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Hello. Welcome to this big room. I think we're here just today. And then back to the usual room on Tuesday.
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Today, I'm going to talk about more mathematics than usual, but it's very applied mathematics.
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Fourier Laurent and Chevy chef are three parallel settings, which are the basis of algorithms used in all areas of computational science.
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In particular, odes and pdes ultimately rely on these ideas.
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And I want to highlight for you the very close relationship between these three in particular.
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These are the basis of spectral methods, but all sorts of expansions.
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And whenever you try to connect something discrete to something continuous, the chances are you're going to use one of these three settings.
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The picture is the unit circle and that's where Z lives.
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So you can imagine a variable Z on the unit circle in the complex plane,
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and then the diameter of that circle would be the unit interval from -1 to 1, and that's where X lives.
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And then theta is the angle to a point Z.
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So if that would be a point Z on the unit circle, then that's the corresponding angle theta and this would be the corresponding value of x.
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So that picture describes how three variables relate one to another.
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You could say, Why do we need all three? Well, the answer is so much mathematics depends on these that all three are truly important.
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Theta is Fourier analysis, Z is complex analysis, and X is real analysis.
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These are huge, huge territories. Of course, once you see the picture, you immediately see various algebraic relations between the two.
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So for example, X is written as one half times Z plus Z inverse.
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And we could also say that x is the cosine of I keep writing theta because in my head I usually use theta.
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But I think in this lecture I intend to use T instead. So if I ever write theta, it's a mistake I meant to say T of course it's really theta.
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But when you're writing codes and so on, theta is an annoying variable.
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So we tend to switch to T. So I'm going to spend the first few minutes actually detailing all three of
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these contexts in a completely parallel fashion to emphasise the parallels.
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So let's begin with Fourier and of course we've all heard of Fourier analysis about 200 years old now.
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So the setting here is that we have a periodic function, so not an arbitrary function, a periodic function,
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and conventionally on an interval of length to pi, I'll call the function capital F of T and t will be the interval from 0 to 2 pi.
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Of course, you could also take minus pi to pi or various other fundamental domains.
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Now, whenever we do numerical things, the chances are we're going to sample the function at a set of points.
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And of course, for periodic functions, equally spaced points are the obvious choice.
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So let's talk about two and plus one.
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Equally spaced, equally spaced points.
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And I will label them to t sub k and that will be equal to two pi k over two n plus one where K ranges between zero and two n.
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So if you draw the picture for that. Notice that because the function is periodic, we only include one end point.
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So here we are on our interval going from 0 to 2 pi.
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And if we had four points, for example, well, sorry, they're going to be five points.
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So if we have five points, for example, they would be here and here and here and here and here, not including there, because that's the same as this.
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So of course, for you now with this corresponds to discrete for analysis, which is what an engineer would typically be doing in signal processing.
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Now, the basic functions that we expand things in, we loosely speak of signs and cosines,
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but it's usually more convenient mathematically to take their complex version.
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The complex exponential. So e to the i k t is the complex exponential with wave number k.
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And of course, from there you can get the cosine and the sine.
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So the cosine would be the real part of this and the sine would be the imaginary part or a better way to say
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it is the cosine is this plus E to the minus k over to the sine is this minus C to the minus Oct to II.
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Now something that we do explicitly or implicitly all the time is trigonometric interpolation.
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And there again, loosely speaking, we're talking about signs and cosines,
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but it's often more convenient to write down the formulas in terms of e to the effect.
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So a trigonometric interpolate, let's say.
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Would be a function I'll call Capital PM and it's a trigonometric polynomial,
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which means it's a finite sum from minus m to m of some coefficient c k times e to the i.
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K t. So that's some from minus end to end.
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Equivalently, you could talk about signs and cosines of Katie.
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Now, when people do numerical integration, which we call quadrature,
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the obvious idea is to interpolate by trigonometric and then integrate the interpolate, and that's called the trapezoidal root.
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It's not obvious that that's the trapezoidal rule, but it is.
