 Good morning, this is our last lecture on the topic of function approximation. In the previous lecture, we saw several ways of function approximation by the use of several families of Eigen functions of different stream level programs. And in particular, we also went through the Fourier series, which provides some additional facility other than what any family of Eigen functions of stream level problem would offer. Towards the end, we also discussed how to extend the idea of Fourier series to a function which is not periodic, which is defined over the entire real line and it is a non-periodic function. For that, we asked this question, how to apply the idea of Fourier series to a non-periodic function over an infinite domain. So, for that, we constructed the Fourier series over a finite interval of length 2 L and then we let the variable L increase and tend to infinity. And as a result, we arrived at this representation of the function which is the Fourier integral. Now in the Fourier integral, this is the coefficient available in the form of an integral. And similarly, this integral over an infinite domain of the continuous variable v. So, these two integrals are the coefficients. And the Fourier integral representation turns out to be a superposition of infinite such components with continuously changing value of e. Now in the case of the Fourier series, this integral was a summation. Now here, the summation becomes a sum of continuous components and therefore, you get an integral. And in the place of the two Fourier coefficients, we have got these in a similar manner in which, as we got it in the case of Fourier series. So, this Fourier integral can be now represented as this integral with the coefficients as coefficient functions. Earlier, these coefficients of these coefficients in the case of Fourier series were a n and b n that is they were indexed. Now these are continuous functions not for n equal to 1, 2, 3, 4 and so on, but p which varies continuously from 0 to infinity. So, in place of Fourier coefficients indexed with an integer variable n, we have got Fourier coefficients which turn out to be continuous functions of the frequency variable p. And these are in a way the corresponding amplitude functions a p and b p which are these two integrals. So, the only difference here is that this is an integral rather than a sum of terms for large number of n's and the coefficients are defined not for different values of n, but for continuous variable frequency variable which is p. Now using this itself, you can also represent this large double integral in a somewhat different manner in which you can push cos p x inside this integral and sin p x inside this integral and then you get this form of the Fourier integral expression which is called the phase angle form in which you find that cos p x cos p v plus sin p x sin p v is replaced with this and this turns out to become a straight forward double integral. Now from this double integral with a little more work, we can define an integral transform and how it is done is here. So, in this cosine, we can replace in this manner cos theta is e to the power i theta plus e to the power minus i theta by 2. So, if we if we substitute this expression for the cosine term there, then that by 2 comes here and inside we have got e to the power i p x minus v plus e to the power minus i p x minus v and then we find here that in this double integral d v for that differential, the corresponding variable v varies from minus infinity to plus infinity and this frequency variable p varies from 0 to infinity. Now this term in the integrand can be omitted if we consider this limit also from minus infinity to infinity that gives us a more symmetric representation. To see that what we do is that we can substitute p equal to minus q. So, that means that for this integral if we try to separately evaluate this component, then in place of minus p we will have q here in place of d p we will have minus d q and therefore, the limit of this integral over p will become from 0 to infinity because of the sign change it will become minus infinity to 0. That is first of all it will become 0 to minus infinity and then because of the sign change we will have the limit reversed. So, now if we recognize this that the integral of this term f v into this turns out to be actually the integral of this same thing over minus infinity to 0 then the addition of these two terms is the addition of an integral from minus infinity to 0 and then from 0 to infinity. That means this integral and this integral together will give the complete integral from minus infinity to plus infinity and that gives us the complex form of the Fourier integral in which it is a single exponential function here with f v and we have this double integral. And then the in the complex form of the Fourier integral this term also rather than cosine and sine has become this exponential function with imaginary index and this c p the amplitude function has become the integral that you get when you take only f v into e to the power minus i p v and plus i p x remains here. So, from the original Fourier integral also we got similar a p and b p now in a complex form the two coefficients combined together to give us the complex coefficient function. Now this actually holds the root of the definition of the Fourier transform let us see that directly here in this chapter we will not go into great detail, but we will just see how the integral transform called Fourier transform gets defined from the complex form of the Fourier integral. So, with a different notation here in place of x we have got p here in place of p the frequency variable we have got w here and so on otherwise it is the same expression and the 1 by 2 pi