Consider a trigonometric polynomial of degree with complex coefficients, represented as a Laurent polynomial where . The Riesz projection of is just the regular part of , the one without negative powers: . Let’s compare the supremum norm of , , with the norm of .

The ratio may exceed . By how much?

The extreme example for appears to be , pictured below together with . The polynomial is in blue, is in red, and the point is marked for reference.

Since has positive coefficients, its norm is just . To compute the norm of , let’s rewrite as a polynomial of . Namely, which simplifies to in terms of . Hence and .

The best example for appears to be vaguely binomial: . Note that the range of is a cardioid.

Once again, has positive coefficients, hence . And once again, is a polynomial of , specifically . Hence and .

I do not have a symbolic candidate for the extremal polynomial of degree . Numerically, it should look like this:

Is the maximum of attained by polynomials with real, rational coefficients (which can be made integer)? Do they have some hypergeometric structure? Compare with the Extremal Taylor polynomials which is another family of polynomials which maximize the supremum norm after elimination of some coefficients.

The theorem is not just about polynomials: it says the Riesz projection is a contraction (has norm ) as an operator .

Proof. Let , the singular part of . The polynomial differs from only by the sign of the singular part, hence by Parseval’s theorem.

Since consists of negative powers of , while does not contain any negative powers, these polynomials are orthogonal on the unit circle. By the Pythagorean theorem, . On the other hand, . Therefore, , completing the proof.

This is so neat. And the exponent is best possible: the Riesz projection is not a contraction from to when (the Marzo-Seip paper has a counterexample).

Suppose is a bounded metric space in which we want to find points at safe distance from one other: for all . Let be the greatest value of for which this is possible (technically, the supremum which might not be attained). We have .

The sequence tends to zero if and only if is totally bounded. From now on, consider a specific metric space which is bounded but not totally bounded: the space of all measurable subsets of with the metric , the measure of the symmetric difference. More specifically, the elements of are equivalence classes of sets that differ by a null subset. Another way to think of it is the subset of which consists of functions that attain only the values and . In this form we identify a set with its characteristic function . The measure space could be replaced by any isomorphic probability space, of which there are many.

The space contains an infinite subset with separation: namely, let be the set of all numbers such that the fractional part of is less than ; or equivalently, the th binary digit of is zero. Each symmetric difference has measure exactly . Therefore, for every .

Upper bound for distances

To get an upper bound for , suppose are -separated. Let be the characteristic function of . By assumption, whenever . Summing over all such pairs shows that . Fix a point and let be the number of sets among that contain . An easy counting argument shows . The latter quantity is bounded by where . Since this bound holds for every , it follows that , which simplifies to

, where .

For example, , , , and so on. Are these estimates sharp? For an affirmative answer it would be enough to demonstrate that by finding a suitable collection of subsets. Note that sharpness requires that almost every point of be covered by exactly of these subsets, and that all pairwise distances between sets must be equal to .

Sharpness of the upper bound

Four subsets with separation are easy to find: , , , and . Thus, .

To find six subsets with separation, some preparation is helpful.

For any set , the map is an isometry of onto . So, there is no loss of generality in assuming that one of our sets is . Then we are to find of measure . In order to have the specified distances, the measure of intersection of any two sets must be .

For example, if , the above says we need three disjoint sets of measure , which is a trivial thing to find. For the sets will no longer be disjoint.

Discretization of the problem

A natural way to proceed is to partition into equal subintervals (or other sets of equal measure). Then we look for sets, each of which consists of subintervals. Their pairwise intersections must consist of subintervals.

The case was done above. If , we want to use subintervals to form sets with subintervals in each. Pairwise intersections must consist of precisely subinterval. Encoding subintervals by their indices 0-9, we can describe such sets as 0123, 3456, 6780, 1479, 2589. And since 10 is a triangular number, there is a neat illustration of these five sets: three sides of the triangle, and two “stars” connecting its center to the sides.

There is no need to translate this back into sets formed by subintervals: as long as the combinatorial problem has a solution, we have the desired conclusion. In this case, the conclusion is .

