On (β,γ)-Chebyshev functions and points of the interval
OOn ( β, γ ) -Chebyshev functions and pointsof the interval Stefano De Marchi ∗ , † , Giacomo Elefante ∗∗ , Francesco Marchetti ∗ ∗ Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, Italy; ∗∗ Département de Mathématiques, Université de Fribourg, Switzerland;
Abstract
In this paper, we introduce the class of ( β, γ ) -Chebyshev functions and corre-sponding points, which can be seen as a family of generalized Chebyshev poly-nomials and points. For the ( β, γ ) -Chebyshev functions, we prove that theyare orthogonal in certain subintervals of [ − , with respect to a weightedarc-cosine measure. In particular we investigate the cases where they becomepolynomials, deriving new results concerning classical Chebyshev polynomi-als of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of ( β, γ ) -Chebyshev points. We also study thebehavior of the Lebesgue constants of the polynomial interpolant at thesepoints on varying the parameters β and γ . Keywords:
Chebyshev polynomials, Chebyshev points, GeneralizedChebyshev points, Lebesgue constant
1. Introduction
Chebyshev polynomials have been long-investigated in scientific literatureand they have been considered in various fields, e.g. in function approxi-mation [35], partial differential equations [38], cryptography [2], distributedconsensus [30], group theory [4], cosmography [11] as well as optimal controlproblems [24]. Different types of Chebyshev polynomials have been studied
Email addresses: [email protected], corresponding author (Stefano DeMarchi ∗ , † ), [email protected] (Giacomo Elefante ∗∗ ), [email protected] (Francesco Marchetti ∗ ) Preprint submitted to Elsevier February 9, 2021 a r X i v : . [ m a t h . NA ] F e b nd many related properties have been reported (for a complete overview theinterested reader may refer to [28, 33]).In the recent literature, Chebyshev polynomials still represent a prolificresearch topic. For example, although generalizations of such polynomialshave been already proposed for example in [26, 31], more recent ones in[5, 13, 22]. Pseudo-Chebyshev functions of rational degree p/q have beenalso recently studied in [12, 14]. As in the standard setting with integerdegrees, it has been proved that the family pseudo-Chebyshev functions sat-isfies a recurrence relation and solves a certain differential equation, similarto the classical Chebyshev polynomials, however it retains an orthogonalityproperty in an interval of the real axis if and only if q = 2 . Furthermore, newidentities about Chebyshev polynomials have been derived also in [7, 39].The zeros of Chebyshev polynomials, the Chebyshev points, are of largeinterest in literature. They represent a preferable choice for interpolation dueto their well conditioning and fast convergence (cf. e.g. [15, 34]). In fact,Chebyshev points retain a logarithmic growth for the norm of the interpolantoperator. i.e. the Lebesgue constant [8, 20]. Furthermore, for their goodproperties, they are widely-adopted, for example, in numerical quadrature[27] or in the solution of differential equations [36].We fix some notations. Let Ω = [ − , and n ∈ N . The Chebyshevpolynomials of the first kind { T n } n =0 , ,... are defined as T n ( x ) = cos( n arccos x ) , x ∈ Ω .T n is indeed an algebraic polynomial of degree n thanks to the change ofvariable x = cos( t ) and the Viète formulae for the cosine. They are a familyof orthogonal polynomials on Ω with respect to the weight function w ( x ) =(1 − x ) − / . The zeros of T n , namely the Chebyshev points (of the first kind),is the set T n = (cid:26) cos (cid:18) (2 j − πn (cid:19)(cid:27) j =1 ,...,n , n ∈ N , which are all reals and inside Ω .Let X n := { x , . . . , x n } be a set of distinct points in Ω and L := { (cid:96) , . . . , (cid:96) n } be the Lagrange polynomials (cid:96) i ( x ) := n (cid:89) j =0 j (cid:54) = i x − x j x i − x j , i = 0 , . . . , n, x ∈ Ω , λ ( X n ; x ) = n (cid:88) i =0 | (cid:96) i ( x ) | , x ∈ Ω , and its maximum over Ω is the corresponding Lebesgue constant Λ( X n , Ω) = max x ∈ Ω λ ( X n ; x ) , which is an indicator both of the conditioning and the stability of the in-terpolation