Lecture 11

[Music] welcome as in the last lecture we were discussing about the convergence of the four year series in terms of the Cesaro ability and what we have seen is that if f is a continuous function at X not then we reach up to the stage where we computed F convolution of FN at x0 minus of f of x0 is less or equal to the integral is broken into two parts one was near zero and another is a from zero so now this is f of x minus of Y minus of f of x and then this is FN of y d y as we have seen that this is uh less or equal to and F is uh this is less than Epsilon by 2 so we have get that Epsilon by 2 then 1X 2 pi integral - pi to Pi because I can enlarge mod Y less than Delta to Y varies over minus pi to Pi that's a bigger set the integral is going to dominate the previous integral and then plus 1 by 2 pi integral mod y greater or equal to Delta now this is the soup of f of x minus of Y I can pull it out and one from here by triangle inequality I get this f n y Dy now for n greater or equal to n subn we have this is less than Epsilon by 2 this is nothing but equal to 1 plus this is uh this is less than Epsilon now which is less than Epsilon by 2 + Epsilon by 2 which is equal to Epsilon so therefore for all n greater or equal to n subn we have have got this that F convolution of FN does f convolution of FN at X this converges to f of x as n goes to Infinity now the second part of the theorem says that if f is a continuous function on minus pi to Pi then this conver is uniform that means it should the N Sub note what we are going to get it is uh not dependent on X now here exactly if you see closely the proof of uh fairs theorem at the point of continuity is that you know if f is a continuous function on this compact interval then it is uniformly continuous therefore the choice of Delta is independent of where our X is therefore we are getting a Delta which is independent of X and rest of the thing and the tail of if we are taking large n then the area covered by mod of F and Y this is going to be small therefore there is uh nowhere we are going to get the dependent t on X because of the uniformly continuity hence if F continuous then as f is uniformly continuous F convolution of FN of X converges to f of x uniformly so this is uh very very interesting and also now one would like to ask if f is not continuous only REM integrable 2 pi periodic 3an integrable function then can we recover our F in some me so this is I can say the corollary if f is reman integrable then 1X 2 piun integral - < to piun mod of f convolution of f n of x minus of f ofx DX this goes to zero as n goes to infinity and the proof is uh standard approximate technique uh which says that for this F integrable there exist a sequence of continuous function the properties of the r and integrable we have integral 1X 2 pi minus pi to Pi mod of f k of xus of f of x DX this goes to zero as K goes to Infinity that is what we had seen in while recalling uh the the remen integration and we also have used this result in the convolution property so therefore what we have - pi to Pi mod of f convolution of FN of x minus of f ofx DX this is I am going to add and subtract minus FK convolution of FN of X Plus FK convolution of FN of x minus F K of X Plus F K of x minus of f ofx I'm adding and subtracting so by triangle inequality this becomes minus Pi 2 pi mod of f convolution of FN of x minus FK convolution of f ofx DX plus FK convolution FN of x minus of f k of X DX + 1X 2 piun integral - < to piun mod of f k of x minus of f ofx DX now as we have seen that this goes to zero as K goes to infinity and this converges as n goes to Infinity for each K FK are continuous function so this converges to zero as n goes to infinity and now we need to worry about the first part let's call this as i1 + I2 + I3 okay so now i1 is equal to 1x 2 pi - < to piun mod of f minus of f k convolution of FN of X DX this we have seen in the properties for convolution is that this is less or equal to 1x 2 pi - piun to piun mod of f minus of FK of X DX plus into 1X 2 piun integral Min - piun to piun mod of FNX which is FNX DX now this by our observation that this is small now this is small for large K and this is small so to be more precise this one is saying that for Epsilon there exist a let me write this as star so star says for Epsilon positive there exist a k KN such that 1 by 2 pi - PK to Pi mod of f k of x - FX DX is less than Epsilon by 3 for all K greater or equal to K KN now for this K KN you can get choose only that K not therefore what we are going to get is that this is less than Epsilon this is 1 so i1 is less than Epsilon by 3 now I2 so therefore and I2 is less than Epsilon by 3 for some n for some n subn for for all n greater or equal to n subn because that's what uh we have seen because uh FK not is a continuous function um by our previous thing it is a uniformly it conver FK for two this is FK convolution of FN this converges to FK KN uniformly therefore I can pull that Epsilon out and this is the length is 2 pi and gets cancelled and and I3 is again less than Epsilon hence the result okay so now what although what we uh we might not have a pointwise convergence if if is just a reman integrable function but certainly under the integral it converges to F because as we know that that if I change uh a function in few points couple of points then obviously integral is not going to change at all so it is uh natural for just for a rem integrable function we need to consider the convergence under the integral sign but however if there is a continuity then we know specifically that at this point we cannot tamper with the F so that therefore it makes sense for talking about the convergence when the function is uh continuous okay so now let's see what are the application what we are going to get uh here so there are few more observation we want to do about the fair kernel so uh now natural thing is that we would like to ask what is the for year coefficient of a f of n now as you can see that this is nothing but summation over mod J less or equal to n 1 - mod J by n and then this is min - piun to Pi by definition 1 by 2 pi Factor will be there e to the power i j minus of n x DX now this is again going to survive when J is equal to n otherwise n is equal to J otherwise it is not going to survive that means J is varying from minus n to n therefore our n has to be less or equal to capital N therefore this is equal to 1 - mod n by n if mod n is less or equal to N and is equal to zero