Examples and counterexamples in mathematics/Real-valued functions of one real variable

PolynomialsEdit

Polynomial with infinitely many rootsEdit

The zero polynomial   every number is a root of P. This is the only polynomial with infinitely many roots. A non-zero polynomial is of some degree n (n may be 0,1,2,...) and cannot have more than n roots since, by a well-known theorem of algebra, if   (for pairwise different  ), then necessarily   for some non-zero polynomial Q of degree  

Integer values versus integer coefficientsEdit

Every polynomial P with integer coefficients is integer-valued, that is, its value P(k) is an integer for every integer k; but the converse is true only for first degree polynomials (linear functions). For example, the polynomial   takes on integer values whenever x is an integer. That is because one of x and x-1 must be an even number. The values   are the binomial coefficients.

More generally, for every n=0,1,2,3,... the polynomial   is integer-valued;   are the binomial coefficients. In fact, every integer-valued polynomial is an integer linear combination of these Pn.

Polynomial mimics cosine: interpolationEdit

The cosine function,   satisfies           also   and   for all x, which gives infinitely many x such that   is one of the numbers   that is, infinitely many points on the graph. A polynomial cannot satisfy all these conditions, because   as   for every non-constant polynomial P; can it satisfy a finite portion of them?

Let us try to find P that satisfies the five conditions           For convenience we rescale x letting   and rewrite the five conditions in terms of Q:           In order to find such Q we use Lagrange polynomials.

Using the polynomial   of degree 5 with roots at the given points 0, 2, 3, 4, 6 and taking into account that   (check it by differentiating the product) we consider the so-called Lagrange basis polynomial   of degree 4 with roots at 2, 3, 4, 6 (but not 0); the division in the left-hand side is interpreted algebraically as division of polynomials (that is, finding a polynomial whose product by the denominator is the numerator  ) rather than division of functions, thus, the quotient is defined for all x, including 0. Its value at 0 is 1. Think, why; see the picture; recall that  .

Similarly, the second Lagrange basis polynomial   takes the values 0, 1, 0, 0, 0 (at 0, 2, 3, 4, 6 respectively). The third one   takes the values 0, 0, 1, 0, 0. And so on (calculate the fourth and fifth). It remains to combine these five Lagrange basis polynomials with the coefficients equal to the required values of Q:

 

Finally,   As we see on the picture, the two functions are quite close for   in fact, the greatest   for these x is about 0.00029.

A better approximation can be obtained via derivative. The derivative of   being   we have           The corresponding derivatives   are close but different; for instance,

 

In order to fix the derivative without spoiling the values we replace   with   where   is a polynomial of degree 4 such that the derivative of   is equal to   for   it means,   since   for these x; so,

  for  

We find such R as before:

 

and get a better approximation   in fact,   If you still want a smaller error, try second derivative and  

Polynomial mimics cosine: rootsEdit

The cosine function,   satisfies   and has infinitely many roots:   A polynomial cannot satisfy all these conditions; can it satisfy a finite portion of them?

It is easy to find a polynomial P such that   and   namely   (check it). What about   and  

The conditions being insensitive to the sign of x, we seek a polynomial of   that is,   where Q satisfies   and   It is easy to find such Q, namely,   (check it), which leads to

  As we see on the picture, the two functions are rather close for   in fact, the greatest   for these x is about 0.028, while the greatest   (for these x) is about 0.056.

The next step in this direction:      

And so on. For every   the polynomial

 

satisfies   and   which is easy to check. It is harder (but possible) to prove that   as   which represents the cosine as an infinite product

 

On the other hand, the well-known power series   gives another sequence of polynomials   converging to the same cosine function. See the picture for Q3;  

Can we check the equality   by opening the brackets? Let us try. The constant coefficient: just 1=1. The coefficient of x2:   that is,   really? Yes,   the well-known series of reciprocal squares is instrumental.

Such non-rigorous opening of brackets can be made rigorous as follows. For every polynomial P, the constant coefficient is the value of P at zero, P(0); the coefficient of x is the value at zero of the derivative, P '(0); and the coefficient of x2 is one half of the value at zero of the second derivative, ½P''(0). Clearly,     and   for all   (as before,  ). The calculation above shows that   as n tends to infinity. What about higher derivative,   does it converge to  ? It is tedious (if at all possible) to generalize the above calculation to k=4,6,...; fortunately, there is a better approach. Namely,   for all complex numbers z, and moreover,   for every R>0. Using Cauchy's differentiation formula one concludes that   (as  ) for each k, and in particular,  

Limit of derivatives versus derivative of limitEdit

 

For   we have   (think, why) as   Nevertheless, the derivative at zero does not converge to 0; rather, it is equal to 1 (for all n) since, denoting   we have    

Thus, the limit of the sequence of functions   on the interval   is the zero function, hence the derivative of the limit is the zero function as well. However, the limit of the sequence of derivatives   fails to be zero at least for   What happens for  ? Here the limit of derivatives is zero, since   (check it; the exponential decay of the second factor outweighs the linear grows of the first factor). Thus,

 

It is not always possible to interchange derivative and limit.

