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 
Review Article
A survey on pseudoChebyshev functions
International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
^{*} Corresponding author: paoloemilioricci@gmail.com
Received:
9
January
2020
Accepted:
11
February
2020
In recent articles, by using as a starting point the Grandi (Rhodonea) curves, sets of irrational functions, extending to the fractional degree the 1st, 2nd, 3rd and 4th kind Chebyshev polynomials have been introduced. Therefore, the resulting mathematical objects are called pseudoChebyshev functions. In this survey, the results obtained in the above articles are presented in a compact way, in order to make the topic accessible to a wider audience. Applications in the fields of weighted best approximation, roots of 2 × 2 nonsingular matrices and Fourier series are derived.
Key words: Grandi curves / PseudoChebyshev functions / Recurrence relations / Differential equations / Orthogonality properties
© P.E. Ricci, Published by EDP Sciences, 2020
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 pseudoChebyshev 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 pseudoChebyshev 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 halfinteger, since in this case, the pseudoChebyshev 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)
Figure 1 Archimedes vs. Bernoulli spiral. 
If θ > 0 the spiral turns counterclockwise, 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 = e^{n}, 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 oddorder 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 .
Figure 2 V_{k} (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue. 
Figure 3 W_{k} (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 4–7.
Figure 4 Rhodonea 
Figure 5 Rhodonea cos(2θ) and cos(3θ). 
Figure 6 Rodoneas and . 
Figure 7 Rodoneas and . 
PseudoChebyshev functions of halfinteger degree
In what follows, we consider the case of the halfinteger degree, which seems to be the most interesting one, since the resulting pseudoChebyshev functions satisfy the orthogonality properties in the interval (−1, 1) with respect to the same weights of the corresponding Chebyshev polynomials.
Definition 1
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
Remark 4
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 pseudoChebyshev functions , , and can be represented, in terms of the third and fourth kind Chebyshev polynomials as follows:(20)
In the considered case of halfinteger degree, the pseudoChebyshev 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.
Figure 8 T_{k+1/2}(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange. 
Theorem 1
The pseudoChebyshev functions satisfy the orthogonality property:(21)where (22)
A few graphs of the functions are shown in Figure 9.
Figure 9 U_{k+1/2}(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange. 
Theorem 2
The pseudoChebyshev 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 pseudoChebyshev 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 pseudoChebyshev 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 pseudoChebyshev functions.
Figure 10 V_{k+1/2}(x), k = 1, 2, 3, 4, 5. (1) Grey; (2) red; (3) blue; (4) orange; (5) violet. 
Figure 11 W_{k+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 pseudoChebyshev functions verify the orthogonality property:(25)(26)
A few graphs of the functions are shown in Figure 11.
Theorem 4
The pseudoChebyshev functions verify the orthogonality property:(27)(28)
Explicit forms
Theorem 5
It is possible to represent explicitly the pseudoChebyshev functions as follows: (29)
Location of zeros
By equation (12), the zeros of the pseudoChebyshev functions and are given by(30)and the zeros of the pseudoChebyshev 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 Rodriguestype formulas are reported in [3].
Links with first and second kind Chebyshev polynomials
Theorem 6
The pseudoChebyshev 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 pseudoChebyshev functions, actually holds in general, as a consequence of the definition T_{k}(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].
PseudoChebyshev functions with general rational indexes
Basic properties of the pseudoChebyshev 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 pseudoChebyshev 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 pseudoChebyshev functions satisfy the differential equation:(38)
so that equation (38) follows.
The following theorems hold:
Theorem 9
The pseudoChebyshev functions satisfy the recurrence relation(39)
Proof. Write equation (39) in the form:then use definition (36) and the trigonometric identity:
Theorem 10
The pseudoChebyshev 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 pseudoChebyshev functions of third and fourth kind
According to equation (10), put by definition:(41)
Theorem 11
The third and fourth kind pseudoChebyshev 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
Combining the above equations, we find:(53)
Links with the pseudoChebyshev 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 socalled 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, [a, b] ≡ [−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 pseudoChebyshev 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 nonsingular complex matrix
The second kind pseudoChebyshev functions can be used in order to find explicit formulas for the roots of a nonsingular 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 multidetermined, we have 2n possible values for the nth root of a 2 × 2 nonsingular matrix.
For example, putting n = 2, we find the representation of the square root of in terms of second kind pseudoChebyshev 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 nonsingular 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 pseudoChebyshev 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 pseudoChebyshev 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 pseudoChebyshev type, verify many properties of the corresponding polynomials and, in the case of halfinteger degree, also the orthogonality properties.
Applications have been shown in the fields of weighted best approximation, roots of 2 × 2 nonsingular matrices and Fourier series.
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All Figures
Figure 1 Archimedes vs. Bernoulli spiral. 

In the text 
Figure 2 V_{k} (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue. 

In the text 
Figure 3 W_{k} (x), k = 1, 2, 3, 4. (1) Violet; (2) brown; (3) red; (4) blue. 

In the text 
Figure 4 Rhodonea 

In the text 
Figure 5 Rhodonea cos(2θ) and cos(3θ). 

In the text 
Figure 6 Rodoneas and . 

In the text 
Figure 7 Rodoneas and . 

In the text 
Figure 8 T_{k+1/2}(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange. 

In the text 
Figure 9 U_{k+1/2}(x), k = 1, 2, 3, 4. (1) Green; (2) red; (3) blue; (4) orange. 

In the text 
Figure 10 V_{k+1/2}(x), k = 1, 2, 3, 4, 5. (1) Grey; (2) red; (3) blue; (4) orange; (5) violet. 

In the text 
Figure 11 W_{k+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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