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So of course we think of the trapezoidal rule as trapezoidal, and then you integrate the trapezoids.
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That turns out to be equivalent to interpolating by trigonometric and integrating the trigonometric interpolate.
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I won't go through that argument. It's a straightforward one. So the trapezoidal rule effectively integrates the trigonometric and turbulent.
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Let's at least make that statement. This is equivalent to integration of P seven.
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More precisely, integration of P seven.
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Well, there are two N plus one points, so I should say the two and plus one point trapezoidal rule is equivalent to integration of that interpolate.
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Now if you want to find roots, let's say root finding, if you want to find the zeros of a trigonometric interpolate.
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Well, I'm not going to go into that just now. But effectively what you do is you can compute the eigenvalues of a matrix.
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So that turns out to be an effective way to find roots of P sub in its so-called companion matrix.
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So root finding and eigenvalue problems are very closely related in numerical practice.
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What happens as you take the limit of K going to infinity?
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I really should say n going to infinity. The limit as NW Goes to Infinity is a 48 series.
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So you have a trigonometric interpolate means a finite degree trigonometric polynomial.
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As that degree goes to infinity, you have 48 series.
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So in that limit you get this famous idea, assuming F is, let's say Lipschitz Continuous or something,
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you can write it as an infinite series where now K goes from minus infinity to infinity of the same thing.
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A k. E to the t. What are the 48 coefficients in that series?
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So a sub k is equal to one over two pi.
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Times the integral from 0 to 2 pi of f little f of t no foreign capital f 50.
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Getting my notations confused. Times e to the minus i k.
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T. D d t. So 48 theories involves 48 coefficients and then infinite series.
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And you all know this from different angles, I'm sure.
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Finally, let me mention the basic principle that people use if they're analysing the convergence of the theories.
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Assuming that your function is analytic so it has convergent Taylor theories.
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The fundamental hypothesis is if you have an analytic function on the interval.
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Then it can be analytically continued into the complex plane into some strip.
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And I will say a little more about that in a moment.
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And that implies that the 48 coefficients decrease geometrically, so they are O of C to the minus absolute value of K for something bigger than one.
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So any time you have an analytic function, it's for you. Coefficients decrease at a geometric rate.
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So that means its trigonometric and turbulence converge at a geometric rate.
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So there on one screen you have a summary of all of 44 theories.
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And let me give you a reference.
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Of course, there are a thousand work from this, but from a numerical point of view, my favourite references as usual something by myself.
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So let me recommend a paper by right and javaid and our to Monto Nellie.
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And myself. And that's it's called extension of fun to periodic functions.
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It's in the Siam Journal of Numerical Analysis, and it's coming out this year.
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You can find it on my web page.
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And in particular, we discuss all this stuff in a way that fits very cleanly with the other cases of Laurent and Cherif.
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So now let's switch to the Laurent setting for analysis is famous to engineers and real mathematicians.
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The Laurent setting is famous to complex mathematicians. This is what you do when you're working in the complex plane.
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And incidentally, my opinion is that mathematically, in some sense, the Laurent case is the fundamental ones.
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Of course, they're all more or less equivalent. But this conceptually, I think, is the basis from which it's easiest to derive the others.
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So now we're going to suppose that we have a function which I will call a fancy F,
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so I'm going to call it like that F of Z and Z is in the unit circle.
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Now, if you have a function defined on the unit circle, then sort of by definition it's periodic in some sense.
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So I don't have to say it's periodic that goes with the territory.
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And then in analogy to the equally spaced points, we now have two and plus one roots of unity.
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So these are the two and plus first roots of the number one.
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And we could write those as z sub k equals e to the i times t some k for k between zero and two m.
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Now corresponding to the complex exponential, you have simply Z to the K, which we could call a mano z to the K.
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So you begin to see why somehow this is a more fundamental territory.
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Z to the K is marginally simpler than E to the T corresponding to a trigonometric and turbulent.
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You have what people would call a laurent polynomial interpolate.
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And well, you should be able to write down the formula just by this close analogy.
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So a laurent polynomial and turbulent I would write as fancy p of z is the sum from minus m
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to end of c sub k z to the k so given and plus one roots of two m plus one roots of unity.
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There is a unique and turbulent of that form. It's a polynomial, except it has some negative powers as well as some positive powers.
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So that's what the word Lawrence polynomial means. Negative as well as positive powers if you want to do quadrature root numerical integration.
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Then again, the obvious formula is the appropriately scaled trapezoidal rule,
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where basically you add up the samples at Roots of Unity and multiply by the obvious factor.
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And that again is equivalent to integrating the interpolate.
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And then what comes next in our list is root finding.
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So that's the same again, I'm not spelling out the matrix involved,
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but if you want to do route finding and for functions that are that sort of live on the unit circle,
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then the way to do it is to set up a eigenvalue problem.
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You find the eigenvalues of a companion matrix.
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A companion matrix is a matrix that's mostly zero except on two diagonals and one row and it contains the coefficient c some k.
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What about the limit? Well, as Kate then goes to infinity, you have a long, long series.
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So you all know. Taylor Theories. Lauren Theories is the same except with negative as well as positive powers of Z.
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So a Lauren series. Means that you have the sum from minus infinity to infinity of a sub k z to the k.
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What are the law on coefficients? Well,
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it's the same formula as above for 48 coefficients transplanted from the TI variable to the Z variable and you get
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a sub k equals one over two pi times the integral over the circle of fancy f of z time z to the minus k minus one.
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And that's an integral over the unit circle. So if you know your complex variables, that's a Kochi integral.
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So Kochi integrals are the same as for a integrals.
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Once you know this stuff, you begin to lose the distinction between the different territories.
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Now, what happens if you have an analytic function?
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Well, now the appropriate region of an electricity is an annulus.
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So let's see on the interval from 0 to 2 pi.
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If you have an analytic function, then it's also analytic in some strip in the complex plan.
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Well, on the unit circle, if you have an analytic function on the unit circle, then it's also analytic in some annulus.
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Surrounding the unit circle, and that would be equivalent to this strip under the E to the i z map, e to the t map.
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So if F is analytic in an unknown annulus.
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Then that implies that the coefficients a sub k decrease geometric same as above o of C to the minus k for C greater than one.
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These two settings are not merely analogous, they're identical.
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They really are the same. Everything is the same. We've just changed variables between Z and now the reference on that.
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This is harder to find numerical references on a lot of people do numerical Fourier analysis.
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Not so many people do numerical lorente analysis, although of course there are some.
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And the paper I would suggest you look at is by my student, Anthony Austin and Peter Craven.
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Yeah. And myself.
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And that's in Siam Journal of Numerical Analysis in 2014.
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And that's called numerical algorithms based on analytic function values at roots of Unity.
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So as I say, these two are identical. The championship chef setting is not quite identical because it has a further symmetry assumption.
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And the picture on that is in these settings I can have an arbitrary function on the circle.
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But when I predict to the to the interval, I'm going to require that my function take the same value there as there.
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See, here's another value, the inverse. They both project to the same value of X.
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And in chubby chef analysis, I want a well-defined function here.
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So that entails a symmetry assumption for Z or T.
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So here we go. Fauria and TV chef were very famous mathematicians.
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I actually don't know anything about. Laurent is not as big a name, I guess.
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Okay, so now we suppose we have a function. Little F, a little X.
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And X is in the unit interval.
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And of course, we don't assume it's periodic. Now,
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the analogy to equally spaced points or roots of unity is what you get when you
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project those equally spaced points or routes of unity down to the interval.
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And those are cheryshev points, and now they're only in plus one of them, not two and plus one of them because pairs project to the same thing.
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So we've end plus one chubby chef points.
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And those are defined by x sub k equals the cosine of k pi over n zero less than or equal to k less than or equal to n.
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Analogous to the binomial or the complex exponential.
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We have the Chevy chef polynomial.
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And the chubby chef polynomial is written with a capital t t sub k of x and that is one half times z to the k plus z to the minus k.
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Another way to say it is, it's the cosine of k times, the arc cosine, the inverse cosine of x.
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So it oscillates like a cosine a tabish polynomial on the unit interval oscillates like a cosine,
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but it's and then it gets huge outside the interval for a polynomial in turbulent.
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Can be written as little pe of nn seven of x is equal to the sum, and now we have a sum from zero to n of c sub k.
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T k of x. If you do Quadrature Well, of course you know the natural method is to integrate the interpolate.
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I've got some data at VHF points. I integrate that interpolate.
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That's my approximate integral. All we need is a name for that algorithm and it's called Crenshaw.
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Curtis. So Crenshaw. Curtis Quadrature is the name we give to integrating the Chevy chef interpolate.
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Root finding. Well, it turns out now that instead of a big diagonal matrix eigenvalue problem,
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you essentially have a tri diagonal matrix eigenvalue problem and the matrix is called a colleague matrix.
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So you do this by computing the eigenvalues of a colleague matrix.
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These names were given at the height of the Cold War when people thought it was funny to talk about
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companion matrices and comrade matrices and colleague matrices in the limit as then goes to infinity.
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You, of course, have a Chubby Chef series.
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So a Chubby Chef series represents a function F of X as an infinite series from zero to infinity of a sum k.
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K of x. I haven't mentioned.
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Why do I have C K's and AK? So the interpreter and I'm using C K's, the series I'm using.
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Well, these are different numbers. As N goes to infinity, those six will approach these.
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But for any finite end, you have coefficients here which will be different from that.
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What are those cheryshev coefficients? Well, it's the same as the 48 or Lawrence thing with the change of variables, and when you do your sums,
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it turns out it's a sub k equals two over pi times the integral from -1 to 1
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of f of x t k of x divided by the square root of one minus x squared the x.
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Finally, what about analytics?
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Well, for periodic functions, if you're analytic on an interval that implies you're analytic in some strip for functions on the unit circle,
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if you're analytic on the circle, that implies you're analytic.
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In some analysts in the Chevy chef case, where, again, we're analytic on an interval but no longer periodic,
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it turns out that the right domain is an ellipse with four at minus one and one.
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And that ellipse is what you get when you take a circle and transplant it by this map.
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So if I take a circle here in the Z plan and apply this tarkowski map to it,
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I get in the lips in the X plane and any function analytic on the unit interval will
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be analytic in some ellipse corresponding to some sufficiently narrow annulus.
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So that's the analytic city setting. If F is analytic in an ellipse.
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With focus at minus one and one. Then that implies that the coefficients decrease geometrically.
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Now. It's a little tedious for me to write out all this stuff for 20 minutes.
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But let me tell you, the amount of practical mathematics codified in those three screens is just extraordinary.
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Here you have the basis of a large part of what people really do on computers, especially when they want high accurate solutions.
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The reference on this is my book Approximation Theory and Approximation Practice.
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At top, I called it approximation theory and approximation practice and I'll pass that around.
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So it's the book with the horse on the cover. This is from Port Meadow, the picture of the horse.
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Okay. Let me show a little bit of that stuff on the computer.
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So we're now working with the code called M 52, which is going to show a little of Fourier series and championship series in seven for convenience.
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So suppose I say Capital F equals ten fun.
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And then the function I will choose will be cosine of e to the sign of t,
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and that's a periodic function for well I put it on minus pi pi in the handout, but it might as well be two pi I think.
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I hope that works. And now in check from the way you do, you signal that you're dealing with something periodic as you type trig.
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So there we're constructing a for a series of four and turbulent to that function.
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And let's plot it I'll say subplot 1 to 1 plot F and I'll plot it in that mode.
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Okay. So there you have a periodic, not especially exciting function.
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Now suppose I say subplot one, two, two and I plot the Fourier coefficients of F.
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So there you are. And you see.
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Zero is the zero wave number, the constant term and you have minus KS and plus K.
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So you always have the symmetry when you write 48 coefficients.
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So there you see the 48 coefficients of that function on an absolute on a large scale and we're plotting their absolute value in seven.
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It always goes down to machine precision and then stops. Of course, mathematically it would keep going forever.
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It's an analytic function which implies it's analytic in some strip,
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which implies that the 48 coefficients decreased geometrically and sure enough, you see a geometric decrease of the Fourier coefficients.
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We have these gallery things built in, so if I say help Cheb gallerie trig, you'll see a collection.
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Things are going slowly of all sorts of built in functions you can play with,
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and we're now going to play with two of them, the AMP signal and the FM signal.
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So suppose I say F equals Chaddock Gallery Trig of F.M. signal.
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That is a preloaded Chevy fun. I'll now say subplot.
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One, two, two, one. And I'll plotted.
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And that is what we see. So it's an FM signal.
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You can sort of see that the amplitude is constant, but the frequency is being modulated.
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Let's look at it's Fourier transform. So I'll say subplot to two, three plot coefficients of S.
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So there you have a lot of coefficients.
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It's a smooth signal. So they're decaying nicely down to machine precision.
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Now let's compare that with an AM signal. So I'll now say F equals cab, dot, gallerie trig and signal and I'll say subplot.
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Two, two, two. So there you have an AM signal where the carrier frequency is constant,
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but the amplitude is being modulated and the you should all think in your head.
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Now what will its four year transform look like? So subplot 2 to 4.
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What you see are peaks at the carrier frequency so that the energy is all near plus or -50 because that's the carrier frequency.
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So to an engineer, there's a lot of interesting stuff there.
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Of course, I'm of a generation when these analogue signals were very important.
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Nowadays everything is so digitised. Who knows how your radio works nowadays?
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Finally, let's do a Chevy chaff analogue. So I'll now say F little s equals headphone of the same thing.
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So that was cosine of E to the sign of x. And of course by default it's the unit interval minus one one.
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If I say subplot one, two, one.
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And plot a little less. You see that?
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I've made a mistake. I meant to put it on a.
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Sorry. I meant to put it on 0 to 2 pi. Yeah.
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So there we have the same function, but now it's represented by a Chevy truck series instead of a 48 series.
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And if I now look at its coefficients, so that would be 1 to 2.
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These will now be Chevy chef rather than 48 coefficients.
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So no longer do we have a zero wave number and then plus and minus we have a zero wave
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number and then just higher and higher degree Chevy chef polynomials again analytically.
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So geometric convergent, that function is analytic in a ellipse and therefore you have geometric convergence of the Chevy Chef series.
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So there's a lot of beautiful mathematics here, which I've been very focussed on in recent years.
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Now let me tell you some more of it.
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So we're now going to focus on TV chef for the rest of today's lecture and tell you a word about Chevy Chef series and interpret.
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Well, the most basic word to say. Is that mathematically cheryshev is essentially the same as for gay.
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There's gifts that extra symmetry condition. Otherwise it's the same.
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But sociologically it's very different in the 48 territory.
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Everyone on earth, given a periodic function and sampling it would of course use equally spaced points.
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This is exactly what everyone would obviously do. Nobody would dream of any other method of representing a periodic function.
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But in the championship territory with non periodic functions, what people dream of is numerically exponentially bad.
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So of course most people don't use championship polynomials.
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They would just use mono x to the one, x to the two. Those are exponentially ill conditioned.
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So numerical methods that work with polynomials in the Multnomah or basis are catastrophically bad.
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If you use TV Chef, everything is just like Fourier analysis and it all works.
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So if you're ever working with polynomials on an interval you must use be here unless the degree of your polynomial is like three or four.
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Okay, so I've told you what the Chevy chef polynomials are and the general formula,
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for example, is t k of x equals the cosine of k times the cosine inverse of x.
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So for example, t zero is one and t one is x and T2 is two x squared minus one and t three is four x cubed minus three x and so on.
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And as I mentioned, they actually oscillate in the unit interval and grow a great deal outside the unit.
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So those are Tabish polynomials. And now as time permits, I want to tell you some theorems about representing functions by Chevy Chef series.
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So they were in one. If F is Lipschitz Continuous, and I'm sure many of you know what that means, and maybe some of you don't.
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You can look it up. It's a little stronger than continuity. It means that basically the slopes are all bounded by a constant.
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Then the TV Chef series, which I've already defined for you, convergence.
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And moreover, it converges absolutely and uniformly.
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These are technical terms. Absolutely means it works.
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Even if you put absolute values on each term, uniformly means that it works even if that the Epsilon two deltas are uniform across the interval.
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The consequences of these are that you can work with to have theories however you want, you can reorder them.
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That doesn't mess up the convergence. You can differentiate them. You can do all sorts of things.
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Turn by turn. Of course, this is an analogue of a theorem in four year analysis,
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which says that if Capital F is Lipschitz Continuous, then the Fourier series converges absolutely and uniformly.
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Ferrum, too, has to do with the geometric rate of convergence.
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Theorem two is that if F is analytic and bounded by some number in absolute value, call it M.
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In the Berenstein Rollups. And I'll say in a moment exactly what I mean by that.
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Then the Chevy Chase coefficient decreased and here's the rate.
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They're bounded by two and row to the minus k.
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So I mentioned earlier that analytic functions have geometrically decreasing cheryshev coefficients.
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Well, here's a theorem to tell you how fast that goes. And now let me tell you what row is.
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There's a unit circle of radius one out here.
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I've drawn a circle of radius row bigger than one.
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So if you take a circle of radius, row bigger than one and map it by this Tarkovsky map, it turns into an ellipse.
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And that's called the Berenstein Rowan. Berenstein worked on this stuff in 1912.
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Now from that found on the size of the Chevy chef coefficient, which of course is analogous to a bound on 48 coefficients,
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you can reach conclusions about the accuracy of approximations.
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So Theorem three. Says if f sub n is the truncated series.
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So we take the Chevy Shuffle series, but we truncated a term n.
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Then the error between F and it's truncated series.
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And when I said the error, I mean the maximum error on the unit interval.
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We could call that the infinity normal error. The maximum error on the unit interval is bounded by two and row to the minus n over row minus one.
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You get that just by adding up all the terms you've truncated away.
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So if you take an infinite TV Chef series with coefficients decreasing at that rate and you chop it at some point, how big is the tail?
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Well, you add up all of these bounds and this is what you get.
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So Chubby Chef series four analytic functions converge at a geometric rate if you truncate them.
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Theorem for. Concerns and turbulence as opposed to truncation.
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So suppose ppm is what we've already talked about the Chevy chef in turbulent.
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As opposed to the truncated series. Well, it turns out you get the same bound except twice as with that factor too out front.
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So f minus pm, the worst error on the unit interval is no greater than four m relative to minus n over rho minus one.
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So that tells you how big the error can be when you do interpolation and cheryshev points.
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If you have an analytic function, you're guaranteed that the entire plinth converge geometrically.
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So the final thing I want to mention is the subject of the other handout.
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So take a look here. This paper by seltzer.
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In 1972, Lagrangian interpolation at the Archbishop Points.
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Seltzer pointed out that there's a beautiful, stable way to compute these polynomial, and it's called the Barry centric formula.
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And here he set that forth. Somebody should have done this 50 years ago, but nobody did it with seltzer in 1972.
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He showed that you really can compute these things even if n is in the millions and this is called the barrier centric interpolation formula.
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Let me write it down. So Theorem five says.
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That piece of em that's the interpolate and cheryshev points is equal to a quotient of one thing by another thing.
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Each of those is a sum from zero to n. There's a prime there, which I'll explain in a second.