process. Note that the Lebesgue constant depends only on thechoice of the interpolation nodes, and therefore many efforts have been madein finding sets of nodes whose Lebesgue constant grows slowly with n ; werefer e.g. to [15, 34].Furthermore, also the set of Chebyshev-Lobatto (CL) points U n +1 = (cid:26) cos (cid:18) jπn (cid:19)(cid:27) j =0 ,...,n , which consists of the zeros of the polynomial T n +1 ( x ) = (1 − x ) n ∂∂x T n ( x ) , x ∈ Ω , is widely-adopted being as well Λ( U n +1 , Ω) = O (log n ) [29]. We refer to therecent survey [23] for further details concerning the Lebesgue constant ofChebyshev, CL and various other sets of points.In this work, we introduce a new family of functions in Ω , namely the ( β, γ ) -Chebyshev functions, which can be considered as a generalization ofChebyshev polynomials. Indeed the family includes the classical Chebyshevpolynomials as a particular case.After investigating such new functions and providing various theoreticalresults, we drive our attention to the corresponding sets of ( β, γ ) -Chebyshevand ( β, γ ) -CL points, analyzing the Lebesgue constant of these points from atheoretical point of view by verifying the results through extensive numericalexperiments. We point out that such points can be characterized as mapped equispaced points, and so they can be studied in the framework of the recentlyproposed fake nodes (cf. [3, 19, 17, 18]).The paper layout and our main contributions follows.3 In Section 2, we introduce what we call the ( β, γ ) -Chebyshev functionsand related points. In particular, in Theorem 1 we prove that suchfunctions are orthogonal in a subinterval of Ω . Moreover, we analyzefor which choice of the parameters β, γ they reduce to polynomials. Indoing so, we show how subsets of classical Chebyshev and CL points areincluded in our general framework. Furthermore, Corollary 2 presentsa new result concerning the orthogonality of standard Chebyshev poly-nomials of the first kind whose degree is a multiple of a fixed naturalnumber.• Section 3 is devoted to show how the new sets of nodes can be obtainedby mapping sets of equispaced points through the Kosloff Tal-Ezer map(cf. [1, 25]). Notice that in Proposition 6 we use this characterizationto link ( β, γ ) -Chebyshev and ( β, γ ) -CL points.• In Section 4, we investigate on the behavior of the Lebesgue constantof the interpolant at these points. More precisely, we show how theparameters β and γ influence the growth of the Lebesgue constantwith respect to the degree n .• Finally, in Section 5 we draw some final considerations and discussfurther developments.
2. The ( β, γ ) -Chebyshev functions and related zeros On Ω , let us consider the ( β, γ ) -Chebyshev functions (of the first kind)defined as T β,γn ( x ) := cos (cid:18) n − β − γ (cid:18) arccos x − γπ (cid:19)(cid:19) , x ∈ Ω , (1)where β, γ ∈ [0 , , β + γ < , n ∈ N . We point out that in general T β,γn is not a polynomial and T , n = T n is the classical Chebyshev polynomials ofthe first kind.The set of zeros of the function T β,γn in Ω β,γ := [ − cos( βπ/ , cos( γπ/ ⊆ Ω , that is T β,γn := (cid:26) cos (cid:18) (2 − β − γ )(2 j − π n + γπ (cid:19)(cid:27) j =1 ,...,n ,
4s what we call the ( β, γ ) -Chebyshev points . Moreover, the extrema points of T β,γn in Ω β,γ are cos (cid:18) (2 − β − γ ) jπ n + γπ (cid:19) , j = 1 , . . . , n − . Similarly, we can define the set of ( β, γ ) -Chebyshev-Lobatto ( ( β, γ ) -CL) pointsas U β,γn +1 := (cid:26) cos (cid:18) (2 − β − γ ) jπ n + γπ (cid:19)(cid:27) j =0 ,...,n . We note that the elements of U β,γn +1 are zeros of the function T β,γn +1 ( x ) = 2 − β − γ n (1 − x ) ∂∂x T β,γn ( x ) , x ∈ Ω . More precisely, U β,γn +1 coincides with the set of zeros of T β,γn +1 if and only if β = γ = 0 . In Figure 1, we display the functions and corresponding pointsfor some values of n, β, γ . Figure 1: The functions T β,γn (solid line) and T β,γn +1 (dashed line), the sets T β,γn (bluecircles) and U β,γn +1 (red crosses), the set Ω β,γ delimited by dotted vertical lines. Left: n = 4 , β = γ = 1 / . Right: n = 5 , β = 3 / , γ = 1 / . Now, we prove a result concerning a symmetric property of ( β, γ ) -Chebyshevfunctions. Proposition 1.