Note the two equivalent definition of the function f; one is piecewise (something for some x, something else for other x, ...), but the other is a single expression   for all these x.

The function f is discontinuous (at 0), and nevertheless it is the limit of continuous functions   This can happen for pointwise convergence (that is, convergence at every point of the considered domain), since the speed of convergence can depend on the point.

Otherwise (when the speed of convergence does not depend on the point) convergence is called uniform; by the uniform convergence theorem, the uniform limit of continuous functions is a continuous function.

It follows that convergence of   (to f) is non-uniform, but this is a proof by contradiction.

A better understanding may be gained from a direct proof. The derivatives   fail to converge uniformly, since   fails to be small (when n is large) for some x close to 0; for instance, try  

 

for all   and   is not zero (unless  ).

In contrast,   uniformly on   that is,   as   since the maximum is reached at   (check it by solving the equation  ) and   And still, it appears to be impossible to interchange derivative and limit. Compare this case with a well-known theorem:

If   is a sequence of differentiable functions on   such that   exists (and is finite) for some   and the sequence   converges uniformly on  , then   converges uniformly to a function   on  , and   for  .

Uniform convergence of derivatives   is required; uniform convergence of functions   is not enough.

Complex numbers, helpful in Sect. "Polynomial mimics cosine: roots", are helpless here, since for   we have   for all  

Monster functionsEdit

Continuous monsterEdit

Everyone knows how to visualize the behavior of a continuous function by a curve on the coordinate plane. To this end one samples enough points   on the graph of the function, plots them on the   plane and, connecting these points by a curve, sketches the graph. However, this seemingly uncontroversial idea is challenged by some strange functions, sometimes called "continuous monsters". Often they are sums of so-called lacunary trigonometric series

  (where the numbers   are far apart).

The most famous of these is the Weierstrass function

  (for appropriate   such that  ).

According to Jarnicki and Pflug (Section 3.1), this is   (and the similar series of sine functions is  ).

 
Weierstrass function

This image (reproduced here by Michael McLaughlin with reference to the above-mentioned Wikipedia article, and here by Richard Lipton without reference) is in fact a graph of the approximation   to the function   obtained by connecting   sample points (which can be seen by inspecting the XML code in the SVG file). It looks like a curve albeit rather strange one. The last two summands are distorted, since the step size   exceeds the period   of   and is close to half the period   of   However, all terms with   contribute at most   that is,   for all   this difference is barely visible unless one zooms the image.

So far, so good. But this monster is not the worst. In order to get a more monstrous lacunary trigonometric series one may try frequencies   increasing faster than   or coefficients decreasing slower than   or both.

Let us try the series

 

these coefficients   are convenient, since their sum is (finite and) evident:   Accordingly,   (since  ), and   for all   (since  ).

Partial sums
 
 
 

Similarly,   and   Accordingly, partial sums   and tails   satisfy     and     for all  

Due to evident symmetries (see the graphs of   on the period  ) it is enough to plot this function on  

 
Partial sum   zoom 4

Looking closely at the graph of   we come to some doubt. At first glance, it is drawn with a thicker pen. But no; some (almost?) vertical lines are thin. So, what do we see here, a curve, or rather, the area between two "parallel" curves?

  and   zoom 40
  and   zoom 400

The graph of   becomes clearly visible after a zoom, but the doubt returns, hardened, with   One more zoom only worsens. We realize that the graph of   on   looks too nice. In particular, it appears that the graph of   crosses the upper side of the red box. So, how to get closer to the graph of  

Fortunately, for some special values of   the exact value of   is evident. First,   (see above), whence   for all integers   due to periodicity:   for all   Similarly,   since   for all odd integers   (in particular,  ). And on the other hand,   for all  

Without the first summand we have the first tail   with the period   Accordingly,   and   for all integers   And on the other hand,   for all   Thus,

Special values and bounds of  
via  
via  
    for all integers  
  for all  

Similarly, for all  

    for all integers  
  for all  
 
Via   zoom 400; also  

Returning to   and   on   we add special values and bounds via   and see that   is much further from   than   from   We have more than 440 points on the red curve, and equally many points on the blue curve.

 
Graph of   zoom 400.

Thus, if the horizontal size of picture is less than 440 pixels, then inevitably the graph of   crosses all the pixels between the red curve and the blue curve! Within the given resolution the graph of   does not look like a curve, but as the area between two parallel curves.

This, in itself, isn't surprising. For example, a 10 megahertz radio wave modulated by a 100 hertz sound may be described by the function   The graph of   on, say,   does not look like a curve, but as the region between two curves,   And on   it looks like a rectangular region. However, zoom ultimately helps; the graph of   on   (or any other 1 microsecond interval) does look like a curve.