otherwise therefore the for year series of FN is summation over mod n less or equal to n 1 - mod n by n e power i n x and this is nothing but a trigonometric polom so the for now from this the forer series of F convolution of FN is now we know that F convolution of FN hat at n this is equal to F hat of N and FN hat at n now this only survive if mod n is less or equal to capital N therefore the convolution F convolution FN foror coefficient is going to survive only only when mod n is less or equal to capital N therefore foror series is going to be a finite sum so this is equal to mod n less or equal to n this is 1 - mod n by n f hat of n e the power I NX again as you can see that F convolution of FN is nothing but a trigonometric polom so this is the trigonometric polom now we have corollary let f and g be continuous function such that F hat at n this is equal to G hat at n for all n belongs to J then f is equal to G so now what it is saying this is the forar transforms uniqueness if F hat of n is equal to G hat of n then F has to be equal to G so in order to prove this it is enough to so if F hat of n is equal to Z for all n v over J then f is nothing but the constant function zero why is that that is if we have this result then we can get the corollary because you consider H equal to f minus of G then H is a continuous function then H hat of n this is equal to F hat of n minus of G hat of n now because F hat is equal to G hat this is equal to zero now if we have that if we assume this statement then this will going to say that H is identically equal to zero this implies that f is equal to G so it now we want to prove this now as you can see F convolution of FN of X this is going to be equal to F is a continuous function f convolution of F and X by this formula this is a trigonometric polinomial so therefore what we had observed that the trigonometric polinomial for your series is the same so now is f of n if we we are putting it to be zero then this is equal to zero and now this is a sequence of continuous function which is zero and we know that this converges to F uniformly and so the sequence of zero is going to converge only to zero thus as F convolution of FN converges to F this implies that f is equal to Z so now this is the uniqueness now once we have the uniqueness we can uh so now as a matter of fact here let's come back to this so now what it is saying is that fair had answered the question if we get F hat of n then we can construct F convolution of FN Capital FN by this so this is going to be F convolution of capital of FN of X so only unnown thing is that fat of n so once we get F of n we can compute F convolution of FN and fair guarante is that this is going to converse to F so we can actually reconstruct our F if we know F hat of n so another uh interesting corollary what we suppose if is a continuous function and F is a continuous function and summation / n mod of fat of n is finite then then summation of f hat of n e the^ i NX that is the foror series of F this converges to f of x that means the foror series if it is converging to some function then this orary is giving us the guarantee that that function is nothing but F itself the proof follows from the uniqueness of the foror coefficient so suppose summation over F hat of n e^ i INX as uh because we have imposed this summation over mod of f f of n is finite therefore this series converges now this converges to some GX now if we compute G hat of n this is equal to 1x 2 pi- < 2 piun summation / n varies over z f hat of uh uh sorry j j e^ I JX and e^ minus of I NX DX now because it is absolutely summable so I can pull out the sum outside the integral so what we have got is nothing but F hat of J 1 by 2 pi - PK to piun e^ i j minus of n X DX now the orthogonality relation of e to the^ i n x will tell that this is going to survive only when J is equal to n so therefore this is this is nothing but equal to Fat of N and this is true for every n all n varies over J therefore by the previous corollary we have f ofx is equal to G of X so we have answer successfully the question what we have asked uh that if the foror series converges then is it going to converge to the function f or not what we have answered if the forer series converges absolutely then it has to converge to F okay so now some other uh implication of the fce theorem is we can right that we know the wus theorem which tell us that if we have a continuous function then it can be uniformly approximated by a sequence of pols that was what the Wess theorem we know now here what Fair theorem is saying is that every 2 pi periodic continuous function can be approximated uniformly by a sequence of trigonometric polinomial what do I mean suppose f is a continuous function then there exist a sequence PN of trigonometric polinomial such that supremum of X Mod of f of x minus PN of X this goes to zero as n goes to Infinity now the proof is uh fairly simple already we have observed this because now what is the candidate for our PN the trigonometric polinomial this is nothing but f convolution with FN of X and what we had seen is that this is a trigonometric polinomial and if f is a continuous function we have seen that this sequence converges un uniformly to F that's what the corollary is so now this is a very uh uh interesting thing is that if we have this then we can get our classical Wess theorem so what does it say I will give a sketch of that uh it says that if f is in C minus pi to Pi let's take then there exist polom sequence of polinomial let's say PN such that supremum of X Mod of f of x minus PN of X this is less than Epsilon so now this by Fair theorem we have F minus of PN X goes to zero so each of this uh PN the trigonometric polinomial is going to look like some more than less or equal to n a n e^ i NX now as you can see e the^ i INX for each fixed n e the power I INX is equal to cos NX + I sin NX now we know the Tor series of the COS NX and I sin NX and we know that this convergence it can be approximated by polom because if you write down the Tor series of uh s and cosine that is nothing but a polom plus the error term and so all these things e n x can be approximated by the polinomial therefore the linear combination finite linear combination of uh this is a n * c ^ I NX can be approximated by the polinomial and hence the wus classical theorem that every continuous function can be approximated uh by the polom so this we can also derive from the fce theorem thank you [Music] [Music] [Music]