that factor has been split into two parts one is here 1 by root over 2 pi and the other is here. Now here this is the same earlier complex coefficient function and, but with this factor here rather than the same function rather than the earlier factor that we had. Now this can be considered as the function representation in which we are talking about a composition that is addition the sum of an infinite number of functions which are these in the form e to the power minus w t by root over 2 pi that is e to the power minus w t divided by root over 2 pi and the coefficients of these superposition coefficients of this superposition turns out to be this that is the amplitude function and the amplitude function varies continuously with the frequency. Now this function representation is in a way the linear combination the composition of a large number of frequency components the frequency components are here and the corresponding coefficients are here corresponding amplitudes are here and this amplitude turns out to be a function of the frequency that you get from here. So this function of frequency which turns out to be the amplitude is the Fourier transform. So amplitude of a frequency component that is this bracketed term here and that is the Fourier transform and this is a function of the frequency variables. So that means a waveform has large number of components of continuous frequencies and now for every frequency this gives us the amplitude value and when with this amplitude value and this frequency we construct the term that that component completely we have got this and such terms infinite number distributed continuously over the frequency variable is the complete function. Now if we rather have this Fourier transform in our hand then that Fourier transform sitting here with e to the power i w t multiplied and d w put here and integrated like this that is in this entire place if we put this Fourier transform then we have got this and this integral will recover the original function that is the inverse Fourier transform. The way we defined earlier Laplace transform and inverse Laplace transform in a similar manner we have got this Fourier transform and inverse Fourier transform Fourier transform and inverse Fourier transform these are quite symmetric in their look here you have got e to the power minus i w t to be multiplied and then integrated here you have e to the power i w t multiplied and then integrated. So this gives you very similar expressions for evaluating the Fourier transform and the inverse Fourier transform. Now these transforms are extremely useful in the solution of differential equations partial differential equations and also in the field of signal processing we will not go into the detail of that because the main agenda of this lecture is another kind of function approximation in which the objective is different than all the function approximation techniques that we have discussed till now. Earlier when we were discussing the interpolatory approximation of functions in that context we studied we discussed three criteria of approximation of functions one was the interpolatory approximation which we studied in that particular lesson and the other was least square approximation which we have studied in several sections in this course. First the linear least square problem which we studied in the module of linear algebra and then we studied the non-linear least square problem which we studied in the context of non-linear optimization and finally the same least square approximation we have been discussing till recently which are least square approximation based on the error value developed through integrals rather than sums of finite number of components finite number of error values. So that least square approximation we have studied already. Now the third criteria of approximation of a function is a minimax approximation in which the objective is not the sum of square all over the sum of squares of errors all over the domain but the objective function to be minimized is the maximum error. So therefore this kind of an approximation is called minimax approximation and one particular family of Eigen functions that is the solutions of the Chebyshe equation and for the Chebyshe polynomials give us specific interesting properties which are directly useful in the case of minimax approximation and therefore this topic of minimax approximation we start with a discussion of Chebyshe polynomials. This differential equation resembles the Legendre equation to a great extent in the case of Legendre equation we had a 2 here minus 2 here we have got minus 1. Now in place of this minus 1 if we had minus 2 then we found that this minus 2 x turned out to be exact derivative of this and therefore these two terms together in the case of Legendre equation turned out to be the perfect differential coefficient of 1 minus x square into y prime. Now in this case that 2 is missing so it will this much right now is not in the self adjoint form right now not in the stream variable form. So if you can if you divide the entire equation with this square root of 1 minus x square then you will find that here then you will have square root of 1 minus x square and here you will have minus x by square root of 1 minus x square and in that case the terms together will be in the self adjoint form and together they will give us the derivative of this combined term and through the division here you will get this. Now there is one interesting point here that if we change the independent variable here and make this substitution x equal to cos theta then this differential equation quite easily boils down to this differential equation is the same differential equation which we saw in the case of Fourier