Open Access
Review
Issue
4open
Volume 3, 2020
Article Number 2
Number of page(s) 19
Section Mathematics - Applied Mathematics
DOI https://doi.org/10.1051/fopen/2020001
Published online 27 April 2020

© P.E. Ricci, Published by EDP Sciences, 2020

Licence Creative Commons
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Introduction

In preceding articles, starting from Bernoulli’s spiral, in the complex form, we have highlithed the obvious connection between the first and second kind Chebyshev polynomials, and the Grandi (Rodhonea) curves.

As these curves exist even for rational index values, we have extended these Chebyshev polynomials to the case of fractional degree [1]. The resulting functions are no more polynomials, but irrational functions that have been called pseudo-Chebyshev functions, because they satisfy many properties of the corresponding Chebyshev polynomials. Actually the irrationality of these functions is due to a constant factor which is multiplied by polynomials.

In subsequent articles [2, 3], recalling the third and fourth kind Chebyshev polynomials, and their connections to the preceding ones, as it is presented in [4], we have also introduced, for completeness, the corresponding pseudo-Chebyshev functions of the third and fourth kind. It has been shown that these families of functions satisfy the recursion and differential equations of the classical Chebyshev polynomials up to the change of the degree (from the integer to the fractional one). The particular case which seems to be the most important one, is when the degree is of the type k + 1/2, that is a half-integer, since in this case, the pseudo-Chebyshev functions satisfy even the orthogonality properties, in the interval (−1, 1), with respect to the same weights of the classical polynomials.

Archimedes vs. Bernoulli spiral

Spirals are described by polar equations and, in a recent work [1], has been highlithed the connection of the Bernoulli spiral, in complex form, with the Grandi (Rhodonea) curves and the Chebyshev polynomials.

The Archimedes spiral [5] (Fig. 1) has the polar equation:(1)

thumbnail Figure 1

Archimedes vs. Bernoulli spiral.

If θ > 0 the spiral turns counter-clockwise, if θ < 0 the spiral turns clockwise. Bernoulli’s (logarithmic) spiral [6] (Fig. 1) has the polar equation(2)

By changing the parameters a and b one gets different types of spirals.

The size of the spiral is determined by a, while the verse of rotation and how it is “narrow” depend on b.

Being a and b positive costants, there are some interesting cases. The most popular logarithmic spiral is the harmonic spiral, also called Fibonacci spiral, in which the distance between the spires is in harmonic progression, with ratio , that is, the “Golden ratio”.

The logarithmic spiral was discovered by René Descartes in 1638, and studied by Jakob Bernoulli (1654–1705).

It is a first example of a fractal. On J. Bernoulli’s tomb there is the sentence: Eadem Mutata Resurgo.

In what follows, we consider a “canonical form” of the Bernoulli spirals assuming a = 1, b = en, that is the simplified polar equation:(3)

The complex Bernoulli spiral

In the complex case, putting(4)and considering the Bernoulli spiral:(5)we find:(6)

Recalling Chebyshev polynomials

Starting from the equations:(7)separating the real from the imaginary part, and putting x = cos t, the first and second kind Chebyshev polynomials are derived:(8)(9)

It is well known that the first kind Chebyshev polynomials [7] play an important role in Approximation theory, since their zeros constitute the nodes of optimal interpolation, because their choice minimizes the error of interpolation. They’re also the optimal nodes for the Gaussian quadrature rules.

The second kind Chebyshev polynomials can be used for representing the powers of a 2 × 2 non singular matrix [8]. Extension of this polynomial falmily to the multivariate case has been considered for the powers of a r × r (r ≥ 3) non singular matrix (see [9, 10]).

Remark 1

Chebyshev polynomials are a particular case of the Jacobi polynomials [11], which are orthogonal in the interval [−1, 1] with respect to the weight . More precisely, the following equations hold:Therefore, properties of the Chebyshev polynomials could be deduced in the framework of hypergeometric functions. However, in this more general approach, the connection with trigonomeric functions disappears.

Remark 2

In connection with interpolation and quadrature problems, two other types of Chebyshev polynomials have been considered. They correspond to different choices of weights:

These were called by Gautschi [12] the third and fourth kind Chebyshev polynomials, and have been used in Numerical Analysis, mainly in the framework of quadrature rules.

Basic properties of the Chebyshev polynomials of third and fourth kind

The third and fourth kind Chebyshev polynomials have been used by many scholars (see e.g. [4, 13]), because they are useful in particular quadrature rules, when singularities of the integrating function occur only at one end of the considered interval, that is (+1 or −1) (see [14, 15]). They have been shown even to be useful for solving high odd-order boundary value problems with homogeneous or nonhomogeneous boundary conditions [13].

The third and fourth kind Chebyshev polynomials are defined in as follows:(10)

Since , as it can be see by their graphs (Figs. 2 and 3), the third and fourth kind Chebyshev polynomials are essentially the same polynomial set, but interchanging the ends of the interval .

thumbnail Figure 2

Vk (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue.

thumbnail Figure 3

Wk (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue.

The orthogonality properties hold [4]:(δ is the Kronecher delta).

One of the explicit advantages of Chebyshev polynomials of third and fourth kind is to estimate some definite integrals aswith the precision degree 2n − 1, by using the n interpolatory points , (k = 1, 2, …, n), in the interval [−1, 1] [13, 14].

The Grandi (Rhodonea) curves

The curves with polar equation:(11)are known as Rhodonea or Grandi curves, in honour of G. Guido Grandi who communicated his discovery to G.W. Leibniz in 1713.

Curves with polar equation: are equivalent to the preceding ones, up to a rotation of radians.

The Grandi roses display

  • n petals, if is odd,

  • 2n petals, if is even.

By using equation (11) it is impossible to obtain, roses with () petals.

Roses with petals can be obtained by using the Bernoulli Lemniscate and its extensions. More precisely,

  • for n = 0, a two petals rose comes from the equation (the Bernoulli Lemniscate),

  • for n ≥ 1, a 4n + 2 petals rose comes from the equation .

A few graphs of Rhodonea curves are shown in Figures 47.

thumbnail Figure 4

Rhodonea

thumbnail Figure 5

Rhodonea cos(2θ) and cos(3θ).

thumbnail Figure 6

Rodoneas and .

thumbnail Figure 7

Rodoneas and .

Pseudo-Chebyshev functions of half-integer degree

In what follows, we consider the case of the half-integer degree, which seems to be the most interesting one, since the resulting pseudo-Chebyshev functions satisfy the orthogonality properties in the interval (−1, 1) with respect to the same weights of the corresponding Chebyshev polynomials.

Definition 1

Let, for any integer :(12)

Note that the above definition holds even for k + 1/2 < 0, taking into account the parity properties of the circular functions.

Recursion for the and

Put for shortness: and . Then, by using definition (12) and the addition formulas for the sine and cosine functions we find:and therefore:that is the same recursion of the classical Chebyshev polynomials:(13)

Furthermore, the initial conditions for the functions are:(14)

Interchanging the roles of and we find that the same recurrence for second sequence holds, but with different initial conditions. We have, precisely:(15)

Differential equations for the and

Even the differential equations satified by the and can be recovered by using a coupled technique. In fact, differentating and , we find:and differentiating again the first equation:that is:

Substituting , we find the differential equation satisfied by the :(16)and interchanging the roles of and we find that the same differential equation is satisfied by the .

Remark 3

The first few are:

Remark 4

The first few are:

The case of the and

Recalling equation (12), sinceandit results that the functions and satisfy the same recursion (13) of the classical Chebyshev polynomials, but with the initial conditions:(17)and(18)

Differential equations for the and

Put for shortness: and . Differentiating the second equation (12), we find:that isand differentiating again, it results:that is(19)and interchanging the roles of and we find that the same differential equation is satisfied by the .

Remark 5

The first few are:and, in general:

Remark 6

The first few are:and, in general:

The pseudo-Chebyshev functions , , and can be represented, in terms of the third and fourth kind Chebyshev polynomials as follows:(20)

In the considered case of half-integer degree, the pseudo-Chebyshev functions satisfy not only the recurrence relations and differential equations analogues to the classical ones, but even the orthogonality properties.

Orthogonality properties of the and functions

A few graphs of the functions are shown in Figure 8.

thumbnail Figure 8

Tk+1/2(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange.

Theorem 1

The pseudo-Chebyshev functions satisfy the orthogonality property:(21)where (22)

A few graphs of the functions are shown in Figure 9.

thumbnail Figure 9

Uk+1/2(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange.

Theorem 2

The pseudo-Chebyshev functions satisfy the orthogonality property:(23)where (24)

Proof. We prove only Theorem 12, since the proof of Theorem 13 is similar.

From the Werner formulas, we have:and

The third and fourth kind pseudo-Chebyshev functions

The results of this section are based on the excellent survey by Aghigh et al. [4]. By using that article, it is possible to derive, in an almost trivial way, the links among the pseudo-Chebyshev functions and the third and fourth kind Chebyshev polynomials.

We recall here only the principal properties, without proofs. Proofs and other properties are reported in reference [3]. In Figures 10 and 11, we show the graphs of the first few third and fourth kind pseudo-Chebyshev functions.

thumbnail Figure 10

Vk+1/2(x), k = 1, 2, 3, 4, 5. (1) Grey; (2) red; (3) blue; (4) orange; (5) violet.

thumbnail Figure 11

Wk+1/2(x), k = 1, 2, 3, 4, 5. (1) Red; (2) blue; (3) orange; (4) violet; (5) grey.

Orthogonality properties of the and functions

A few graphs of the functions are shown in Figure 10.

Theorem 3

The pseudo-Chebyshev functions verify the orthogonality property:(25)(26)

A few graphs of the functions are shown in Figure 11.

Theorem 4

The pseudo-Chebyshev functions verify the orthogonality property:(27)(28)

Explicit forms

Theorem 5

It is possible to represent explicitly the pseudo-Chebyshev functions as follows: (29)

Location of zeros

By equation (12), the zeros of the pseudo-Chebyshev functions and are given by(30)and the zeros of the pseudo-Chebyshev functions and are given by(31)furthermore, the functions always vanish at the end of the interval [−1, 1].

Remark 7

More technical properties as the Hypergeometric representations and the Rodrigues-type formulas are reported in [3].

Links with first and second kind Chebyshev polynomials

Theorem 6

The pseudo-Chebyshev functions are connected with the first and second kind Chebyshev polynomials by means of the equations: (32)

Proof. The results follow from the equations:(33)(see [4]), by using definition (12).

Remark 8

Note that the first equation, in (29), extends the known nesting property verified by the first kind Chebyshev polynomials:(34)

This property, already considered in [16] for the first kind pseudo-Chebyshev functions, actually holds in general, as a consequence of the definition Tk(x) = cos(k arccos(x)). Note that this composition identity even holds for the first kind Chebyshev polynomials in several variables [10], as it was proven in [17].

Pseudo-Chebyshev functions with general rational indexes

Basic properties of the pseudo-Chebyshev functions of the first and second kind

We put, by definition:(35)(36)where p and q are integer numbers, ().

Note that definitions (35) and (36) hold even for negative indexes, that is for p/q < 0, according to the parity properties of the trigonometric functions.

The following theorems hold:

Theorem 7

The pseudo-Chebyshev functions satisfy the recurrence relation(37)

Proof. Write equation (37) in the form:then use definition (35) and the trigonometric identity:

Theorem 8

The first kind pseudo-Chebyshev functions satisfy the differential equation:(38)

Proof. Note that

so that equation (38) follows.

The following theorems hold:

Theorem 9

The pseudo-Chebyshev functions satisfy the recurrence relation(39)

Proof. Write equation (39) in the form:then use definition (36) and the trigonometric identity:

Theorem 10

The pseudo-Chebyshev functions satisfy the differential equation(40)

Proof. Differentiating, and differentiating again equation (36) we find subsequently:so that equation (40) follows.

Basic properties of the pseudo-Chebyshev functions of third and fourth kind

According to equation (10), put by definition:(41)

Theorem 11

The third and fourth kind pseudo-Chebyshev functions are related to the first and second kind ones by the equations: (42)

Proof. It is sufficient to use the addition formulas for the cosine and sine functions.

Therefore, we can derive the equations:(43)

Some general formulas

Putting:(44)and using the cosine or sine addition formulas, we find:(45)(46)

Particular results

(47) (48) (49) (50) (51) (52)

Combining the above equations, we find:(53)

Links with the pseudo-Chebyshev functions

The third and fourth kind Chebyshev polynomials are defined as follows:(54)(55)

Therefore, we find the equations:(56)(57)

Applications

P.L. Chebyshev initiated his 40 years research on approximation with two articles, in connection with the mechanism theory: “Théorie des mécanismes connus sous le nom de parallelélogrammes” (1854) and “Sur les questions de minima qui se rattachent à la représentation approximative des fonctions” (1859) [18]. The origin of these researches is to be found in his desire to improve J. Watt’s steam engine. In fact, the study of a mechanism that converts circular into linear motion, improving the results of Watt, has led him to new problems in approximation theory (the so-called uniform optimal approximation) whose solution makes use of the first kind Chebyshev polynomials.

The problem is posed in these terms:

Assigned a continuous function in , among all monic polynomials of degree , to find the best uniform approximation, i.e. such that:

is the uniform norm, (also called minimax norm, or Chebyshev norm). The existence of the minimum was proved by Kirchberger [19] and Borel [20].

Assuming, without essential restriction, [ab] ≡ [−1, 1], and putting f(x) ≡ 0, the solution is given by the first kind (monic) Chebyshev polynomial :

A characteristic property of the best approximation is the “alternating property” (or “equioscillation property”), according to which the best approximation of the zero function attains its maximum absolute value, with alternating signs, at least at points of the given interval.

Weighted minimax approximation by pseudo-Chebyshev functions

In what follows, we deal with a minimax property in [−1, 1] of the type:where is a weight function. The “alternating property” still characterizes the solution of the problem [15].

By nothing that the functionattains its maximum absolute value, with alternating signs, at the points , (n = 0, 1, …, k), it results that

Theorem 12

The minimax approximations to zero on [−1, 1], by monic polynomials of degree n, weighted by , is given by .

In a similar way, as the functionattains its maximum absolute value, with alternating signs, at the points , (n = 0, 1, …, k), it results that

Theorem 13

The minimax approximations to zero on , by monic polynomials of degree n, weighted by , is given by .

Roots of a 2 × 2 non-singular complex matrix

The second kind pseudo-Chebyshev functions can be used in order to find explicit formulas for the roots of a non-singular 2 × 2 complex matrix.

Let be such a matrix, and denote respectively by and the trace and the determinant of . Then in [8] the second kind Chebyshev polynomials have been used in order to prove the equation:(58)where n is an integer number and denotes the 2 × 2 identity matrix. Actually this equation still holds when n is replaced by 1/n, in the form:(59)

Obviously, as the roots in the preceding equation are multi-determined, we have 2n possible values for the n-th root of a 2 × 2 non-singular matrix.

For example, putting n = 2, we find the representation of the square root of in terms of second kind pseudo-Chebyshev functions:(60)and recalling the initial conditions (17), so thatwe recover the known equation:(61)since the roots of d have two (dependent) determinations. Therefore, we find, in total, four possible values for the square root of .

By using the results in [9] an extension to the roots of a 3 × 3 non-singular complex matrix could be obtained, provided that the matrix eigenvalues are explicitly known. However, the resulting formula is much more involved, and will be reported in another paper.

Representation of the Dirichlet kernel

The Dirichlet kernel can be expressed in terms of the pseudo-Chebyshev functions.

Theorem 14

The representation formula of the Dirichlet kernel holds: (62)

Proof. From the well known equationit follows:

Summation of trigonometric series

Consider now a trigonometric series of a , -periodic function f, that is:where and are the Fourier coefficients of .

By Carleson’s theorem [21], the above series converges in mean and even pointwise, up to a set of Lebesgue measure zero.

Writing the partial sums of the above series,(63)we find the the result:

Theorem 15

The partial sums of a Fourier series can be represented, in terms of the pseudo-Chebyshev functions, in the form: (64)

Conclusion

The growth of living organisms is often described by the Bernoulli’s logarithmic spiral, which is one of the first examples of fractals. The study of natural forms is commonly associated with mathematical entities like extensions of Lamé’s curves, Grandi’s roses, Bernoulli’s lemniscate, etc.

In this article it has been shown that innumerable plane forms can be described by means of polar equations that extend some of the above mentioned geometrical entities. Moreover, the consideration of Grandi’s roses in the case of fractional indexes gives rise, in a natural way, to mathematical functions that generalize to the case of fractional degree the classical Chebyshev polynomials. The resulting functions, called of pseudo-Chebyshev type, verify many properties of the corresponding polynomials and, in the case of half-integer degree, also the orthogonality properties.

Applications have been shown in the fields of weighted best approximation, roots of 2 × 2 non-singular matrices and Fourier series.

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All Figures

thumbnail Figure 1

Archimedes vs. Bernoulli spiral.

In the text
thumbnail Figure 2

Vk (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue.

In the text
thumbnail Figure 3

Wk (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue.

In the text
thumbnail Figure 5

Rhodonea cos(2θ) and cos(3θ).

In the text
thumbnail Figure 6

Rodoneas and .

In the text
thumbnail Figure 7

Rodoneas and .

In the text
thumbnail Figure 8

Tk+1/2(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange.

In the text
thumbnail Figure 9

Uk+1/2(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange.

In the text
thumbnail Figure 10

Vk+1/2(x), k = 1, 2, 3, 4, 5. (1) Grey; (2) red; (3) blue; (4) orange; (5) violet.

In the text
thumbnail Figure 11

Wk+1/2(x), k = 1, 2, 3, 4, 5. (1) Red; (2) blue; (3) orange; (4) violet; (5) grey.

In the text

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