In contrast, for   zoom never helps, it only worsens. In fact, it ultimately leads to a graph that looks like a rectangular region. This shows the monstrous nature of   On the other hand, the height of the (nearly) rectangular region converges to zero as zoom tends to infinity; this shows the continuous nature of  

 
Random sample of points on the graph of  

Visualization by a region-like graph leaves much to be desired. Unable to draw a curve-like graph, we still can do more. We can choose at random many values of   compute the corresponding values   of the given function, and draw the points   This picture shows a sample of   random points on the graph of the tail   in the first quarter of the period of this tail (which is sufficient due to the evident symmetries, as was noted). Truth be told, the function   was used here as a satisfactory approximation of   and each   was specified by more than   ternary (that is, base 3) digits in order to compute  

As before, the red line and the blue line are bounds of the given function   (via  ), and the points on these curves are special values. A wonder: all the   random points are far from these bounds! Usually   points is enough to get a satisfactory picture of the maximum value of the function. But for this monster,   points are few.

Generally, for the sum of a lacunary trigonometric series, it is quite a challenge, to find its maximum and minimum (on a given interval) even approximately, say, with relative error less than 10%. Our choice of frequencies   in concert with our use of cosine rather than sine, allow us to find unusually high values of the sum. In order to fully appreciate this good luck, try to maximize such function as   or to minimize   and you will realize, why Weierstrass preferred cosine functions and frequencies of the form   where   is a positive odd integer. Usual numerical optimization methods fail because local extrema are numerous and very sharp. By continuity, a monster function is close to its maximum in some neighborhood of its maximizer, but this neighborhood is very small; a random point has very little chance of getting into such neighborhood. The vast majority of values are far from extreme.

 
Increasing rearrangement of sample points on the graph of  

In order to investigate the distribution of values of the tail   on its period, one may took the   values   in the first quarter of the period (used above), together with the   opposite numbers   (these are values in the second quarter), sort the list of all these   numbers in ascending order, and treat the result as values of a new function at equally spaced points. Here is the result plotted, see the red curve. This is a numeric approximation of the so-called monotone (increasing) rearrangement of the given function.

Significantly, the result is quite close to the famous normal distribution  , and the red curve is quite close to the corresponding quantile function   indicated by the black points.

(Here  ) A wonder: the function is monstrous, but its monotone rearrangement is nice. Why so?

The given function is the sum of many summands (of the form  ); and "The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution."

Choosing a point   at random according to the uniform distribution we may treat the summands as random variables. However, one of the conditions is independence; and our summands are dependent, moreover, functionally dependent, since   (More generally,     being the Chebyshev polynomial.) Nevertheless, probabilistic approach to lacunary trigonometric series exists and bears fruit; in particular, appearance of normal distribution in this context is a well-know phenomenon.

But clearly, the normal approximation must fail somewhere, since our function is bounded (by  ), while a normal random variable is not.

The word "central" in the name of the central limit theorem is interpreted in two ways: (1) of central importance in probability theory; (2) describes the center of the distribution as opposed to its tails, that is, large deviations.

 
Increasing rearrangement of   zoom 40.

In the normal distribution, deviations above   appear with the probability   accordingly, for   the inequality   holds on intervals of total length   and a random sample of size   should contain about   such values. This is indiscernible on the previous picture, but clearly visible on the zoomed picture.

 
Increasing rearrangement of   log scale.

For   the normal probability is only   and   is only   in order to see what happens for   and   we need larger samples and logarithmic scale. This last picture presents a sample of size   (that took a computer several hours). We see that the distribution is close to normal at   but moves away from normal at   Hopefully, the approximate normality follows from some moderate deviations theory that applies at   while the departure from normality follows from some large deviations theory that applies at   and further; but for now, in spite of the recent progress in the probabilistic approach to lacunary series, such theories are not available, and the situation near the big red question mark on the picture remains unknown. It is natural to guess that is this domain the probability of   deviation is smaller than its normal approximation, therefore, smaller than  

Assuming that this conjecture is true, and taking into account that   and   we conclude that the function   exceeds   of its maximum on intervals of total length less (probably, much less) than  

But, again, this monster is not the worst. In order to get a more monstrous lacunary trigonometric series one may try frequencies   increasing faster than   or coefficients decreasing slower than   or both. Unexpectedly, or not so unexpectedly, such functions, being more monstrous analytically, are more tractable probabilistically. (In the paper by Delbaen and Hovhannisyan, note the coefficients   for   in Remark 2.3; note also the "big gaps" theorems 1.4, 2.15, 2.16 for  ) This is a special case of a general phenomenon formulated by De Bruijn as follows:

It often happens that we want to evaluate a certain number [...] so that the direct method is almost prohibitive [...] we should be very happy to have an entirely different method [...] giving at least some useful information to it. And usually this new method gives (as remarked by Laplace) the better results in proportion to its being more necessary [...]