Let n ∈ N > and ν ∈ [0 , . Then, for x ∈ Ω , T ν, n ( x ) = ( − n T ,νn ( − x ) , and T ν, n +1 ( x ) = ( − n T ,νn +1 ( − x ) . roof. By using the identity arccos ( − x ) = π − arccos x and the additionformula for the cosine, we get T ,νn ( − x ) = cos (cid:18) n − ν (cid:18) π − arccos x − νπ (cid:19)(cid:19) = cos (cid:18) nπ − n − ν arccos x (cid:19) = cos( nπ ) cos (cid:18) n − ν arccos x (cid:19) = ( − n T ν, n ( x ) , because of the fact that sin( nπ ) = 0 for any n ∈ N > .Moreover, for x ∈ Ω , T ν, n +1 ( x ) = 2 − β − γ n (1 − x ) ∂∂x T ν, n ( x )= ( − n − β − γ n (1 − x ) ∂∂x T ,νn ( − x ) = ( − n T ,νn +1 ( − x ) . This concludes the proof.
Corollary 1.
In the hypotheses of Proposition 1, we have ¯ x ∈ T ν, n if and only if − ¯ x ∈ T ,νn , ¯ x ∈ U ν, n if and only if − ¯ x ∈ U ,νn . These properties are a direct consequence of the results in Proposition 1.In Figures 2 and 3 we show the above symmetric properties for somevalues of n, β, γ . Figure 2: The functions T β,γn (solid line) and T β,γn +1 (dashed line), the sets T β,γn (bluecircles) and U β,γn +1 (red crosses), the set Ω β,γ delimited by dotted vertical lines. Left: n = 5 , β = 1 / , γ = 0 . Right: n = 5 , β = 0 , γ = 1 / . .00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.001.000.750.500.250.000.250.500.751.00 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.001.000.750.500.250.000.250.500.751.00 Figure 3: The functions T β,γn (solid line) and T β,γn +1 (dashed line), the sets T β,γn (bluecircles) and U β,γn +1 (red crosses), the set Ω β,γ delimited by dotted vertical lines. Left: n = 6 , β = 4 / , γ = 0 . Right: n = 6 , β = 4 / , γ = 0 . This family of functions satisfies a recurrence formula.
Proposition 2.
The functions { T β,γn } n =0 , ,... satisfy T β,γ ( x ) = 1 , T β,γ ( x ) = cos (cid:18) − β − γ (cid:18) arccos x − γπ (cid:19)(cid:19) ,T β,γn +1 ( x ) + T β,γn − ( x ) = 2 T β,γ ( x ) T β,γn ( x ) for x ∈ Ω .Proof. Letting θ = arccos x − γπ , x ∈ Ω , by using the addition formulae forthe cosine we get cos (cid:18) n + 1) θ − β − γ (cid:19) + cos (cid:18) n − θ − β − γ (cid:19) = 2 cos (cid:18) θ − β − γ (cid:19) cos (cid:18) nθ − β − γ (cid:19) , which concludes the proof.Furthermore, we have the following orthogonality result. Theorem 1.
The functions { T β,γn } n =0 , ,... are orthogonal on Ω β,γ with respectto the weight function w β,γ ( x ) = 2(2 − β − γ ) √ − x , x ∈ Ω β,γ , having (cid:90) Ω β,γ T β,γr ( x ) T β,γs ( x ) w β,γ ( x ) d x = if r (cid:54) = s,π if r = s = 0 ,π if r = s (cid:54) = 0 . roof. As well-known, (cid:90) π cos( rθ ) cos( sθ ) d θ = if r (cid:54) = s,π if r = s = 0 ,π if r = s (cid:54) = 0 . Then, the result follows by the change of variable x = cos((2 − β ) θ/ γπ/ that maps Ω into Ω β,γ . In what follows, we analyze under which choices of the parameters thefunction T β,γn is a polynomial. We remark that thanks to Proposition 1, wedo not need to discuss the case β = 0 , γ > . β > , γ = 0 Consider β = p/q, p, q ∈ N > and independent of n . From (1), we thenrequire − pq ∈ N > ⇐⇒ (2 q − p ) m = 2 q for a given m ∈ N > . Hence, we obtain p = 2 q ( m − m and β m := 2( m − m = 2 − m . (2)Therefore, for x ∈ Ω T β m , n ( x ) = cos (cid:18) n − β m arccos x (cid:19) = cos( mn arccos x ) = T mn ( x ) , which implies T β m , n = (cid:26) cos (cid:18) (2 j − π mn (cid:19)(cid:27) j =1 ,...,n , U β m , n +1 = (cid:26) cos (cid:18) jπmn (cid:19)(cid:27) j =0 ,...,n . We observe that β m ∈ [1 , for every m ∈ N > . From Theorem 1 we get thefollowing corollary which shows that the Chebyshev polynomials (of the firstkind) with degree and weight function that are a multiple of a fixed m ∈ N > ,satisfy this additional orthogonality property.8 orollary 2. Let m ∈ N > , the polynomials (cid:8) T β m , n (cid:9) n =0 , ,... = { T mn } n =0 , ,... are orthogonal on Ω β m , with respect to the weight function w β m , ( x ) = m √ − x , x ∈ Ω β m , . Proof.
The proof directly follows from Theorem 1.Let us now consider the case β depending linearly on n . In particular, n − pq ∈ N > ⇐⇒ (2 q − p ) m = 2 qn, m ∈ N > , that is, p = 2 q ( m − n ) m and β m,n := 2( m − n ) m = 2 − nm , (3)where we assume n < m so that β m,n ∈ ]0 , . Then, for a fixed m ∈ N > and n = 0 , . . . , m − the corresponding functions and points are T β m,n , n ( x ) = cos (cid:18) n − β m,n arccos x (cid:19) = cos( m arccos x ) = T m ( x ) , and T β m,n , n = (cid:26) cos (cid:18) (2 j − π m (cid:19)(cid:27) j =1 ,...,n , U β m,n , n +1 = (cid:26) cos (cid:18) jπm (cid:19)(cid:27) j =0 ,...,n . (4)Where T β m,n , n and U β m,n , n +1 are subsets of classical Chebyshev and CL points(see Theorem 2 below). The case m = n + 1 is of particular interest for usand is discussed in Section 4. β > , γ = β In view of (2), we take β = γ = β m . (5) Proposition 3.
Let m, n ∈ N > and let β = γ = β m . If n ∈ N > , then T βm , βm n = T mn . If n ∈ N > \ N > , then T βm , βm n = (cid:40) T mn if m − ∈ N > , − T mn if m ∈ N > . If n is odd, then T βm , βm n = T mn if m − ∈ N > , − T mn if m − ∈ N > \ N > , − sin( mn arccos x ) if ( m − n + 1 ∈ N > , sin( mn arccos x ) if ( m − n − ∈ N > . Proof.
For x ∈ Ω we have T βm , βm n ( x ) = cos (cid:18) mn (cid:18) arccos x − β m π (cid:19)(cid:19) = cos (cid:18) mn arccos x − ( m − nπ (cid:19) = cos( mn arccos x ) cos (cid:18) ( m − nπ (cid:19) ++ sin( mn arccos x ) sin (cid:18) ( m − nπ (cid:19) . Then the three cases follow by evaluating the sine and cosine for the corre-sponding values of m, n ∈ N > .We note that the zeros of T βm , βm n in Ω are, therefore, mn + 1 Chebyshev-Lobatto points in the case where n is odd and ( m − n ± ∈ N > , otherwisethey are mn Chebyshev points.Concerning (cid:0) β m , β m (cid:1) -Chebyshev and (cid:0) β m , β m (cid:1) -CL points, we obtain T βm , βm n = (cid:26) cos (cid:18) (2 j − π mn + ( m − π m (cid:19)(cid:27) j =1 ,...,n , U βm , βm n +1 = (cid:26) cos (cid:18) jπmn + ( m − π m (cid:19)(cid:27) j =0 ,...,n . orollary 3. Let m ∈ N > be an odd number. The polynomials (cid:8) T βm , βm n (cid:9) n =0 , ,... = (cid:40) { T mn } n =0 , ,... if m − ∈ N > , { ( − n T mn } n =0 , ,... if m − ∈ N > \ N > , are orthogonal in Ω βm , βm with respect to the weight function w βm , βm ( x ) = m √ − x , x ∈ Ω βm , βm . Proof.
The proof directly follows from Theorem 1 and Proposition 3.Finally, in view of (3), we consider β = γ = β m,n , n < m. Proposition 4.
Let m ∈ N > be fixed and let β = γ = β m,n with n < m . If m − n ∈ N > , then T βm,n , βm,n n = T m . If m − n ∈ N > \ N > , then T βm,n , βm,n n = − T m . If m − n + 1 ∈ N > , then T βm,n , βm,n n ( x ) = − sin( m arccos x ) for x ∈ Ω . If m − n − ∈ N > , then T βm,n , βm,n n ( x ) = sin( m arccos x ) for x ∈ Ω .Proof. The proof is similar to that of Proposition 3.
Let β = 1 − p /q , γ = 1 − p /q , p , p ∈ N , q , q ∈ N > . In general, T β,γn is a polynomial if n p q + p q ∈ N > , that is p q + p q | q q n ⇐⇒ ( p q + p q ) m = 2 q q n, m ∈ N > . It is worthwhile to point out a particular choice of β and γ for which the ( β, γ ) -Chebyshev (Lobatto) points result in subsets of Chebyshev (Lobatto)points. 11 heorem 2. Let T n = { t j } j =1 ,...,n and U n +1 = { u j } j =0 ,...,n be the set ofChebyshev and CL points respectively. Moreover, let κ , κ ∈ N , κ := ( κ , κ ) , and β κ = κ n + κ + κ , γ κ = κ n + κ + κ . Then, T β κ ,γ κ n = T n + κ + κ \ { t , . . . , t κ − , t n + κ +1 , . . . , t n + κ + κ } , U β κ ,γ κ n +1 = U n + κ + κ +1 \ { u , . . . , u κ − , u n + κ +1 , . . . , u n + κ + κ } . Proof.
We present the proof only for ( β, γ ) -CL points because for the ( β, γ ) -Chebyshev points is similar. U β κ ,γ κ n +1 = cos (cid:18) − β κ − γ κ n jπ + γ κ π (cid:19) , j = 0 , . . . , n, = cos (cid:32) nn + κ + κ n jπ + κ πn + κ + κ (cid:33) , = cos (cid:18) ( j + κ ) πn + κ + κ (cid:19) , = cos (cid:18) l πn + κ + κ (cid:19) , l = κ , . . . , n + κ . With the notation of Theorem 2 and recalling the analysis carried out inSection 2.2.2, we highlight β = γ = β n +2 ,n − nn + 2 = 2 n + 2 = 2 κ n + κ + κ , with κ = κ = 1 and in this case, we obtain indeed the sets U βn +2 ,n , βn +2 ,n n +1 = U n +3 \ {± } , T βn +2 ,n , βn +2 ,n n = T n +2 \ { t , t n +2 } . (6)We notice that the set U βn +2 ,n , βn +2 ,n n +1 has already been investigated in theliterature (cf. [8, 23]). 12 . The ( β, γ ) -Chebyshev (Lobatto) points are mapped equispacedpoints We show that the ( β, γ ) -Chebyshev (Lobatto) points can be obtained bymapping equispaced points via the so-called Kosloff Tal-Ezer (KTE) map[1, 25] M α ( x ) := sin( απx/ απ/ , α ∈ ]0 , , x ∈ Ω . (7)Letting S n := (cid:26) − j − n (cid:27) j =1 ,...,n , then the Chebyshev points of first kind are T n = M (cid:0) S n (cid:1) := (cid:8) M ( s j ) | s j ∈ S n (cid:9) j =1 ,...,n . Moreover, if E n := (cid:26) − jn − (cid:27) j =0 ,...,n − then the Chebyshev-Lobatto points are U n = M (cid:0) E n (cid:1) := (cid:8) M ( e j ) | e j ∈ E n (cid:9) j =0 ,...,n . Similarly for the ( β, γ ) -Chebyshev (Lobatto) points we can prove the follow-ing result. Proposition 5.