Next up, . Following the above method, we partition into equal subintervals and look for sets, each of which consists of subintervals. Pairwise intersections must consist of subintervals each. Can this be done?

Think of a cube. It has vertices and faces, and . In the set {faces and vertices}, we choose subsets of cardinality as follows:

{face, all its vertices, and the opposite face}. There are 6 such sets. Any two of them have exactly 2 elements in common.

{all faces} is our 7th set, it has 2 elements in common with each of the above.

Thus, . But it is concerning that and required different combinatorial constructions. What happens next?

For , we partition into equal subintervals and look for sets, with subintervals in each. Pairwise intersections must consist of subintervals. I don’t see any neat construction here.

Summary so far

The results obtained so far for can be summarized as follows: there exists a 0-1 matrix of size in which the dot product of two rows is when the rows are distinct and otherwise. Specifically,

The matrix is best written in block form according to the faces-vertices interpretation. Here

and

Does exist? Perhaps not.

Combinatorics of finite sets

In combinatorial terms, we are looking for a -uniform -intersecting family of subsets of , with specific values

, , , .

Explanation of terms: , being -uniform means every set has elements, and being -intersecting means all pairwise intersections have elements.

Thus began my fishing expedition into the literature on combinatorics of finite sets. Surprisingly, it ended with a pivot back to where it started: measurable subsets of an interval.

Fractional intersecting uniform families

A recent (2018) preprint Uniform sets in a family with restricted intersections (Yandong Bai, Binlong Li, Jiuqiang Liu, Shenggui Zhang) relates the combinatorial problem back to subsets of an interval. I quote the relevant result (Theorem 2.5) with slight changes of notation.

Let be the real interval and let be the Lebesgue measure on . If is a family of subsets of such that

for all , and

for all distinct ,

then we call a fractional -intersecting -uniform -family of . For given real numbers and integer , let be the largest real number such that there exists a fractional -intersecting -uniform -family of .

Theorem 2.5 Let be two real numbers and let be an integer. Then

where ,

and is the solution of the system

.

This looks complicated, but with our values , , both and turn out to be equal to . This is an exceptional case when the linear system is degenerate: it amounts to . The final result is as we wanted. Yay.

So, the upper bounds for distances are indeed sharp, meaning that subsets of the unit interval can be separated by measure-distance of , where . But dividing the intervals into equal parts may not be enough to construct such subsets: in general one has to use a larger number of parts, some integer multiple of .

The authors remark that if are integers, then a partition with exists in the combinatorial realm of . However, this is not “if and only if”. Indeed, which takes integer values when , and is fractional (less than 1) when . But we do have a combinatorial solution when . So it is not quite clear for what values of there exists a -uniform -intersecting family of subsets of . Perhaps an answer could be found by following the construction for this specific case, when the formulas magically simplify.

(This applies more generally to holomorphic functions, but polynomials suffice for now.) The quantity makes sense for but is not actually a norm when .

When restricted to polynomials of degree , the Hardy norm provides a norm on , which we can try to visualize. In the special case the Hardy norm agrees with the Euclidean norm by Parseval’s theorem. But what is it for other values of ?

Since the space has real dimensions, it is hard to visualize unless is very small. When , the norm we get on is just the Euclidean norm regardless of . The first nontrivial case is . That is, we consider the norm

Since the integral on the right does not depend on the arguments of the complex numbers , we lose nothing by restricting attention to . (Note that this would not be the case for degrees .)

One easy case is since we can find for which the triangle inequality becomes an equality. For the values numerics will have to do: we can use them to plot the unit ball for each of these norms. Here it is for :

And :

And which is pretty close to the rotated square that we would get for .

In the opposite direction, brings a surprise: the Hardy -norm is strictly convex, unlike the usual -norm.

it’s a strange shape whose equation involves the complete elliptic integral of second kind. Yuck.

Well, that was 6 years ago; with time I grew to appreciate this shape and the elliptic integrals.

Meanwhile, surprises continue: when , the Hardy norm is an actual norm, with a convex unit ball.

Same holds for :

And even for :

And here are all these norms together: from the outside in, the values of are 0.01, 0.2, 0.5, 1, 2, 4, 10, 100. Larger makes a larger Hardy norm, hence a smaller unit ball: the opposite behavior to the usual -norms .

I think this is a pretty neat picture: although the shapes look vaguely like balls, the fact that the range of the exponent is instead of means there is something different in the behavior of these norms. For one thing, the Hardy norm with turns out to be almost isometrically dual to the Hardy norm with : this became the subject of a paper and then another one.

Adding an edge to a connected graph (between two of its existing vertices) makes the graph “even more” connected. How to quantify this? The Poincaré inequality gives a way to do this: for any function with zero mean,

where the sum on the left is over the vertex set , the sum on the right is over the edge set , and on the right are the endpoints of an edge. Suppose is the smallest possible constant in this inequality, the Poincaré constant of the graph. Adding an edge to the graph does not change the sum on the left but may increase the sum on the right. Thus, the new Poincaré constant satisfies : greater connectivity means smaller Poincaré constant.

It is more convenient to work with the reciprocal of , especially because , the smallest positive eigenvalue of the graph Laplacian. Let be the corresponding eigenvalue after an edge is added. The above shows that . But there is also an upper bound on how much this eigenvalue (or any Laplacian eigenvalue) can grow. Indeed, adding an edge amounts to adding the matrix

(with a bunch of zero rows/columns)

to the Laplacian matrix . The eigenvalues of are and . As a consequence of the Courant-Fischer minimax formulas, the ordered eigenvalues satisfy for every .

Both of the above bounds are sharp. If an diagonal edge is added to the 4-cycle,

the smallest positive eigenvalue remains the same (). The reason is that a typical eigenfunction for takes on values (in cyclic order) and adding an edge between two vertices with the same value of such eigenfunction does not affect the Poincaré constant.

On the other hand, adding one more edge increases from to .

Normalized Laplacian

Another form of the Poincaré inequality involves the vertex degree: when summing over vertices, we give each of them weight equal to the number of neighbors.

provided

Here , the smallest positive eigenvalue of the normalized Laplacian , the matrix with on the diagonal, where off-diagonal entries are if , and otherwise. The sum of the normalized Laplacian eigenvalues is equal to (the number of vertices), regardless of how many edges are there. So we cannot expect all them to grow when an edge is added. But how do they change?

Let be the corresponding eigenvalue after an edge is added.
Here are two examples for which :

Completing a 3-path to a 3-cycle increases the eigenvalue from 1 to 3/2.

Completing a 4-path to a 4-cycle increases the eigenvalue from 1/2 to 1.

This pattern does not continue: “5-path to 5-cycle” results in a smaller increase, which is not even largest among 5-vertex graphs. It appears that the above examples are the only two cases of , with all other graphs having . (I do not have a proof of this.)

The opposite direction

Adding an edge can also make smaller (thus, the weighted Poincaré constant becomes larger, showing that it may not be as useful for quantifying the connectivity of a graph). The smallest example is adding an edge to the star graph on 4 vertices:

here but .

Let us analyze the star graphs on vertices, for general . In an earlier post we saw that , so it remains to find . Let us order the vertices so that is the center of the star and is the added edge. Write for brevity. Then the normalized Laplacian matrix is

where all rows numbered consist of in the first column and in the th column. This shows that has rank , hence is an eigenvalue of multiplicity . Also, is a simple eigenvalue as for any connected graph. Less obviously, is an eigenvalue with the corresponding eigenvector – this eigenvalue comes from the submatrix in row/columns 2 and 3, where the surrounding values in these row/columns are nice enough to cancel out. We are left with two eigenvalues to find, and we know their sum is . This means the characteristic polynomial of is of the form

where the constant remains to be found. I am going to skip to the answer here: . The quadratic formula delivers the last two eigenvalues:

The smaller of these is that we were looking for:

as

Thus, adding an edge to a star graph results in negative which decreases to as . Exhaustive search for shows that the star graph has the smallest value of among all graphs on vertices. It is reasonable to expect this pattern to continue, which would imply in all cases.

provides reasonable approximations to the function near zero: for example, with we get

The quality of approximation not being spectacular, one can try to improve it by using rational functions instead of polynomials. In view of the identity

one can get with . The improvement is substantial for negative :

Having rational approximation of the form makes perfect sense, because such approximants obey the same functional equation as the exponential function itself. We cannot hope to satisfy other functional equations like by functions simpler than .

However, the polynomial is not optimal for approximation except for . For degree , the optimal choice is . In the same plot window as above, the graph of is indistinguishable from .

This is a Padé approximant to the exponential function. One way to obtain such approximants is to replace the Taylor series with continued fraction, using long division:

Terminating the expansion after an even number of apprearances gives a Padé approximant of the above form.

This can be compared to replacing the decimal expansion of number :

with the continued fraction expansion

which, besides greater accuracy, has a regular pattern: 121 141 161 181 …

The numerators of the (diagonal) Padé approximants to the exponential function happen to have a closed form:

which shows that for every fixed , the coefficient of converges to as . The latter is precisely the Taylor coefficient of .

In practice, a recurrence relation is probably the easiest way to get these numerators: begin with and , and after that use . This relation can be derived from the recurrence relations for the convergents of a generalized continued fraction . Namely, and . Only the first relation is actually needed here.

Using the recurrence relation, we get

(so, not all coefficients have numerator 1…)

The quality of approximation to is best seen on logarithmic scale: i.e., how close is to ? Here is this comparison using .

For comparison, the Taylor polynomial of fifth degree, also on logarithmic scale: where .

The standard normal probability density function has inflection points at where which is about 60.5% of the maximum of this function. For this, as for other bell-shaped curves, the inflection points are also the points of steepest incline.

This is good to know for drawing an accurate sketch of this function, but in general, the Gaussian curve may be scaled differently, like , and then the inflection points will be elsewhere. However, their relative height is invariant under scaling: it is always 60.5% of the maximum height of the curve. Since it is the height that we focus on, let us normalize various bell-shaped curves to have maximum :

So, the Gaussian curve is inflected at the relative height of . For the Cauchy density the inflection is noticeably higher, at of the maximum:

Another popular bell shape, hyperbolic secant or simply , is in between with inflection height . It is slightly unexpected to see an algebraic number arising from this transcendental function.

Can we get inflection height below ? One candidate is with large even , but this does not work: the relative height of inflection is . Shown here for :

However, increasing the power of in the Gaussian curve works: for example, has inflection at relative height :

More generally, the relative height of inflection for is for even . As , this approaches . Can we go lower?

Well, there are compactly supported bump functions which look bell-shaped, for example for . Normalizing the height to makes it . For this function inflection occurs at relative height about .

Once again, we can replace by an arbitrary positive even integer and get relative inflection height down to . As increases, this height decreases to where is the golden ratio. This is less than which is low enough for me today. The smallest for which the height is less than is : it achieves inflection at .

In the opposite direction, it is easy to produce bell-shaped curve with high inflection points: either or will do, where is slightly larger than . But these examples are only once differentiable, unlike the infinitely smooth examples above. Aside: as , the latter function converges to the (rescaled) density of the Laplace distribution and the former to a non-integrable function.

As for the middle between two extremes… I did not find a reasonable bell-shaped curve that inflects at exactly half of its maximal height. An artificial example is with but this is ugly and only smooth.

The algebraic connectivity of a graph is its smallest nontrivial Laplacian eigenvalue. Equivalently, it is the minimum of edge sums over all functions normalized by and . Here are vertex/edge sets, and means the difference of the values of at the vertices of the edge .

It is clear from the definition that adding edges to cannot make smaller; thus, among all graphs on vertices the maximal value of is attained by the complete graph, for which . For any other graph which can be shown as follows. Pick two non-adjacent vertices and and let , , and elsewhere. This function is normalized as required above, and its edge sum is at most since there are at most edges with a nonzero contribution.

What if we require the graph to have degree vertex at most , and look for maximal connectivity then? First of all, only connected graphs are under consideration, since for non-connected graphs. Also, only the cases are of interest, otherwise the complete graph wins. The argument in the previous paragraph shows that but is this bound attained?

The case is boring: the only two connected graphs are the path and the cycle . The cycle wins with versus .

When , one might suspect a pattern based on the following winners:

The structure of these two is the same: place points on a circle, connect each of them to nearest neighbors.

But this pattern does not continue: the 8-vertex winner is completely different.

This is simply the complete bipartite graph . And it makes sense that the “4 neighbors” graph loses when the number of vertices is large: there is too much “redundancy” among its edges, many of which connect the vertices that were already connected by short paths.

In general, when , the complete bipartite graph achieves and therefore maximizes the algebraic connectivity. The fact that follows by considering graph complement, as discussed in Laplacian spectrum of small graphs. The complement of is the disjoint union of two copies of the complete graph , for which the maximal eigenvalue is . Hence .

When is odd, we have a natural candidate in for which the argument from the previous paragraph shows . This is indeed a winner when :

The winner is not unique since one can add another edge between two of the vertices of degree 2. This does not change the , however: there is a fundamental eigenfunction that has equal values at the vertices of that added edge.

Same for : the complete bipartite graph shares the maximum value of algebraic connectivity with two other graphs formed by adding edges to it:

However, the family does not win the case in general: we already saw a 4-regular graph on 7 vertices with , beating . Perhaps wins when is odd?

I do not have any other patterns to conjecture, but here are two winners for : the cube and the “twisted cube”.

The cube is twisted by replacing a pair of edges on the top face with the diagonals. This is still a 3-regular graph and the algebraic connectivity stays the same, but it is no longer bipartite: a 5-cycle appears.

There is a useful sequential characterization of continuity in metric spaces. Let be a map between metric spaces. If for every convergent sequence in we have in , then is continuous. And the converse is true as well.

Uniformly continuous functions also have a useful property related to sequences: if is uniformly continuous and is a Cauchy sequence in , then is a Cauchy sequence in . However, this property does not characterize uniform continuity. For example, if , then Cauchy sequences are the same as convergent sequences, and therefore any continuous function preserves the Cauchy-ness of sequences—it does not have to be uniformly continuous.

Let us say that two sequences and are equivalent if the distance from to tends to zero. The sequential characterization of uniform continuity is: is uniformly continuous if and only if for any two equivalent sequences and in , their images and are equivalent in . The proof of this claim is straightforward.

In the special case when is a constant sequence, the sequential characterization of uniform continuity reduces to the sequential characterization of continuity.

A typical example of the use of this characterization is the proof that a continuous function on a compact set is uniformly continuous: pick two equivalent sequences with non-equivalent images, pass to suitable subsequences, get a contradiction with continuity.

Here is a different example. To state it, introduce the notation .

Removability Theorem. Let be continuous. Suppose that there exists such that for every , the restriction of to is uniformly continuous. Then is uniformly continuous on .

This is a removability result because from having a certain property on subsets of we get it on all of . To demonstrate its use, let with the standard metric, , and . The uniform continuity of on follows immediately from the derivative being bounded on that set (so, is Lipschitz continuous there). By the removability theorem, is uniformly continuous on .

Before proving the theorem, let us restate the sequential characterization in an equivalent form (up to passing to subsequences): is uniformly continuous if and only if for any two equivalent sequences and there exist equivalent subsequences and , with the same choice of indices in both.

Proof of the theorem. Suppose and are equivalent sequences in . If , then as well, and the continuity of at implies that both and converge to , hence are equivalent sequences. If , then by passing to a subsequence we can achieve for some constant . By the triangle inequality, for sufficiently large we have . Since is uniformly continuous on , it follows that and are equivalent.

Consider this linear differential equation: with boundary conditions and . Nothing looks particularly scary here. Just one nonconstant coefficient, and it’s a simple one. Entering this problem into Wolfram Alpha produces the following explicit solution:

I am not sure how anyone could use this formula for any purpose.

Let us see what simple linear algebra can do here. The differential equation can be discretized by placing, for example, equally spaces interior grid points on the interval: , . The yet-unknown values of at these points are denoted . Standard finite-difference formulas provide approximate values of and :

where is the step size, in our case. Stick all this into the equation: we get 4 linear equations, one for each interior point. Namely, at it is

(notice how the condition is used above), at it is

and so on. Clean up this system and put it in matrix form:

This isn’t too hard to solve even with pencil and paper. The solution is

It can be visualized by plotting 4 points :

Not particularly impressive is it? And why are all these negative y-values in a problem with boundary condition ? They do not really look like they want to approach at the left end of the interval. But let us go ahead and plot them together with the boundary conditions, using linear interpolation in between:

Or better, use cubic spline interpolation, which only adds another step of linear algebra (see Connecting dots naturally) to our computations.

This begins to look believable. For comparison, I used a heavier tool: BVP solver from SciPy. Its output is the red curve below.

Those four points we got from a 4-by-4 system, solvable by hand, pretty much tell the whole story. At any rate, they tell a better story than the explicit solution does.

Graphics made with: SciPy and Matplotlib using Google Colab.

A natural way to measure the nonlinearity of a function , where is an interval, is the quantity which expresses the deviation of from a line, divided by the size of interval . This quantity was considered in Measuring nonlinearity and reducing it.

Let us write where the supremum is taken over all intervals in the domain of definition of . What functions have finite ? Every Lipschitz function does, as was noted previously: . But the converse is not true: for example, is finite for the non-Lipschitz function , where .

The function looks nice, but is clearly unbounded. What makes finite? Note the scale-invariant feature of NL: for any the scaled function satisfies , and more precisely . On the other hand, our function has a curious scaling property where the linear term does not affect NL at all. This means that it suffices to bound for intervals of unit length. The plot of shows that not much deviation from the secant line happens on such intervals, so I will not bother with estimates.

The class of functions with is precisely the Zygmund class defined by the property with independent of . Indeed, since the second-order difference is unchanged by adding an affine function to , we can replace by with suitable and use the triangle inequality to obtain

where . Conversely, suppose that . Given an interval , subtract an affine function from to ensure . We may assume attains its maximum on at a point . Applying the definition of with and , we get , hence . This shows . The upshot is that is equivalent to the Zygmund seminorm of (i.e., the smallest possible M in the definition of ).

A function in may be nowhere differentiable: it is not difficult to construct so that is bounded between two positive constants. The situation is different for the small Zygmund class whose definition requires that as . A function is differentiable at any point of local extremum, since the condition forces its graph to be tangent to the horizontal line through the point of extremum. Given any two points we can subtract the secant line from and thus create a point of local extremum between and . It follows that is differentiable on a dense set of points.

The definitions of and apply equally well to complex-valued functions, or vector-valued functions. But there is a notable difference in the differentiability properties: a complex-valued function of class may be nowhere differentiable [Ullrich, 1993]. Put another way, two real-valued functions in need not have a common point of differentiability. This sort of thing does not often happen in analysis, where the existence of points of “good” behavior is usually based on the prevalence of such points in some sense, and therefore a finite collection of functions is expected to have common points of good behavior.

The key lemma in Ullrich’s paper provides a real-valued VMO function that has infinite limit at every point of a given set of measure zero. Although this is a result of real analysis, the proof is complex-analytic in nature and involves a conformal mapping. It would be interesting to see a “real” proof of this lemma. Since the antiderivative of a VMO function belongs to , the lemma yields a function that is not differentiable at any point of . Consider the lacunary series . One theorem of Zygmund shows that when , while another shows that is almost nowhere differentiable when . It remains to apply the lemma to get a function that is not differentiable at any point where is differentiable.