Asymptotically sharp inequalities for polynomials involving mixed Gegenbauer norms

1. Introduction and main result

In this work, we are concerned with the inequality (1) ‖ f ( ν ) ‖ β ⩽ C n ( ν ) ( α , β ) ‖ f ‖ α for all f ∈ P n , bounding the norm of the νth derivative of a polynomial from above in terms of the norm of the polynomial itself. Such an inequality is called a Markov-type inequality. Here and in the following, P n denotes the space of algebraic polynomials with complex coefficients of degree at most n, and ‖ · ‖ α is the Gegenbauer norm, given by (2) ‖ f ‖ α 2 = ∫ − 1 1 | f ( t ) | 2 ( 1 − t 2 ) α d t . The term C n ( ν ) ( α , β ) is a constant depending on n, ν, α, and β, but not on f. We want to find the smallest constant such that inequality (1) holds for every polynomial f of degree at most n. Let γ n ( ν ) ( α , β ) be this constant. If D ν : P n ( α ) → P n ( β ) denotes the operator that sends a polynomial of degree at most n to its νth derivative, where P n ( α ) and P n ( β ) are the spaces P n equipped with the norm ‖ · ‖ α and ‖ · ‖ β , respectively, we find that the constant is just the operator norm (spectral norm) of this operator. In general it is not possible to give exact expressions for this norm, but we can ask for the asymptotic behavior when n goes to infinity.

Inequalities of this type have first been studied by Andrei Andreevich Markov for the maximum norm on a bounded interval, in answer to a problem stated by the chemist Dimitri Ivanovich Mendeleev. The latter already solved it in the case n = 2 , ν = 1 . Markov then solved it for arbitrary degree of the polynomial (see, e.g., [2,3]), and found ‖ D f ‖ ∞ ⩽ n 2 ‖ f ‖ ∞ for all real f ∈ P n , where ‖ · ‖ ∞ denotes the maximum norm on [ − 1 , 1 ] . Now, one might get the impression that to find the constant for higher derivatives can be determined by repeatedly applying this inequality. This would yield ‖ D ν f ‖ ∞ ⩽ n 2 ( n − 1 ) 2 ⋯ ( n − ν + 1 ) 2 ‖ f ‖ ∞ for all real f ∈ P n . While this inequality is still correct, it does not provide the best possible, i.e., smallest, constant. Markov’s younger brother Vladimir took on to this problem and showed that ‖ D ν f ‖ ∞ ⩽ n 2 ( n 2 − 1 ) ( n 2 − 2 2 ) ⋯ ( n 2 − ( ν − 1 ) 2 ) ( 2 ν − 1 ) ! ! ‖ f ‖ ∞ for all f ∈ P n . This constant is best-possible with the Chebychev polynomials being extremal for the maximum norm.

The first to consider this type of inequality with the maximum norm replaced by a Hilbert space norm was Erhard Schmidt [13]. Among the Laguerre and Hermite norms, he considered the Legendre norm ‖ f ‖ 2 = ∫ − 1 1 | f ( t ) | 2 d t on both sides of the inequality, which is just a special form of the Gegenbauer case, with α = β = 0 . For ν = 1 he proved γ n ( 1 ) ( 0 , 0 ) ∼ 1 π n 2 . The expression a n ∼ b n means that the quotient a n b n converges to 1 as n goes to infinity.

Lawrence F. Shampine [14,15] later investigated second order derivatives for the Laguerre and Legendre norms. For the latter, he found γ n ( 2 ) ( 0 , 0 ) ∼ n 4 4 μ 0 2 , where μ 0 ≈ 1.8751041 is the smallest positive solution of the equation 1 + cos μ cosh μ = 0 .

Finally, Peter Dörfler [8,9] gave estimates in the Laguerre case. The approach both took was taking the matrix representation of the differential operator in a basis of orthonormal polynomials. The result was a special Toeplitz matrix, and one now had to find the spectral norm (which coincides with the operator norm) or at least give good estimates. In this way, they obtained the results mentioned. Almost 20 years nothing significant happened. Then, in 2008, Dörfler wrote a letter to Albrecht Böttcher and asked him how one could determine the spectral norm of the occurring Toeplitz matrices and thus make some progress after such a long time of stagnation. Böttcher immediately saw that the problem can be solved by using an old (and then already forgotten) trick used by Harold Widom in the 1960s in another context [17,18]. The idea is to consider the integral operator K n on L 2 ( 0 , 1 ) with the piecewise constant kernel k n ( x , y ) = a ⌊ n x ⌋ , ⌊ n y ⌋ for an ( n × n ) -matrix A n = ( a j k ) j , k = 0 n − 1 . Then, the identity ‖ A n ‖ ∞ = n ‖ K n ‖ ∞ holds. Here and in what follows, we denote by ‖ A ‖ ∞ the spectral norm if A is a matrix, and the operator norm if A is an operator. If one can show that after appropriate scaling the operators K n converge uniformly to some operator K, i.e., ‖ n − μ K n − K ‖ ∞ → 0 for n → ∞ , then one has ‖ A n ‖ ∞ ∼ ‖ K ‖ ∞ n μ + 1 . Using this, Böttcher and Dörfler settled not only the problem for the Laguerre case but also for the Gegenbauer case. In fact, they later discovered that Shampine [14,15] also made use of the trick consisting in passing from matrices to integral operators. However, the limitation of Shampine’s original approach was that he considered the operator ( D ν ) ∗ D ν , which gets very complex for higher values of ν. But, as it was shown by Böttcher and Dörfler, one can go a lot further and substitute the norms by their weighted analogues. Along with the similar result for the Laguerre case, for the Gegenbauer case they found γ n ( ν ) ( α , α ) ∼ ‖ G ν , α , α ‖ ∞ n 2 ν , where G ν , α , α is the integral operator on L 2 ( 0 , 1 ) given by ( G ν , α , α f ) ( x ) = 1 2 ν − 1 Γ ( ν ) ∫ x 1 x 1 / 2 + α y 1 / 2 − α ( y 2 − x 2 ) ν − 1 f ( y ) d y . First and foremost, this proved that the limit γ n ( ν ) ( α , α ) n − 2 ν exists as n → ∞ . This was previously not even known for α = 0 .

In 2009, Jürgen Prestin brought the attention of Böttcher and Dörfler to the problem of using two different weights in the inequality. This concerns two different norms, and as changes of norms may improve error estimates, this could be applied in approximation theory and numerical analysis. A very simple case is β = α + ν . Then, the matrix representation consists of a single diagonal. Therefore, γ n ( ν ) ( α , α + ν ) = n ! ( n − ν ) ! Γ ( n + 2 α + ν + 1 ) Γ ( n + 2 α + 1 ) ∼ n ν . This result has been established by Guessab and Milovanović, see [10].

After handling the problem for the Laguerre case [4], in [6], Böttcher and Dörfler treated this for the Gegenbauer case. First, they found that if β − α < ν − 1 / 2 , γ n ( ν ) ( α , β ) ∼ 1 2 ν + α − β ‖ L ν , α , β ∗ ‖ ∞ n 2 ν + α − β , where L ν , α , β ∗ is the integral operator on L 2 ( 0 , 1 ) given by ( L ν , α , β ∗ f ) ( x ) = 1 Γ ( ν − β + α ) ∫ 0 x x − α / 2 y β / 2 ( x − y ) ν − β + α − 1 f ( y ) d y . One might ask why a different operator comes into play. Actually, the original integral operator appearing here is much more closely related to the operator G ν , α , α from above. But since the operator 2 β − α − ν L ν , α , β and the corresponding G ν , α , β are unitarily equivalent and good estimates for the norm of the operator L ν , α , β were already present, it is convenient to work with this operator instead. Additionally, it shows how closely related these two seemingly different problems in fact are.

Of course, the problem of finding the norm of the matrix has now just been transfered to finding the norm of an integral operator, and usually one goes the other direction to determine the norm of an operator. However, good bounds are known for this operator. In particular, if β − α = ν − 1 , the operator takes on a much simpler form. Then, it is known from [5] that the norm of this operator is just 2 / ( ν + 1 ) times the inverse of the smallest positive zero of the Bessel function J ( α − 1 ) / ( ν + 1 ) . Since for the very special values υ = − 1 / 2 and υ = 1 / 2 , the Bessel functions J υ take the elementary form J − 1 / 2 ( x ) = 2 π x cos x , J 1 / 2 ( x ) = 2 π x sin x , this implies in particular γ n ( 1 ) ( 0 , 0 ) ∼ 1 π n 2 , γ n ( 1 ) ( 2 , 2 ) ∼ 1 2 π n 2 , γ n ( 2 ) ( − 1 2 , 1 2 ) ∼ 2 3 π n 3 , γ n ( 2 ) ( 5 2 , 7 2 ) ∼ 1 3 π n 3 , γ n ( 3 ) ( 3 , 5 ) ∼ 1 4 π n 4 , γ n ( 4 ) ( 7 2 , 13 2 ) ∼ 1 5 π n 5 . These examples are taken from [6].

One thing that will become more clear later, is that the methods for determining the best constant in inequality (1) vary tremendously depending on the relation of the occurring parameters, ω : = β − α − ν . First, if ω < 0 , the method indicated above can be used. For ω < − 1 / 2 this has been done by Böttcher and Dörfler. Note that this restriction comes from the method used and is not inherent to the problem. Since the convergence of a sequence of integral operators has to be shown in the operator norm, this is in general really hard, if at all possible in a direct way. However, with this restriction, the operator is a Hilbert-Schmidt operator and it suffices to show the convergence in the Hilbert-Schmidt norm, which is much more easy to achieve. Since the convergence in the Hilbert-Schmidt norm implies the convergence in the operator norm, the result follows. A detailed proof of this convergence can be found in [12]. In [7] it was shown that the integral operator in question, under less stringent restrictions, still belongs to some Schatten class, raising the hopes that the convergence can also be shown for − 1 / 2 ⩽ ω < 0 .

The second case is that for ω being a non-negative integer, the investigated matrix has a nice banded structure. Hence, it is relatively simple to estimate the norm. This has already been done by Böttcher and Dörfler in [6], too. Apart from the small gap mentioned before, the last case is for ω > 0 and not an integer. This is what we will tackle here. Therefore, we can present our main result of this paper.

3. An upper bound on the norm of the matrix

We are now ready to prove our claim. First, we want to recall the result given by Böttcher and Dörfler in [6]. Assume that β = α + m with some non-negative integer m ⩾ ν . Then, making use of the fact that the matrix is now banded with bandwidth independent of n, they could derive an upper bound by writing the matrix as a sum of diagonal matrices. The norm of each diagonal is just the maximum element on each diagonal which can be estimated from above by n ν m − ν ℓ 1 2 m − ν ( 1 + O ( 1 / n ) ) for the ν + 2 ℓ th diagonal. Summing this up leads to an upper bound on the norm of n ν ( 1 + O ( 1 / n ) ) . In order to show a lower bound, they proved that the matrix scaled by n − ν converges entrywise to the adjoint of a Toeplitz matrix with the symbol 2 − ( m − ν ) ( 1 − z 2 ) m − ν . Due to the special structure of the matrix, the convergence in the norm follows. From the Banach–Steinhaus theorem, we get that the limit inferior of the spectral norm of the scaled down matrix is not smaller than the spectral norm of this Toeplitz matrix. Knowing that ‖ T ∗ ( 2 − ( m − ν ) ( 1 − z 2 ) m − ν ) ‖ ∞ = max | z | = 1 | 2 − ( m − ν ) ( 1 − z 2 ) m − ν | = 2 − ( m − ν ) 2 m − ν = 1 , they conclude that γ n ( ν ) ( α , α + m ) ∼ n ν .

Looking at this result, we observe that the constant is independent on α and β, or the difference, provided this difference is non-negative. Now one might guess that the constant stays the same for the non-integral cases in between. This is indeed true, and we are going to show this by an interpolation result, going back to Stein [16], at least for the upper bound.

OPEN IN VIEWER

Fig. 1.

The possible parameter set for α , β > − 1 , β − α ⩾ 0 .

To apply this interpolation theorem, we first have to take a closer look at the norms in question and how they relate to each other for different choices of parameters. For α > − 1 , let us introduce the weight function u ( t , α ) : = ( 1 − t 2 ) α / 2 . Now we can write the norm (2) as ‖ f ‖ α 2 = ∫ − 1 1 | f ( t ) | 2 u 2 ( t , α ) d t = ∫ − 1 1 | f ( t ) u ( t , α ) | 2 d t = ‖ f u ( · , α ) ‖ L 2 ( − 1 , 1 ) 2 , where ‖ · ‖ L 2 ( − 1 , 1 ) denotes the usual L 2 -norm on ( − 1 , 1 ) . Thus, we can modify the parameter without actually using another norm. Before we utilize this, we want to illustrate what we are trying to achieve. For α , β > − 1 and β − α ⩾ 0 the set of possible parameters is indicated in Fig. 1. The diagonal lines indicate the pairs ( α , β ) that have an integral difference. As it becomes immediately clear from the sketch is that for each pair ( α , β ) with non-integral difference, satisfying the conditions mentioned above, there exist pairs ( α , β 0 ) and ( α , β 1 ) with β 0 − α ∈ N 0 and β 1 − α ∈ N 0 , as well as β ∈ ( β 0 , β 1 ) , α > − 1 . But for such differences we already have good upper estimates. This is where the interpolation theorem of Stein comes into play. However, there is a small catch. The original version of Stein is formulated for infinite dimensional operators only, whereas the differential operator is finite dimensional on P n . But to dispel any doubts, we give the following lemma which brings things to the finite dimensional case. The underlying idea was already used in [11] to show the corresponding estimate in the Laguerre case. In [12] it was then used in a more general form. Although we presently do not really need it, it can be made even more universal, which could be of further use for future problems.

We note that we could even allow a larger class of weight functions on the right side of the inequality, i.e., not necessarily of the same type as the weight function on the left-hand side, but stick to this type for reasons of readability. Also, we are not restricted to the subclass of polynomials of degree at most n but could choose any finite dimensional subspace of the corresponding L 2 .

By a simple calculation we verify that with Γ = ( − 1 , ∞ ) and A = D ν being the operator that maps a polynomial of degree at most n to its νth derivative, and ( κ n ) n being the sequence of non-negative integers, both u ( t , α ) = ( 1 − t 2 ) α / 2 and the constant C n ( α , β ′ ) = γ n ( ν ) ( α , β ′ ) for β ′ with β ′ − α ∈ N 0 satisfy the conditions of the lemma. Thus, γ n ( ν ) ( α , β ) ⩽ n ν ( 1 + O ( 1 / n ) ) for all α , β > − 1 , β − α ⩾ ν , as claimed.

Finding a lower bound is a little more involved. Unlike in the case of integral differences, the matrix under consideration is not banded anymore but a full upper triangle. Hence, some shortcuts that could be made then cannot be done here. In order to give a lower bound anyway, we reuse a trick that was applied for the Laguerre case [11]. The main idea is the following. Given some vector of norm one, the norm of the matrix applied to this vector always gives a lower bound on the spectral norm of the matrix itself. Our job now is to find a vector in such a way that the norm gets as big as possible while retaining the ability to handle it in a reasonable way.

From (3) we see that the matrix has a chessboard structure. We may assume, without loss of generality, that N = n − ν + 1 is an even number, since the constants are monotonously increasing, i.e., γ n − 1 ( ν ) ( α , β ) ⩽ γ n ( ν ) ( α , β ) ⩽ γ n + 1 ( ν ) ( α , β ) . Then, there is a permutation matrix U n such that A n = U n E n 0 0 F n U n , where A n denotes the upper non-null block of C n , E n = ( e j k ) j , k = 0 N / 2 − 1 , and F n = ( f j k ) j , k = 0 N / 2 − 1 with e j k = c 2 j , 2 k + ν ( ν ) ( α , β ) , f j k = c 2 j + 1 , 2 k + ν + 1 ( ν ) ( α , β ) . Clearly, ‖ C n ‖ ∞ = ‖ A n ‖ ∞ = max { ‖ E n ‖ ∞ , ‖ F n ‖ ∞ } . Setting ω = β − α − ν , e j k takes the form e j k = 2 − ω Γ ( 2 k + ν + 1 ) ( 2 k + α + ν + 1 / 2 ) Γ ( 2 k + 2 α + ν + 1 ) Γ ( 2 j + 2 β + 1 ) ( 2 j + β + 1 / 2 ) Γ ( 2 j + 1 ) × Γ ( α + ν + k + j + 1 / 2 ) Γ ( β + 1 + k + j + 1 / 2 ) ( − 1 ) k − j ω k − j .

Taking a closer look at these entries, we observe that the biggest contribution to the spectral norm comes from the entries closer to the main diagonal and the more off-diagonal the elements are, the more negligible they become. Furthermore, we have Γ ( α + ν + k + j + 1 + 1 / 2 ) Γ ( β + 1 + k + j + 1 + 1 / 2 ) = α + ν + k + j + 1 / 2 ω + 1 + α + ν + k + j + 1 / 2 Γ ( α + ν + k + j + 1 / 2 ) Γ ( β + 1 + k + j + 1 / 2 ) . Hence, these terms are always decreasing for growing k or j, independently of α and β, given that β − α ⩾ ν . Therefore, we estimate each of these occurrences from below by Γ ( α + ν + N − 3 / 2 ) Γ ( β + 1 + N − 3 / 2 ) . Similarly to the way this was done in [11] for the Laguerre case, we define the vectors v + = ( v j + ) j = 0 N / 2 − 1 , v − = ( v j − ) j = 0 N / 2 for α ⩾ 1 / 2 and α < 1 / 2 , respectively, as follows: v j + = ( − 1 ) j ( N + α + ν − 3 / 2 ) ( 2 j + α + ν + 1 / 2 ) × ∏ ℓ = j + 1 N / 2 − 1 ( 2 ℓ + ν − 1 ) ( 2 ℓ + ν ) ( 2 ℓ + 2 α + ν − 1 ) ( 2 ℓ + 2 α + ν ) : j ⩾ N / 2 − μ 0 : otherwise v j − = ( − 1 ) j ( N − 2 μ + α + ν + 1 / 2 ) ( 2 j + α + ν + 1 / 2 ) × ∏ ℓ = N / 2 − μ + 1 j ( 2 ℓ + 2 α + ν − 1 ) ( 2 ℓ + 2 α + ν ) ( 2 ℓ + ν − 1 ) ( 2 ℓ + ν ) : j ⩾ N / 2 − μ 0 : otherwise with some μ : = μ ( N ) ∈ N , μ ≪ N / 2 − 1 . In the following, we will only work with v + , and note that the main steps of the proof can be applied to the case where we need v − in an analogous fashion.

We can write the first factor in v j + as N + α + ν − 3 / 2 2 j + α + ν + 1 / 2 = ∏ ℓ = j + 1 N / 2 − 1 2 ℓ + α + ν + 1 / 2 2 ℓ + α + ν − 3 / 2 . Joining the products and putting α = δ + 1 / 2 , δ > − 3 / 2 into the factors in the entries of v + , the term under the square root becomes 2 ℓ + ν − 1 2 ℓ + ν − 1 + δ · 2 ℓ + ν 2 ℓ + ν + 2 δ · 2 ℓ + ν + 1 2 ℓ + ν + 1 + 2 δ . Each fraction is smaller than one for δ ⩾ 0 , i.e., for α ⩾ 1 / 2 , and strictly larger than one for δ < 0 , i.e., for α < 1 / 2 . Thus, for the values of α where these vectors are applied, the estimates ‖ v + ‖ 2 2 ⩽ μ and ‖ v − ‖ 2 2 ⩽ μ hold.

There are two important reasons for the choice of these vectors. First, it is simple enough so that we can get a good upper estimate on the norm of the vector. Secondly, we can eliminate all terms from the sum that contain only the index k and have no dependence on j. This will come in handy when we are going to sum over the entries. The term under the first square root of e j k multiplied with the corresponding entry v k + then becomes Γ ( 2 k + ν + 1 ) ( 2 k + α + ν + 1 / 2 ) Γ ( 2 k + 2 α + ν + 1 ) N + α + ν − 3 / 2 2 k + α + ν + 1 / 2 × ∏ ℓ = k + 1 N / 2 − 1 ( 2 ℓ + ν − 1 ) ( 2 ℓ + ν ) ( 2 ℓ + 2 α + ν − 1 ) ( 2 ℓ + 2 α + ν ) = ( N + α + ν − 3 / 2 ) Γ ( 2 k + ν + 1 ) ( 2 k + ν + 1 ) ⋯ ( N + ν − 2 ) Γ ( 2 k + 2 α + ν + 1 ) ( 2 k + 2 α + ν + 1 ) ⋯ ( N + 2 α + ν − 2 ) = ( N + α + ν − 3 / 2 ) Γ ( N + ν − 1 ) Γ ( N + 2 α + ν − 1 ) . Therefore, for α ⩾ 1 / 2 , we get the following first estimate on the norm of E n : ‖ E n ‖ ∞ 2 ⩾ ‖ E n v + ‖ 2 2 ‖ v + ‖ 2 2 ⩾ 2 − 2 ω Γ ( N + ν − 1 ) ( N + α + ν − 3 / 2 ) Γ ( N + 2 α + ν − 1 ) Γ 2 ( α + ν + N − 3 / 2 ) Γ 2 ( β + 1 + N − 3 / 2 ) × μ − 1 ∑ j = 0 N / 2 − 1 Γ ( 2 j + 2 β + 1 ) ( 2 j + β + 1 / 2 ) Γ ( 2 j + 1 ) ( ∑ ℓ = max { 0 , N / 2 − j − μ } N / 2 − 1 − j ω ℓ ) 2 .

Now, the next step is to evaluate the remaining sum. First, we notice two things. The inner sum is almost the Taylor sum for the function ( 1 + z ) ω , evaluated at z = 1 . However, it is cut of at some point and does in general not run from ℓ = 0 . At some point, this sum begins to alternate, thus making it hard to get a simple lower estimate. Next, looking at the term Γ ( 2 j + 2 β + 1 ) ( 2 j + β + 1 / 2 ) Γ ( 2 j + 1 ) = 2 j + 2 β 2 j · 2 j + 2 β − 1 2 j − 1 · 2 j + β + 1 / 2 2 j + β − 3 / 2 Γ ( 2 j + 2 β − 1 ) ( 2 j + β − 3 / 2 ) Γ ( 2 j − 1 ) , we notice that it is increasing due to our assumption β − α > ν , and thus β > ν + α > ν − 1 ⩾ 0 . So, to get a lower estimate, we could just take the minimum of all these terms, which is attained for j = 0 . But again, this would be to rough and would thus not lead to the desired expression. We still can manage this but have to work a little harder. In order to achieve this, we again follow an idea first employed for the Laguerre case in [11].

Taking a close look at the inner sum, we notice that, for small j, each of the remaining summands ω ℓ has a very small absolute value that is rapidly decreasing at least for ℓ ⩾ ⌈ ω ⌉ . Since we take only the squared value of the sum, this expression is positive. Therefore, we could just ignore these terms to get a lower estimate. We can even go so far that the sum always starts at ℓ = 0 then. So the whole sum now runs from j = N / 2 − μ . Now we can go back to our earlier estimate on the quotient. Now the minimum over all these terms is attained for j = N / 2 − μ , and thus evaluates to Γ ( N − 2 μ + 2 β + 1 ) ( N − 2 μ + β + 1 / 2 ) Γ ( N − 2 μ + 1 ) .

It remains to show that the inner sum can be estimated in a nice way. Here, we just give the general idea and refer to [11] or [12], where this is explained in more detail. When there are enough summands, i.e., at least ⌈ ω ⌉ + 1 , the last term in each sum is either positive or negative. By allowing additional terms of the same type and doing so in pairs until infinity, the sums where the last term was positive are decreasing, while the sums where the last term was negative are increasing. This is due to the fact that the sign of these terms alternates and the absolute value is decreasing. Now that the sums run to infinity, their value is just 2 ω . The main problem here is that, in order to show a lower bound, this only works for the first case and not for the second. As was shown in [11], we can overcome this defect by adding all the sums which have at most ⌈ ω ⌉ summands. They are big enough such that in sum we still can make the lower estimate in this case. For each case there are ⌊ ( μ − ⌈ ω ⌉ ) / 2 ⌋ terms of this kind. Therefore, this part is not smaller than 2 · ⌊ μ − ⌈ ω ⌉ 2 ⌋ · 2 2 ω .

Putting all of the above together, we now arrive at the estimate ‖ E n ‖ ∞ 2 ⩾ 2 − 2 ω Γ ( N + ν − 1 ) ( N + α + ν − 3 / 2 ) Γ ( N + 2 α + ν − 1 ) · Γ 2 ( α + ν + N − 3 / 2 ) Γ 2 ( β + 1 + N − 3 / 2 ) × μ − 1 · 2 · ⌊ μ − ⌈ ω ⌉ 2 ⌋ Γ ( N − 2 μ + 2 β + 1 ) ( N − 2 μ + β + 1 / 2 ) Γ ( N − 2 μ + 1 ) · 2 2 ω .

The last step, and the main reason for introducing the variable μ, is that we let μ tend to infinity, but in a lower order than N, in such a way that 2 ⌊ μ − ⌈ ω ⌉ 2 ⌋ ∼ μ and N − 2 μ ∼ N . Following this, we get an asymptotic lower bound of N ν for the norm of the matrix E n . For F n , the same approach delivers the same bound. Consequently, considered asymptotically, the upper and lower bounds for the norm of C n are the same, and thus γ n ( ν ) ( α , β ) ∼ n ν , whenever β − α − ν ⩾ 0 also if β − α is not an integer.