A survey on pseudo-Chebyshev functions

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 pseudo-Chebyshev 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 non-singular matrices and Fourier series are derived.


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 (À1, 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: q ¼ e n h ; ðn 2 N Þ: ð3Þ The complex Bernoulli spiral In the complex case, putting and considering the Bernoulli spiral: we find: Recalling Chebyshev polynomials Starting from the equations: separating the real from the imaginary part, and putting x = cos t, the first and second kind Chebyshev polynomials are derived: U nÀ1 x ð Þ :¼ sinðn arccosxÞ 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 Â 2 non singular matrix [8]. Extension of this polynomial falmily to the multivariate case has been considered for the powers of a r Â r (r ! 3) non singular matrix (see [9,10]). Remark 1. Chebyshev polynomials are a particular case of the Jacobi polynomials P ða;bÞ n ðxÞ [11], which are orthogonal in the interval [À1, 1] with respect to the weight ð1 À xÞ a ð1 þ xÞ b . More precisely, the following equations hold: T n ðxÞ ¼ P ðÀ1=2;À1=2Þ n ðxÞ; U n ðxÞ ¼ P ð1=2;1=2Þ n ðxÞ: 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. 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 À1) (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 ½À1; 1 as follows: Since W n ðxÞ ¼ ðÀ1Þ n V n ðÀxÞ, 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 ½À1; 1.
One of the explicit advantages of Chebyshev polynomials of third and fourth kind is to estimate some definite integrals as with the precision degree 2n À 1, by using the n interpolatory points x k ¼ cos ð2kÀ1Þp 2nþ1 , (k = 1, 2, . . ., n), in the interval [À1, 1] [13,14].

The Grandi (Rhodonea) curves
The curves with polar equation: 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: q ¼ sinðnhÞ are equivalent to the preceding ones, up to a rotation of p=ð2nÞ radians. The Grandi roses display n petals, if n is odd, 2n petals, if n is even.
Roses with 4n þ 2 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 q ¼ cos 1=2 ð2hÞ; (the Bernoulli Lemniscate), for n ! 1, a 4n + 2 petals rose comes from the equation q ¼ cos 1=2 ½ð4n þ 2Þh.
A few graphs of Rhodonea curves are shown in Figures 4-7.

