t-\sin 2 t-\frac{2}{3} \sin 3 t$

Linear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Joseph L.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Heather Z.

Kayleah T.

Samuel H.

Joseph L.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications