This is post is related to Extremal Taylor polynomials where it was important to observe that the Taylor polynomials of the function do not have zeros in the unit disk. Let’s see how far this generalizes.
The function has the rare property that all zeros of its Taylor polynomial have unit modulus. This is clear from
In this and subsequent illustrations, the zeros of the first 50 Taylor polynomials are shown as blue dots, with the unit circle in red for reference.
When the exponent is less than -1, the zeros move inside the unit disk and begin forming nice patterns in there.
When the exponent is strictly between -1 and 1, the zeros are all outside of the unit disk. Some of them get quite large, forcing a change of scale in the image.
Why does this happen when the exponent approaches 1? The function is its own Taylor polynomial, and has the only zero at -1. So, when , the Taylor polynomials are small perturbations of . These perturbations of coefficients have to create additional zeros, but being small, they require a large value of to help them.
For a specific example, the quadratic Taylor polynomial of is , with roots . When , one of these roots is near (as it has to be) and the other is large.
Finally, when and is not an integer, we get zeros on both sides of the unit circle. The majority of them are still outside. A prominent example of an interior zero is produced by the first-degree polynomial .
Suppose is a holomorphic function in the unit disk such that in the disk. How large can its Taylor polynomial be in the disk?
We should not expect to be bounded by 1 as well. Indeed, the Möbius transformation has Taylor expansion , so in this case. This turns out to be the worst case: in general is bounded by 5/4 in the disk.
For the second-degree polynomial the sharp bound is , attained when ; the image of the unit circle under the extremal is shown below. Clearly, there is something nontrivial going on.
Edmund Landau established the sharp bound for in his paper Abschätzung der Koeffizientensumme einer Potenzreihe, published in Archiv der Mathematik und Physik (3) 21 in 1913. Confusingly, there are two papers with the same title in the same issue of the journal: one on pages 42-50, the other on pages 250-255, and they appear in different volumes of Landau’s Collected Works. The sharp bound is in the second paper.
By rotation, it suffices to bound , which is . As is often done, we rescale a bit so that it’s holomorphic in a slightly larger disk, enabling the use of the Cauchy integral formula on the unit circle . The Cauchy formula says . Hence
It is natural to use now, which leads to
Here we can use the geometric sum formula and try to estimate the integral of on the unit circle. This is what Landau does in the first of two papers; the result is which is the correct rate of growth (this is essentially the Dirichlet kernel estimate from the theory of Fourier series). But there is a way to do better and get the sharp bound.
First idea: the factor could be replaced by any polynomial as long as the coefficients of powers up to stay the same. Higher powers contribute nothing to the integral that evaluates , but they might reduce the integral of .
Second idea: we should choose to be the square of some polynomial, , because can be computed exactly: it is just the sum of squares of the coefficients of , by Parseval’s formula.
Since is the -th degree Taylor polynomial of , it is natural to choose to be the -th degree Taylor polynomial of . Indeed, if , then as desired (asymptotics as ). The binomial formula tells us that
The coefficient of here can be written out as or rewritten as which shows that in lowest terms, its denominator is a power of 2. To summarize, is bounded by the sum of squares of the coefficients of . Such sums are referred to as the Landau constants,
A number of asymptotic and non-asymptotic formulas have been derived for , for example Brutman (1982) shows that is between 1 and 1.0663.
To demonstrate the sharpness of the bound , we want and on the unit circle. Both are arranged by taking which is a Blaschke product of degree . Note that the term can also be written as . Hence which is simply when . Equality holds in all the estimates above, so they are sharp.
Here are the images of the unit circle under extremal Taylor polynomials and .
These polynomials attain large values only on a short subarc of the circle; most of the time they oscillate at levels less than 1. Indeed, the mean value of cannot exceed the mean of which is at most 1. Here is the plot of the roots of extremal : they are nearly uniform around the circle, except for a gap near 1.
But we are not done…
Wait a moment. Does define a holomorphic function in the unit disk? We are dividing by here. Fortunately, has no zeros in the unit disk, because its coefficients are positive and decreasing as the exponent increases. Indeed, if with , then has constant term and other coefficients , , … , . Summing the absolute values of the coefficients of nonconstant terms we get . So, when these coefficients are attached to with , the sum of nonconstant terms is strictly less than in absolute value. This proves in the unit disk. Landau credits Adolf Hurwitz with this proof.
In fact, the zeros of (Taylor polynomials of ) lie just outside of the unit disk.
The zeros of the Blaschke products formed from are the reciprocals of the zeros of , so they lie just inside the unit circle, much like the zeros of (though they are different).
For some reason I wanted to construct polynomials approximating this piecewise constant function :
Of course approximation cannot be uniform, since the function is not continuous. But it can be achieved in the sense of convergence of graphs in the Hausdorff metric: their limit should be the “graph” shown above, with the vertical line included. In concrete terms, this means for every there is such that for the polynomial satisfies
How to get such explicitly? I started with the functions when is large. The idea is that as , the limit of is what is wanted: when , when . Also, for each there is a Taylor polynomial that approximates uniformly on . Since the Taylor series is alternating, it is not hard to find suitable . Let’s shoot for in the Taylor remainder and see where this leads:
Degree polynomial for
Degree polynomial for
Degree polynomial for
Degree polynomial for
Degree polynomial for
The results are unimpressive, though:
To get within of the desired square-ness, we need . This means . Then, to have the Taylor remainder bounded by at , we need . Instead of messing with Stirling’s formula, just observe that does not even begin to decrease until exceeds , which is more than . That’s a … high degree polynomial. I would not try to ask a computer algebra system to plot it.
The more terms of Taylor series we use, the more resemblance we see between the Taylor polynomial and the sine function itself. The first-degree polynomial matches one zero of the sine, and gets the slope right. The third-degree polynomial has three zeros in about the right places.
The fifth-degree polynomial will of course have … wait a moment.
Since all four critical points are in the window, there are no real zeros outside of our view. Adding the fifth-degree term not only fails to increase the number of zeros to five, it even drops it back to the level of . How odd.
Since the sine Taylor series converges uniformly on bounded intervals, for every there exists such that . Then will have the same sign as at the maxima and minima of the latter. Consequently, it will have about zeros on the interval . Indeed, the intermediate value theorem guarantees that many; and the fact that on will not allow for extraneous zeros within this interval.
Using the Taylor remainder estimate and Stirling's approximation, we find . Therefore, will have about real zeros at about the right places. What happens when is too large for Taylor remainder estimate to be effective, we can't tell.
Let's just count the zeros, then. Sage online makes it very easy:
sineroots = [[2*n-1,len(sin(x).taylor(x,0,2*n-1).roots(ring=RR))] for n in range(1,51)]
The up-and-down pattern in the number of zeros makes for a neat scatter plot. How close is this data to the predicted number ? Pretty close.
The slope of the blue line is ; the (ir)rationality of this number is unknown. Thus, just under a quarter of the zeros of are expected to be real when is large.
The actual number of real zeros tends to exceed the prediction (by only a few) because some Taylor polynomials have real zeros in the region where they no longer follow the function. For example, does this:
Richard S. Varga and Amos J. Carpenter wrote a series of papers titled Zeros of the partial sums of and in which they classify real zeros into Hurwitz (which follow the corresponding trigonometric function) and spurious. They give the precise count of the Hurwitz zeros: for the sine and for the cosine. The total number of real roots does not appear to admit such an explicit formula. It is the sequence A012264 in the OEIS.