Let β, γ ∈ [0 , , β + γ < and let Ω β,γ := [ − β, − γ ] .Moreover, let S β,γn := (cid:26) − γ − (2 − β − γ )(2 j − n (cid:27) j =1 ,...,n , and E β,γn := (cid:26) − γ − (2 − β − γ ) jn − (cid:27) j =0 ,...,n − . Then, we have T β,γn = M (cid:0) S β,γn (cid:1) , U β,γn = M (cid:0) E β,γn (cid:1) . roof. It is sufficient to observe that sin (cid:18) π (cid:18) − γ − (2 − β − γ )(2 j − n (cid:19)(cid:19) = cos (cid:18) (2 − β − γ )(2 j − π n + γπ (cid:19) and sin (cid:18) π (cid:18) − γ − (2 − β − γ ) jn − (cid:19)(cid:19) = cos (cid:18) (2 − β − γ ) jπ n −
1) + γπ (cid:19) . The set of ( β, γ ) -Chebyshev and ( β, γ ) -CL points are linked together asfollows. Proposition 6.
Let β, γ ∈ [0 , , β + γ < . Then T β,γn = U β + − β − γ n ,γ + − β − γ n n . Proof.
In view of Proposition 5, it is sufficient to prove the identity S β,γn = E β + − β − γ n ,γ + − β − γ n n . To simplify the notation we denote as ρ = 2 − β − γ , then for all j = 1 , . . . , n ,we get − γ − ρ (2 j − n = 1 − γ − ρ n − j − n − (cid:18) ρ − ρn (cid:19) ; − ρ (2 j − n + ρ n = 1 − jn − ρ (cid:18) − n (cid:19) ; ρ (1 − j ) n = 1 − jn − ρ (cid:18) n − n (cid:19) and this concludes the proof. Remark 1.
The result in Proposition 6 allows us to restrict to the set of ( β, γ ) -CL points. Moreover, we recall a remarkable property of classicalChebyshev points, that is T n = U n , n n . . Lebesgue constant for ( β, γ ) -CL nodes In this section we analyze the behavior of the Lebesgue constant of the ( β, γ ) -CL points on Ω .If β and γ are small enough , we can think of the ( β, γ ) -CL points asperturbed CL points and the next theorem shows that the Lebesgue constantof ( β, γ ) -CL points grows logarithmically. Theorem 3.
Letting β, γ ∈ [0 , , β + γ < , δ := max { β, γ } and U β,γn +1 theset of n + 1 ( β, γ ) -CL points, n ∈ N . If δ < πn (2 + π log( n + 1)) , then Λ( U β,γn +1 , Ω) = O (log n ) . Proof.
In [32], it has been proved that the set U n +1 under a maximal pertur-bation (cid:15) such that (cid:15) < n (2 + π log( n + 1)) (8)retains the logarithmic growth of the Lebesgue constant.In our setting, taking (cid:101) u j ∈ U β,γn +1 and u j ∈ U n +1 , we consider (cid:15) j := | (cid:101) u j − u j | , j = 0 , . . . , n as the perturbation of the j -th point, that is (cid:15) j = (cid:12)(cid:12)(cid:12)(cid:12) cos (cid:18) (2 − β − γ )2 n jπ + γπ (cid:19) − cos (cid:18) jπn (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) β + γ n jπ − γπ (cid:12)(cid:12)(cid:12)(cid:12) ≤ δπ . Then, thanks to (8), if δπ < n (2 + π log( n + 1)) , the Lebesgue constant grows logarithmically, as claimed.Two particular cases deserve to be analyzed.15 .1. Case β > , γ = 0 We point out that we do not need to consider the case β = 0 and γ > by virtue of the symmetric property stated in Corollary 1.We start by considering the special case, ¯ β n = 2 /n , which yields to theset of points (cf. (4)) U ¯ β n , n = U n +1 \ {− } . Theorem 4.
Let ¯ β n = 2 /n , n ∈ N > . Then for the associate Lebesguefunction we have λ (cid:0) U ¯ β n , n ; − (cid:1) = 2 n − . Proof.
Let U n = { u j } j =0 ,...,n − . Then, for i = 0 , . . . , n − and x ∈ Ω , wehave (cid:96) i ( x ) = n − (cid:89) j =0 j (cid:54) = i x − u j · n − (cid:89) j =0 j (cid:54) = i u i − u j = n − (cid:89) j =0 j (cid:54) = i x − u j · ( u i − u n ) n (cid:89) j =0 j (cid:54) = i u i − u j = n − (cid:89) j =0 j (cid:54) = i x − u j · ( u i + 1) 2 n − n ( − i σ i , with σ i = 1 / if i = 0 and σ i = 1 otherwise (see e.g. [37, p. 37]). Therefore, (cid:96) i ( −
1) = n − (cid:89) j =0 j (cid:54) = i ( − − u j ) · n − n ( − i ( u i + 1) σ i = n − (cid:89) j =0 ( − − u j ) · − − u i · n − n ( − i ( u i + 1) σ i . We notice that n − (cid:89) j =0 ( − − u j ) = n − (cid:89) j =0 ( u n − u j ) ( n + 1) -barycentric weight related to the Lagrangeinterpolant at the CL nodes U n +1 . Hence, n − (cid:89) j =0 ( − − u j ) = 2 − n +2 n ( − n . This leads to | (cid:96) i ( − | = (cid:12)(cid:12)(cid:12)(cid:12) − − u i · − n +2 n ( − n · n − n ( − i ( u i + 1) σ i (cid:12)(cid:12)(cid:12)(cid:12) = 2 σ i . Finally, λ (cid:16) U ¯ β n , n ; − (cid:17) = n − (cid:88) i =0 | (cid:96) i ( − | = n − (cid:88) i =0 σ i = 2( n −
1) + 1 . Conjecture 1.
In view of Theorem 4, we claim that the maximum of theLebesgue function is attained in x = − , that is Λ (cid:0) U ¯ β n , n , Ω (cid:1) = λ (cid:0) U ¯ β n , n ; − (cid:1) = 2 n − . The statements in Theorem 4 and Conjecture 1 are displayed in Figure 4for some values of n .As supported by extensive numerical tests, the Lebesgue constant Λ (cid:0) U β, n , Ω (cid:1) passes from a logarithmic to a linear growth with n by increasing the valueof β from β = 0 to β = ¯ β n . Moreover, as β > ¯ β n gets larger, the growthbecomes exponential. We show this behavior in Figure 4 (right). Figure 4: Left: the function λ (cid:0) U ¯ β n , n ; · (cid:1) with n = 5 (black), n = 6 (red) and n = 7 (blue). Right: varying n = 5 , . . . , , the Lebesgue constant Λ (cid:0) U β j,n , n , Ω (cid:1) with β j,n = j/ (10 n ) , j = 0 , . . . , . The linear case β ,n = ¯ β n is displayed using a dashed line. emark 2. Let n ∈ N be fixed. From numerical experiments we notice thatthere exists a β (cid:63)n ∈ [0 , /n [ such that for all β ∈ [0 , , β (cid:54) = β (cid:63)n , Λ (cid:0) U β (cid:63)n , n , Ω (cid:1) < Λ (cid:0) U β, n , Ω (cid:1) . Furthermore, Λ (cid:0) U