series. Now from here we can very easily see that the solutions of this will be cosine n x and sin n x they will be the linearly independent solutions of this differential equation. Now in particular the solution sin n x will be very complicated functions but the other family of solutions cosine n x that is for different values of n cos 0 that is 1 then cos x sorry not x but theta. So the solutions of this will be cosine theta and sin theta. So the term sin theta when finally expressed in the suffix back by putting theta equal to cos inverse x will turn out to be quite complicated function. On the other hand the other solution that is y equal to cos theta will turn out to be polynomial sin x. So you can see that so that is why we are particularly interested in the polynomial solutions of this differential equation and note that this differential equation also will define a singular stream level problem just like the Legendre equation because here the function r x which appears here is 0 at x equal to minus 1 and at x equal to 1. That means you do not need any boundary conditions to define the stream level problem. So this differential equation will define a singular stream level problem over this interval Legendre equation also did the same thing. Now we are particularly interested in the polynomial solutions of this particular stream level problem and one family of solutions of that will turn out to be polynomial solutions. Now the way we discovered this fact in the case of Legendre polynomials it is possible to discover the same fact for Chebyshev polynomials also in the same manner by consideration of the series solution and noting that for integer values of n the series terminate and get you polynomial equations polynomial functions. The same conclusion will be reached through this arrangement in which we replace x with cos theta and solutions of this turn out to be cosine n theta sin n theta. We discard that sin n theta because that is not going to give us polynomial solutions but cosine n theta will give us polynomial solutions and for all values of n cosine n theta will have a value 1 when we take x as 1. So the closed form expression for that particular family of solutions which are going to turn out to be polynomial solutions is this that is t n x is cosine n theta theta is cos inverse x. Now if we put n equal to 0 then we get t 0 which is cos 0 that is 1. If we put n equal to 1 then we get cos cos inverse x which is x. If we put n equal to 2 then we have got cos 2 theta we know that cos 2 theta is twice cos square theta minus 1. Similarly, cos 3 theta will be 4 cos cube theta minus 3 cos theta and so on. So we can also very easily prove that the family this family of functions t 0 t 1 t 2 t 3 etcetera will satisfy this three term recurrence relation that is t k plus 1 will be twice x t k minus t k minus 1 which means that after evaluating the two of them we can evaluate others in terms of polynomials directly rather than writing multiple angle formulas of cosines from this recurrence relation. What other properties these polynomials have these solutions of the stream level problem have. Now that with a little exercise we can establish a lot of properties of these polynomials. In the exercises of chapter 39 and chapter 40 of the textbook we have the steps to establish many of these properties. So it is expected that the student will go through those exercises and execute those steps to learn this the learn the derivation of these properties first hand. Here we will just summarize the immediate observations immediate properties for proceeding further. Coefficient in a sebaceous polynomials are all integers which is different from the Legendre polynomials. In particular the leading coefficient of t n will be t 2 to the power n minus 1 for even n that is for t 2 t 4 t 6 t n x is an even number of n function because they will have only the even powers. While for odd n it is an odd function this is the same as the Legendre polynomials in the case of Legendre polynomials also we noticed this property. All the sebaceous polynomials will have the value 1 at x equal to 1 that is obvious because at x equal to 1 cos inverse x will be 0. So for all values of n n into 0 is 0 so cos of 0 will be 1. So value of this all these polynomials for x equal to 1 will turn out to be 1 that means all of them are bunched at 1. Then we find that at minus 1 the value will turn out to be this minus 1 or 1 depending upon whether n is odd or even this also is same as Legendre polynomials. Legendre polynomials that means also had this property and this is a very interesting property that is the value the absolute value of t n x over this interval is always less than equal to 1. This also is there in the case of Legendre polynomials. Again zeros also the distribution of zeros of a sebaceous polynomial also shares a similarity with Legendre polynomials. The zeros of the sebaceous polynomials are all real and lie within this interval. But the locations of the zeros of sebaceous polynomials polynomials turn out to be turn out to have very particular significance there at these locations. And these locations are called sebaceous accuracy points why that we will explore further. Further another interesting thing is that the zeros are interlaced by the zeros of t n the n th sebaceous polynomial are interlaced by those of t n plus 1. That means the n th degree sebaceous polynomial will have n zeros and every zero of the t n function will turn out to be n plus 1. And closed by two consecutive zeros of the next order sebaceous polynomials. This interlacing property is also interesting. Now we find that these three properties except this part the sebaceous polynomials share with the Legendre polynomials. And this next