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You have in the numerator minus one to the J times f at the j championship point
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divided by x minus x j and in the denominator you have minus one to the j.
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Divided by x. Minus x. J. This is called the virus centric interpolation formula.
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So it's a mathematical formula which is kind of a curious one,
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because what you seem to see here is a function with a lot of polls in the numerator and a function with a lot of polls in the denominator.
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And yet the end result is a polynomial. Well, that is true.
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It's a mathematical fact that it's equal to a polynomial.
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Moreover, it is provably numerically stable.
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So this formula really does work. And this is what Chevy Fun did for many, many years.
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This is the formula that led to the creation of Chevy Fun, actually.
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I have to explain what the Prime means. So when I say the sum of something from K equals zero to m prime,
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I mean that we take one half times D, not plus one half times DRM and then all the others.
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Like that. So the prime means you multiply the terms by one half.
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Prime. Prime. So this amazing formula enables you to interpolate by polynomials in TV, Chef points.
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To any degree, it's as stable as Fourier analysis.
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It's just the same, really. So anything you can do with a discrete 48 transform, you can also do with Chevy Chef points for non periodic functions.
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If you look at this a little more, the more you look at it, the more puzzling it seems.
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Because what happens when x is equal to x j?
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Well, in that case, you bypass the formula and you say part of x j is equal to the data value f of x j.
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But what if x is very close to x j.
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We're going to have something huge here with tremendous cancellation errors and something huge here with tremendous cancellation errors.
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But it turns out the cancellation errors cancel and it really is provably numerically stable.
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So this is both fast. If there were then operations not in squared and numerically stable.
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So this really is the right way to interpolate. And I just want to finish by showing you that.
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Oh, so the back page of the handout is a paper by myself and Jean-Paul Baru.
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Baru is a professor at the University of Fribourg in Switzerland, and he is a great authority on the various centric methods.
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So if you want to read about it, maybe our survey paper is a good place to do that.
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So let me finish with this final demonstration showing Ben Stein ellipses.
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So this one's called. I'm 53. Ben Stein So what I'll do is I'll set F equal to a famous function.
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I run a function it's called. So it's one divided by one plus 25 x squared.
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And now I'll say a subplot. One, three, one plot f y lim minus one, two.
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So there you have the wrong assumption. It's analytic, but it's troublesome for certain methods of approximation.
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Let's look at its TV chef coefficient. So I'll say subplot 132 plot coefficients of f line width.
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Point six. Really? Oh. So there you see what looks like a solid triangle.
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Actually, all the odd coefficients have zero and the even coefficients are non-zero.
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So we're getting this kind of effect. But you see the totally clean geometric rate of convergence of the cheryshev coefficients.
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Finally, the last command will show the Bernstein Ellipse.
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So suppose I now say subplot. One, three, three.
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Plot, region. Plot.
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Region of death plots. The Bernstein Ellipse associated with this function axis equal grid arm.
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So well, there you see the unit interval and then that ellipse is the region of an electricity of this function
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because the wrong function has poles at plus or minus I over five and that ellipse is bounded by that.
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Let's now finally confirm the rate of convergence.
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It turns out. Oh, well, let's let's plot those polls. I say hold on.
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And I'll say plot -0.2.
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I'd point to I as red dots. Oops.
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00:49:48,490 --> 00:49:59,729
What did I do? So you can see the poles of that function are right there on the very fine lips they bound the region of analytically.
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So that's the largest ellipse inside which this function is analytic.
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Finally, let's compute the Bernstein value of row.
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It turns out it's one plus the square root of 26 divided by five.
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And let's see how our prediction matches the data.
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So I'll go back to the middle plot. And now I'll say plot k against Ro to the minus K as a dashed red line.
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So what this is going to do is I've figured out the value of row corresponding to that ellipse of analytically.
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And you remember the prediction is the coefficients decrease geometrically and the constant involved is a row.
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So there you can see the prediction is beautifully matched by the.