β, n , Ω (cid:1) is monotonically decreasing for β ∈ [0 , β (cid:63)n [ and in-creasing for β ∈ ] β (cid:63)n , (see Figure 5). For some values of β , the growth of Λ (cid:0) U β, n , Ω (cid:1) is slower than the growth of the Lebesgue constant related to theclassical CL points. Figure 5: Fixed n = 40 , the Lebesgue constant Λ (cid:0) U β, n , Ω (cid:1) varying β ∈ [0 , /n ] (left) and β ∈ [1 /n, /n ] (right). β = γ Let now ¯ δ n = 2 / ( n + 1) . Recalling (6), we have that (see e.g. [9]) Λ (cid:0) U ¯ δ n , ¯ δ n n , Ω (cid:1) = n. As we show in Figure 6, considerations similar to those in Section 4.1 canbe drawn, with ¯ δ n playing the role of ¯ β n .18 .00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.001234567 5 10 15 20 25 30 35 40020406080100 Figure 6: Left: the function λ (cid:0) U ¯ δ n , ¯ δ n n ; · (cid:1) with n = 5 (black), n = 6 (red) and n = 7 (blue). Right: varying n = 5 , . . . , , the Lebesgue constant Λ (cid:0) U δ j,n ,δ j,n n , Ω (cid:1) with δ j,n = j/ (10( n + 1)) , j = 0 , . . . , . The linear case δ ,n = ¯ δ n is displayed using a dashed line. Remark 3.
Let n ∈ N be fixed. From numerical experiments we notice thatthere exists δ (cid:63)n ∈ [0 , / ( n + 1)[ such that ∀ δ ∈ [0 , , δ (cid:54) = δ (cid:63)n , Λ (cid:0) U δ (cid:63)n ,δ (cid:63)n n , Ω (cid:1) < Λ (cid:0) U δ,δn , Ω (cid:1) . Moreover, Λ (cid:0) U δ,δn , Ω (cid:1) is monotonically decreasing for δ ∈ [0 , δ (cid:63)n [ and increas-ing for δ ∈ ] δ (cid:63)n , . In Figure 7 we plot the Lebesgue constant for differentvalues of δ . The behavior is slightly different from the results shown in Fig-ure 6, in fact the minimum is achieved just before the blowing up. Figure 7: Fixed n = 40 , the Lebesgue constant Λ (cid:0) U δ,δn , Ω (cid:1) varying δ ∈ [0 , / ( n + 1)] (left)and δ ∈ [1 / ( n + 1) , / ( n + 1)] (right).
5. Conclusions
In this work, we introduced ( β, γ ) -Chebyshev functions and points, whichcan be considered a generalization of classical Chebyshev polynomials and19oints. In particular, for some choices of the parameters we showed that ( β, γ ) -Chebyshev functions are orthogonal polynomials in Ω β,γ ⊆ [ − , fora proper weight function (see Theorem 1), thus they may be used in Gaus-sian quadrature formulae or via Newton-Côtes formulae similarly to whathas been done in [16].Furthermore, we characterized ( β, γ ) -Chebyshev points as mapped equis-paced points via KTE map and we analyzed their related Lebesgue constantsshowing that, for certain small values of the parameters, they preserve thelogarithmic growth, as for the classical CL points, providing alternative setsfor stable polynomial approximation. This construction suggests a naturalextension to the tensor product polynomial approximations. Moreover, forpolynomial interpolation of total degree, we can obtain good interpolationnodes having quasi-optimal approximation properties like the well-knowntwo-dimensional Padua points in [ − , (see [6, 10]) or, in higher dimen-sions, the Lissajous points [21] .
6. Acknowledgments
This research has been accomplished within the Rete ITaliana di Ap-prossimazione (RITA) and the thematic group on Approximation Theory andApplications of the Italian Mathematical Union. We received the support ofGNCS-IN δ AM and were partially funded by the ASI - INAF grant “ArtificialIntelligence for the analysis of solar FLARES data (AI-FLARES)”and theNATIRESCO BIRD181249 project.
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