property extremar of sebaceous polynomials have a very unique property which no other familiar functions possesses. And that is all extremar of the sebaceous polynomial t n are of equal magnitude magnitude is unity. And they alternate in sign and occur at these values of x one maximum next minimum next maximum next minimum all these maximum and minimum of the same absolute value 1. That is maximum value 1 then minimum value minus 1 then maximum value 1 minimum value minus 1 maximum value 1 minimum value minus 1 and so on. So, this is a very unique property and this property is called the minimax property. We will see the significance of this property. Orthogonality and norms. So, this is the integral here is equal to 0 for m not equal to n. That is the outcome of the result on the stream level problem here you see that the function p x is 1 by root over 1 minus x square. So, that is why two distinct solution that is two solutions of this stream level problem corresponding to different values of n that is different Eigen values distinct Eigen values will turn out to be orthogonal with respect to the weight function 1 by root over 1 minus x square. So, the statement of orthogonality of Shobishi polynomials turns out to be like this. So, this is the statement on orthogonality and the definition of norms will come when we put m equal to n. And for m equal to n which is not 0 we get it pi by 2 and for m equal to n which is 0 we get pi. This is similar to what we found in the case of the Fourier coefficients. Now with these properties we can have a close look at one such sample Shobishi polynomials between minus 1 and 1. So, between minus 1 and 1 the Shobishi polynomials varies like this. You see the zeros and extrema of this turn out to have a uniform distribution over the theta variable say for theta equal to 0 x is 1. So, the value of Shobishi polynomials is 1 then for theta equal to 30 degree pi by 6 x is here cos of cos 30 degree that is root 3 by 2. So, at that we have got a 0 of the Shobishi polynomials. So, then 0 30 degree minus 60 degree at theta equal to 60 degree we have got this negative value minus 1. So, here was 1 extrema value maximum next minimum. So, at 30 degree at 0 we had maximum value at 30 degree we had a 0 at 60 degree we have the minimum value minus 1 then at 90 degree x equal to 0 we have got t 3 x also 0. So, that is a second 0 from this side then again at 120 degree we have got the maximum value at 150 degree we have got a 0 at 180 degree we have got this extrema value. So, the zeros and extrema are distributed like this. The extrema values have the same absolute value 1 these maxima are at value 1 these minima are at value minus 1. This is a distinct property which is not shared by any other familiar functions and as you compare it with say Legendre polynomials you will find that this is the graph of the 8th order Legendre polynomials and 8th order Shobishi polynomials super imposed super imposed. So, this is the graph of p 8 Legendre polynomials and this is the graph of the Shobishi polynomials. You can see that the Shobishi polynomials has this equal ripple property that is equal amplitude oscillations. So, from here it goes all the way down in the case of Legendre polynomials as you proceed towards the center of the domain near 0 the amplitude tends to come down compared to here it comes down towards the center. In the case of Shobishi polynomials every extrema is at the extreme value of minus 1 or 1. So, 8th order polynomial it has got going down going up going down like 8 such trends you have and all the zeros you can see here 1 2 3 4 5 6 7 8 and they are distributed uniformly over the theta variable cos inverse x. So, being cosines and polynomials at the same time Shobishi polynomials possess a wide set of interesting properties which have this unique application in minimax approximation. The most striking property is this equal ripple oscillations leading to the minimax property. We will see this in two stages first what is this minimax property of Shobishi polynomials. The important result is that among all polynomials p and x of degree n with the leading coefficient equal to unity why this is needed because if we want to compare two polynomials then we can make no comparison if one of the polynomials can be multiplied by an arbitrary number. So, we need to fix something for both the polynomials to make a worthwhile comparison. So, we say that we will fix the coefficient of the largest degree term that is we are talking about monic polynomials. So, the leading coefficient will have unity that means x to the power n plus a 1 x to the power n minus 1 plus a 2 into x to the power n minus 2 and so on. So, the leading coefficient is 1. So, among all such polynomials of degree n with the leading coefficient unity that is among all monic polynomials of degree n this is the one which has the least deviation from 0. Now, why this factor because we already know that T n x has the leading coefficient which is 2 to the power n minus 1. So, to make it monic we must multiply it with 2 to the power 1 minus n. So, the odd divide with 2 to the power n minus 1. So, therefore, the monic version of the n th order is this and among all monic polynomials of n th degree the claim is that this one will have least deviation from 0 over this interval. So, mathematically we can say that maximum over this domain this interval of absolute value of p n x is always greater than equal to the corresponding value for this particular function. So, all other polynomials will deviate more from 0 and this will deviate least and that least deviation turns out to be this. Now, this equality part is easy to establish because we already know that T n x will deviate between minus 1 and 