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Point) In the real vector space c"(R) with inner product(f,9} L; f)gtz) dx.consider the orthogonal set of polynomials {u1, U2, Uz} , whereU1 = 1, U2 = %, and U...

Question

Point) In the real vector space c"(R) with inner product(f,9} L; f)gtz) dx.consider the orthogonal set of polynomials {u1, U2, Uz} , whereU1 = 1, U2 = %, and Uz = 223Let v = f(z) be the function defined by S 0 f(z) ifz < 0, 31 if > 0.Compute the following inner products_(u1, U1) (u2, U2) (u3, Uz) (v,u1) (v, U2) (v, U3)21/38/453/21/4(b) Find the second degree generalized Fourier approximation fz(z) of f(z).f(x)

point) In the real vector space c"(R) with inner product (f,9} L; f)gtz) dx. consider the orthogonal set of polynomials {u1, U2, Uz} , where U1 = 1, U2 = %, and Uz = 22 3 Let v = f(z) be the function defined by S 0 f(z) ifz < 0, 31 if > 0. Compute the following inner products_ (u1, U1) (u2, U2) (u3, Uz) (v,u1) (v, U2) (v, U3) 21/3 8/45 3/2 1/4 (b) Find the second degree generalized Fourier approximation fz(z) of f(z). f(x)



Answers

Let $V$ be the vector space of polynomials over $\mathbf{R}$ of degree $\leq 2$ with inner product defined by $\langle f, g\rangle=\int_{0}^{1} f(t) g(t) d t .$ Find a basis of the subspace $W$ orthogonal to $h(t)=2 t+1$.

We're giving the functions of zero F one of two and a three everyone to find an orthogonal bases for the subspace respond by these functions. So let's begin by computing the first element of the basis which we call g zero of tea. And we can just put this to be f zero off the the constant function. One thing we're going to use the Grand Street process. We might as well just compute now The entire dinner part off G zero and zero, which were mind is the integral from 0 to 2 pi off zero vax that multiplies. Jeez, relax. The X, of course, is the integral from 0 to 2 by one and of course, is Dubai. Now we start by computing the projection off F one on 20 and these F one against zero dividing G zero against you zero of Oh, these are mistake. They should be Gee, zero 15. There were no matter. So these one over two pi which is the norm of zero as we computed previously, and they need to go from 0 to 2 pi ng off if one of x so cause I novaks the multiplies g zero back. So one So the X And, of course it is zero because these integral zero Therefore we can put G one off T too mean f one of T minus the projection off F one on 20 That, of course, is the call sign of tea. And again we compute now in a product off G one against you one which is the girl from zero to to buy off co sign squared off teen Well, Max, the X and this is exactly by now, repeat the process. We compute the projection off two on 20 Same ideas. He's one divided by two buying because he's a normal zero. The integral from 0 to 2 pi off co sign squared the vax D X, which is which is a pie and then is divided by one multiplied by one over two to buying. So these one off and then the projection left to own, did you? One similarities becomes one off one divided by pine, which is the norm off Juwan that multiplies in there About zero from zero to Dubai Off course in cute of axe, Jax. Everything multiplying G one of teams off Christine but now it is all integral 00 Therefore, we we found our third element of the basest G to Off T, which, of course, given by after 15 minus the projection off G to well over half two on two G zero and Juwan Under Fourth is is co sign squared of T minus one off and again, let's compute G two against you two. I need to go from 0 to 2 by off GTO X Square DX, and you can compete with the program T's turns out to be exactly by fourth. Finally, to find the fourth element of the basis, we need to compute the projection off three on tow, the other elements of the basis. So let's quickly do this. So the projection into G zero turns out to be exactly zero than the projection of After You Wanted You Won. It's given by again these entirely multiplies concern of tea and teases Exactly 3/4 off course 90. And finally, the projection of a 300. You, too, is again given by she's integral. That multiplies. Do you two off team and again they're all integral zero. Luckily, so we can to find your three off Dean as after a of T minus. All these projections that were just computed, which therefore is because I'm cute off t minus 3/4 co sign of tea and now the function J 0 to 1 g 23 Arnulfo colonel basis for the space that we were wearing.

In the question. We have to prove the falling properties. If let's g it's maybe a personal and Fitch bless G h. This will be first property second. See it? It's as it was to see and fetch We're see is really and third property toe grooviest G. It wants to G. If not, these properties will be through one on the basis off the one property that we know that this if g is a full system integration from A to B and fix GXE and don't work from red function the into DX we're the Blue X is the positive way function was a great function. And in this property we have to assume that the function f g and h our continuous on the new us Hey, so me So now going towards the solution for a park f bless g using the property, it will be quite toe. Let me going from Hey Toby effects less gxe and do at checks into the blue X which is the way function The X, which will be Puerto the main from maple be infects checks w x dx less. Yeah, Toby DX had chicks w x NeeIix so f busy. It will be the Puerto, in fact, less de And so our first proper case who want no going towards the second property. See if at using the form like will be Quito Limit going from a to B. See if next had chicks no view. Let's be X Next is as C is the real number. So it s common which is equals to and fix. And it checks w x bx, which will be it Puerto see into f at that is our second property is also proven now going towards the third property that is G f equals two using the form level we call to limit some May Toby GXE and fixed made X on the Buicks. That is, as these are in America play so GX into effects will be Puerto effects and g x w x the x, which is a cool story if G and the third probably is also do well. Thank you


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