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 (À1, 1) with respect to the same weights of the corresponding Chebyshev polynomials. Definition 1. Let, for any integer k: 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 T kþ 1 2 ðxÞ and W kþ 1 2 ðxÞ Put for shortness: y k :¼ T kþ 1 2 and u k :¼ W kþ 1 2 . 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: Furthermore, the initial conditions for the T kþ 1 2 ðxÞ functions are: Interchanging the roles of y k and u k we find that the same recurrence for second sequence holds, but with different initial conditions. We have, precisely: Differential equations for the T kþ 1 2 ðxÞ and W kþ 1 2 ðxÞ Even the differential equations satified by the y k and u k can be recovered by using a coupled technique. In fact, differentating y k and u k , we find: and differentiating again the first equation: that is: Substituting u 0 k , we find the differential equation satisfied by the y k : and interchanging the roles of y k and u k we find that the same differential equation is satisfied by the u k .
Remark 3. The first few T kþ 1 2 ðxÞ are: Remark 4. The first few W kþ 1 2 ðxÞ are: The case of the U kÀ 1 2 ðxÞ and V kþ 1 2 ðxÞ Recalling equation (12), since it results that the functions U kÀ 1 2 ðxÞ and V kþ 1 2 ðxÞ satisfy the same recursion (13) of the classical Chebyshev polynomials, but with the initial conditions: and Differential equations for the U kÀ 1 2 ðxÞ and V kþ 1 2 ðxÞ Put for shortness: z k :¼ U kÀ 1 2 and w k :¼ V kþ 1 2 . Differentiating the second equation (12), we find: and differentiating again, it results: and interchanging the roles of z k and w k we find that the same differential equation is satisfied by the w k .
Remark 5. The first few U kÀ 1 2 ðxÞ are: and, in general: Remark 6. The first few V kþ 1 2 ðxÞ are: and, in general: The pseudo-Chebyshev functions T kþ 1 2 ðxÞ, U kÀ 1 2 ðxÞ, V kþ 1 2 ðxÞ and W kþ 1 2 ðxÞ can be represented, in terms of the third and fourth kind Chebyshev polynomials as follows: 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 T kþ1=2 ðxÞ and U kþ1=2 ðxÞ functions A few graphs of the T kþ 1 2 functions are shown in Figure 8. Theorem 1. The pseudo-Chebyshev functions T kþ1=2 ðxÞ satisfy the orthogonality property: where h; k are integer numbers; A few graphs of the U kþ 1 2 functions are shown in Figure 9.
Theorem 2. The pseudo-Chebyshev functions U kþ 1 2 ðxÞ satisfy the orthogonality property: where h; k are integer numbers;  Proof. We prove only Theorem 12, since the proof of Theorem 13 is similar. From the Werner formulas, we have: h 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.
Orthogonality properties of the V kþ1=2 ðxÞ and W kþ1=2 ðxÞ functions A few graphs of the V kþ 1 2 functions are shown in Figure 10. Theorem 3. The pseudo-Chebyshev functions V kþ1=2 ðxÞ verify the orthogonality property: A few graphs of the W kþ 1 2 functions are shown in Figure 11. Theorem 4. The pseudo-Chebyshev functions W kþ1=2 ðxÞ verify the orthogonality property: Explicit forms Theorem 5. It is possible to represent explicitly the pseudo-Chebyshev functions as follows: Location of zeros By equation (12), the zeros of the pseudo-Chebyshev functions T kþ 1 2 ðxÞ and V kþ 1 2 ðxÞ are given by  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: Proof. The results follow from the equations: (see [4]), by using definition (12). h Remark 8. Note that the first equation, in (29), extends the known nesting property verified by the first kind Chebyshev polynomials: This property, already considered in [16] for the first kind pseudo-Chebyshev 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].

Pseudo-Chebyshev functions with general rational indexes
Basic properties of the pseudo-Chebyshev functions of the first and second kind We put, by definition: where p and q are integer numbers, (q 6 ¼ 0). 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 T p q ðxÞ satisfy the recurrence relation T p q þ1 ðxÞ ¼ 2 x T p q ðxÞ À T p q À1 ðxÞ: Proof. Write equation (37) in the form: then use definition (35) and the trigonometric identity: The first kind pseudo-Chebyshev functions T p q ðxÞ satisfy the differential equation: ð1 À x 2 Þ y 00 À x y 0 þ p q 2 y ¼ 0: Proof. Note that so that equation (38) follows. h The following theorems hold: Theorem 9. The pseudo-Chebyshev functions U p q ðxÞ satisfy the recurrence relation U p q þ1 ðxÞ ¼ 2 x U p q ðxÞ À U p q À1 ðxÞ: ð39Þ Proof. Write equation (39) in the form: then use definition (36) and the trigonometric identity: The pseudo-Chebyshev functions U p q ðxÞ satisfy the differential equation Proof. Differentiating, and differentiating again equation (36) we find subsequently: Combining the above equations, we find: Links with the pseudo-Chebyshev functions The third and fourth kind Chebyshev polynomials are defined as follows: ðxÞ T nþ 1 2 ðxÞ; ð54Þ Therefore, we find the equations: 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 ½a; b, among all monic polynomials p n ðxÞ 2 P n of degree n, to find the best uniform approximation, i.e. such that: jf ðxÞ À p n ðxÞj ¼ jjf ðxÞ À p n ðxÞjj 1 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] [À1, 1], and putting f(x) 0, the solution is given by the first kind (monic) Chebyshev polynomial T n ðxÞ=2 nÀ1 : jjT n ðxÞ=2 nÀ1 jj 1 ¼ min: 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 k þ 1 points of the given interval.

Weighted minimax approximation by pseudo-Chebyshev functions
In what follows, we deal with a minimax property in [À1, 1] of the type: min p n 2Pn jjwðxÞðf ðxÞ À p n ðxÞÞjj 1 ¼ min; where wðxÞ is a weight function. The "alternating property" still characterizes the solution of the problem [15].