1. That means the absolute value of the deviation of T n x is 1. So, absolute value of the deviation of this factor into T n x will be this factor itself. This unit equality part is quite obvious for the inequality that is all other polynomials will deviate more we need to do something. So, for that we first assume that suppose there is some other polynomial other than this which has even less deviation. So, if there exists some monic polynomial p n x of degree n which deviate less which deviate less than this. If we assume that like that then what we can do is that we consider the values of this candidate polynomial at the extrema of the Chebyshev polynomials. Now, already know that the n th degree Chebyshev polynomial has n plus 1 alternate extrema positive negative positive negative and so on. So, each with this absolute value that we already know. So, at those extrema points of the Chebyshev polynomial we consider the values of this. Now, these points are giving us the maximum deviation of this which is 2 to the power 1 minus n and we know we have assumed that this candidate function deviates less. So, if it deviates less then it will never be able to cross the extrema value of this. So, that means that if we consider this difference then this part in the difference will always dictate the sign of the difference polynomial q n x. So, if this is positive then whatever be the value of this the difference will be positive. If this is negative then whatever be the value of p n the difference will be negative because this fellow's value has been assumed to be less than the extrema value of this and we are evaluating at the extrema points. Then this difference polynomial will have the same sign as the first part. Now, we already know the nature of the signs of this in the sequence that is n plus 1 locations n plus 1 alternating extrema this function has. So, its sign will be alternating at these n plus 1 locations positive negative positive negative and like that. So, with alternating signs at n plus 1 locations in sequence p n x and therefore q n x also will have n intervening zeros. So, because this q n x has the same alternation of sign as this because this one is going to dictate the sign of q n. So, q n with alternating signs at n plus 1 locations will have n intervening zeros positive here negative here. So, in between q n x must have a 0. So, between n plus 1 extrema of alternating sign it will not extrema of this difference function, but with alternating signs at n plus 1 locations in sequence it will have n intervening zeros. But then that is something very awkward because both of these were monic polynomials which means that in both of these cases the coefficient of x n was unity and in that case in the difference the x n term will get cancelled and this polynomial is actually an n minus 1 degree polynomial. But then we have just now found that it has n intervening zeros how can an n minus 1 degree polynomial have n zeros. So, this leads to a contradiction which means that this assumption of a monic polynomial of less variation less deviation is certainly wrong. So, through contradiction we get this result that among all polynomials of degree n it is the Chebyshev polynomial the scaled version of it to make it a monic polynomial deviates least and this turns out to be the deviation. Now how do we utilize this property? Let us explore with a linear combination of the Chebyshev polynomials will have this series which is called the Chebyshev series for which we can construct the coefficients a 0 a 1 a 2 a 3 etcetera as we have earlier studied that is how to construct the generalized Fourier coefficient for this Chebyshev series. There are the Chebyshev coefficients for which we have these expressions. Now typically we would not be able to construct this infinite series. So, we typically try to represent a function with a truncated series only up to a finite term and this approximation is called the Chebyshev economization. We are representing the function in an economical manner now the leading error term in that case will be the next term that is up to the term T n we are already incorporating in this truncated series. The next term which will be the leading error term will be this and we know from the foregoing discussion that among other among all the candidates for the polynomials this will deviate least from 0 over this interval and therefore it is qualitatively similar to the error function. Then the question arises that with this how to develop a Chebyshev series approximation? Do we try to find out so many Chebyshev polynomials and then evaluate the coefficients and then construct this series the truncated series while that could be an option. The good news is that even that much computation is not really needed to be done there is a little shortcut to do that. So, first of all this is a little bit of pre processing involved if the required domain of the function description function representation is not minus 1 to 1, but it is a to b then we always can scale it scale the variable T to x with this in which x belongs to this. Then onwards we work with this interval of x the economized series this one which we are constructing that will give the minimax deviation of the leading error term which will deviate least and that deviation will be the will have the equal ripple property which every Chebyshev polynomial has. Now, since we are discussing with reference to the leading error term now for a moment if we concentrate on this leading error term only and pretend that this happens to be the actual error and assuming this to be the actual error we know that there will be n plus 1 values of x where this actual error officially turns out to be 0 why because T n plus 1 has n 