IEEE Circuits and Systems Magazine - Q2 2019 - 61

Unlike Chebyshev, E. I. Zolotarev's interest were equiripple polynomial
and rational approximations by itself.
Chebyshev polynomials of first kind represent no approximating polynomial of a useful filter because of no
band selectivity, in a compound form, they are used in
several PERAs which are discussed later. Further, a set
of Chebyshev polynomials Tm (w), m = 0, 1, ... n forms a
base, and, consequently, any polynomial Pn (w) of the
degree n can be expressed besides its natural power
series in w also in a form of an expansion into Chebyshev polynomials
n

/

m =0

b (m) w m =

n

/

a (m) Tm (w) .

(2)

m =0

An advantage of the Chebyshev representation of polynomials (2) is a dramatically smaller dynamic range
of the coefficients a (m) compared to the coefficients
b (m), especially for polynomials of the degree greater
than n . 40. It is also worth noting that a counterpart to Tn (w), namely the Chebyshev polynomial of
second kind
U n (w) =

sin 6(n + 1) arccos (w)@
, -1 # w # 1
sin ^arccos (w) h

(3)

T17 (w)

T17 (w)

represents no equiripple approximation (Fig. 3). Interestingly, the function 1 - w 2 U n (w) is equirippled, however, it is no longer a polynomial. A modification of U n (w),
namely the antiderivate of its compound form is useful
in the PERA of two constants in two disjoint intervals
which is outlined in Sec. V.

20
10
0
-10
-20
-1 -0.8 -0.6 -0.4 -0.2 0
w

0.2 0.4 0.6 0.8

1

Figure 3. chebyshev polynomial U 17 (w).

1
0.5
0
-0.5
-1

1
0.5
0
-0.5
-1

U17 (w )

Pn (w) =

Nex t milestone in the first period in the PER A
emerged in 1877. Another celebrity in the approximation theory, namely a student of P. L. Chebyshev and
also a representative of the Sankt Peterburg School of
Number Theory, E. I. Zolotarev (1847-1878), presented in [17] a generalization of the Chebyshev polynomial of first kind which represents a PERA of a single
constant in two disjoint intervals. Unlike Chebyshev,
Zolotarev's interest were equiripple polynomial and
rational approximations by itself. Zolotarev stated
four approximation problems and the above mentioned approximation in form of an equiripple polynomial Z p, q (w, \) is a solution of his first approximation
problem, for details see pp. 1-26 in [17]. Polynomial
Z p, q (w, \) is sketched in Fig. 5. By comparison of Fig. 2
bottom and Fig. 5, we can see that the third lobe from
the left in Fig. 2 is pulled up in Fig. 5 forming a main
lobe. This main lobe causes the selectivity of a related
filter. Zolotarev expressed the solution of his first approximation problem in terms of the Jacobi elliptic Eta
function H (z, \) [18]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
w
(a)

1

0

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
acos(w)/π = ωT/π
(b)

Figure 2. chebyshev polynomial T17 (w) of an argument w
(top) and of a transformed argument w (bottom).

second QUArTer 2019

Figure 4. egor Ivanovich Zolotarev.

Ieee cIrcUITs And sysTeMs MAgAZIne

61
IEEE Circuits and Systems Magazine - Q2 2019

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