0s. So, there are n values of x from minus 1 to 1 in that interval of interest in which this function is 0. So, these are at the 0s of the n plus 1th Chebyshev polynomial. So, at these values of x we find that we if we take the exact actual function value at these points then there will be the correct values through which we can interpolate a function. So, that means that we can take these points these values of x and at these values of x which turn out to be the 0s of this we evaluate the function we have got the values of the function at n plus 1 values of x. And we know that if this happens to be the actual error then the polynomial that will be interpolated through these n plus 1 points will deviate least. So, that means that after we have got the 0s of this function this Chebyshev polynomial of n plus 1 is d z and got the Chebyshev accuracy points the so called Chebyshev accuracy points. Then the polynomial interpolated through these values at the Chebyshev accuracy points through any other means possibly with the Lagrange interpolation itself will give us the Chebyshev series approximation in terms of the actual function that we developed. Because here we were seeing that this series will have 0 error at those values which are those values of x which are roots of this 0s of this. And now we are telling that through these same points with we can construct a polynomial which is through Lagrange interpolation. Now n plus 1 values of an n s degree polynomial fixes the polynomial uniquely. So, therefore whether we construct the series by finding out the Chebyshev coefficients a 0 a 1 a 2 etcetera or whether we construct that by Lagrange interpolation over those points where we know the correct value of the function it will lead to the same polynomial. And therefore from the foregoing theory of Chebyshev polynomials we can just take at which points we give the correct values of the function and then from the values at those n plus 1 points we simply construct the polynomial through Lagrange interpolation. So, this is called Chebyshev Lagrange approximation. Now this much on approximation with Chebyshev polynomials themselves. Now the minimax property goes beyond because here this assumption has worked that the leading error is the important component of the error. But that will not be always the case because in the actual complete series there could be components there could be contributions from t n plus 2 t n plus 3 and so on. So, if those components those contributions are significant then even though the approximation resulting from Chebyshev Lagrange consideration gives us qualitatively similar error values. But still it will not be the perfect minimax approximation even the perfect minimax approximation perfect minimax approximation we can construct and that is based on this very important property very important theorem of by Chebyshev that is before that make note that the situations in which the minimax approximation is desirable are those in which we want to develop the approximation once and keep it for use in future. A priori we do not know at which point the function value will be needed. And therefore it will be a good idea if we can say something regarding the uniform quality control that the function evaluated by this method will never be too long too much mistaken that is there is a limit on the maximum deviation that it may have. So, minimax approximation gives us that guarantee that the maximum deviation is limited the maximum deviation is minimized. And the interesting thing is that that approximation is going to have this minimax property which has constant amplitude of the variation. And this is the Chebyshev minimax theorem that is how do you recognize that a particular represent has this minimax property. And criterion for recognizing that minimax approximation is the same as the maximum deviation going to be same in absolute value at all the extrema that is equal ripple oscillation is directly related to the minimax property. So, the Chebyshev minimax theorem says that of all polynomials of degree up to n a polynomial p x is the minimax polynomial approximation of a function that is it minimizes this maximum deviation. If and only if there are n plus two values of x where this difference assumes alternate maxima and minima. So, where that this difference takes its extreme values of the same magnitude and alternate sign that means the exact properties that we saw in Chebyshev polynomial that is it has extreme maximum and minima values of the same absolute value 1 1 minus 1 1 minus 1 they are they were the extrema. And that actually confers on the Chebyshev minimax property. Similarly, for any approximation the approximation being minimax is equivalent to its equal ripple characteristics. In the following we will not really have a formal proof of this theorem, but we will in a way in a schematic manner we will see the line of proof which also gives us the seed of an algorithm to construct that minimum minimax polynomial approximation. Suppose you have got a function f x and for which you have got a polynomial approximation of the fourth degree which is the polynomial p x and the error function is f x minus p x. Now, the theorem says that p x is a minimax polynomial approximation of f x if the error function f x minus p x has n plus 2 that is 6 extrema of equal magnitude and alternating sign. So, d minus d d minus d d minus d 6 extrema of same magnitude and opposite signs if f x minus p x has that characteristic then p x is the minimax polynomial approximation of f x this is the statement of the Chebyshev theorem. Now, suppose the polynomial that we have constructed does not have this property that is it lacks that property only by a little that is 5 extrema are of the same value d minus d d minus d this particular extrema is not at minus d, but at a little above that is this is not really the farthest extrema value that we were looking for that means here in this case there is a gap. So, in that case this is not the minimax polynomial approximation. Now, if there is any such gap now this idea of utilizing any gap to reduce the deviation at the other extrema gives us the conviction that if not a formal proof of the minimax theorem and also in a way a very fruitful algorithm. Say suppose for being the minimax polynomial approximation this extrema had to be here at value minus d now there is a gap it does not reach here. So, we say that we might increase this deviation that is rather than this value we can increase the deviation that is take this minimum even lower and at the cost of this deviation we try to reduce the deviation at other extrema then it will reduce the maximum value of the error. So, for that what we do if this gap is of magnitude epsilon then we construct a polynomial with largest possible value epsilon by 2. So, that means it is bounded within y plus minus epsilon by 2 we consider a polynomial of that type which is bounded in its value within plus minus epsilon by 2 that kind of a polynomial we can call as delta p x that is the polynomial by which p x has to be changed. And we impose the condition that this delta p x other than having no value beyond this plus minus epsilon by 2 band is positive here at this extrema negative here at this extrema then positive here at this extrema positive here at this extrema and again negative here at this extrema. So, at all the 5 extrema minimum maximum maximum at these maxima we are asking for positive positive positive and positive values of delta p and at the 2 minima here and here we are asking for negative and negative values for delta p. Now note that for satisfying these requirements what delta p has to do delta p has to cross how many times from positive to negative to positive and then finally negative here. Since the largest magnitude extrema here there is a pair of continuous maxima without there being a minimum in between between this and this this point and this point m and n the function the polynomial delta p did not have to change its sign. So, it had to change its sign once here second time here and third time here. So, thrice changing sign is perfectly fine for a 4 degree polynomial. So, we can certainly construct a 4 degree polynomial which is positive here negative here positive positive and negative here. So, that kind of a delta p we can construct and we can have a hold over its maximum value also because if its maximum value turns out to be larger than epsilon by 2 then you can multiply the polynomial throughout by a small number to bring it within epsilon by 2 and this limiting of its absolute value within epsilon by 2 will ensure that whatever damage the polynomial p x is cause through the addition of delta p will not damage this point too much beyond as other values come closer as other values come closer to 0 at at other extremum points. This one will not go beyond epsilon by 2. So, the best improvement at other point that is possible is epsilon by 2 and even after the watch damage at this point it will not go beyond epsilon by 2. So, that means through this exercise we can develop a suitable delta p which when added to the current p will give us a new polynomial in which all these extrema will be reduced in their magnitude all these extrema will be reduced in magnitude and perhaps this extremum will be increasing in its magnitude, but never beyond what will be the resulting value of the other extrema. That means that any gap like this can be utilized to reduce the deviation at the other critical extrema. Now, consider the same situation if this point were also here. In that case in order to reduce all the extrema we would need the values to come down at these three maxima and to come up at these three minima if there were no gap, but that would not be possible because in that case delta p would be required to be positive negative positive negative positive negative. A fourth degree polynomial can never show never have so much of so many of oscillations. So, therefore, if there is no gap like this then that will mean that we cannot improve the approximation further. On the other hand any gap that exists like this we can always utilize that gap to reduce the variation at the other point. So, this is the schematic demonstration of the truth of the minimax polynomial. In other standard books on approximation theory you can have a look at the formal proof. Incidentally this same reasoning turns out to be the basis of the construction of the minima minimax polynomial approximation and that algorithm is called Ramesh algorithm. The general Ramesh algorithm is actually for rational approximation, rational function approximation, but the same theme can be utilized for approximation with polynomials also and for this minimax polynomial approximation we will find that the Chebyshev polynomial turns out to be a very good starting point with the help of Chebyshev Lagrangian approximation as we have seen sometime back. So, you will find that in the light of this general minimax theorem you will find that T n plus 1 x the n plus 1 th degree Chebyshev polynomial which we considered earlier was actually qualitatively similar to this complete error function with its equal ripple properties. Now, this completes our module on approximation theory. Next couple of lectures we will devote on a quick recapitulation of the very important partial differential equation and after that couple of lectures on partial differential equations we will consider complex analysis. Thank you.