PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 233–238 DOI: 10.2298/PIM1410233S

ON A CONVERGENT PROCESS OF BERNSTEIN

László Szili and Péter Vértesi

Abstract. Bernstein in 1930 defined a convergent interpolation process based on the roots of the Chebyshev polynomials. We prove a similar statement for certain Jacobi roots.

1. Introduction. Preliminary results

1.1. In 1930, Bernstein [1] (cf. [2], too) defined the following convergent interpolatory process on the roots of

T_{n}(x) = cos(n arccosx) = cosnϑ, −1 ⩽ x ⩽ 1, 0 ⩽ ϑ ⩽ π, n = 1, 2, …

(Chebyshev polynomials); the roots are

(1.1)  x_{kn} = cosϑ_{kn} = cos \frac{2k − 1}{2n} π, k = 1, 2, …, n; n = 1, 2, . . ..

Let l, q be natural numbers; for simplicity we suppose that n = 2lq. We divide the nodes into q rows as follows.

\begin{matrix}x_{1n} & x_{2n} & … & x_{2l,n} \\ x_{2l+1,n} & x_{2l+2,n} & … & x_{4l,n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{2l(q−1)+1,n} & x_{2l(q−1)+2,n} & … & x_{2lq,n} \end{matrix}

If f ∈ C (the set of continuous functions on [−1, 1]) and

ℓ_{kn} (T, x) = \frac{T_{n}(x)}{T′_{n}(x_{kn})(x − x_{kn})} , k = 1, 2, …, n; n = 1, 2, …

are the Lagrange fundamental polynomials based on (1.1) we define the following interpolatory polynomials Q_{nl} if l = 1, 2 and 3.

(1.2)  Q_{n1} (f, x) ≡ Q_{n1} (f) = {f_{1}(ℓ_{1} + ℓ_{2})} + {f_{3}(ℓ_{3} + ℓ_{4})}
+ {f_{5}(ℓ_{5} + ℓ_{6})} + ⋯+ {f_{n−1} (ℓ_{n−1} + ℓ_{n})},

2010 Mathematics Subject Classification: 41A05.

Key words and phrases: interpolation, Bernstein process, Jacobi roots.

Dedicated to Professor Giuseppe Mastroianni on the occasion of his retirement.

(1.3) Q_{n2} (f, x) ≡ Q_{n2} (f)
= {f_{1}(ℓ_{1} + ℓ_{4}) + f_{2}(ℓ_{2} − ℓ_{4}) + f_{3}(ℓ_{3} + ℓ_{4})}
+ {f_{5}(ℓ_{5} + ℓ_{8}) + f_{6}(ℓ_{6} − ℓ_{8}) + f_{7}(ℓ_{7} + ℓ_{8})}
+ {f_{9}(ℓ_{9} + ℓ_{12}) + f_{10}(ℓ_{10} − ℓ_{12}) + f_{11}(ℓ_{11} + ℓ_{12})} + …
+ {f_{n−3} (ℓ_{n−3} + ℓ_{n}) + f_{n−2} (ℓ_{n−2} − ℓ_{n}) + f_{n−1} (ℓ_{n−1} + ℓ_{n})},

Q_{n3} (f, x) ≡ Q_{n3} (f) (1.4)

= {f_{1}(ℓ_{1} + ℓ_{6}) + f_{2}(ℓ_{2} − ℓ_{6}) + f_{3}(ℓ_{3} + ℓ_{6}) + f_{4}(ℓ_{4} − ℓ_{6}) + f_{5}(ℓ_{5} + ℓ_{6})}
+ {f_{7}(ℓ_{7} + ℓ_{12}) + f_{8}(ℓ_{8} − ℓ_{12}) + f_{9}(ℓ_{9} + ℓ_{12})
+ f_{10}(ℓ_{10} − ℓ_{12}) + f_{11}(ℓ_{11} + ℓ_{12})} + ⋯
+ {f_{n−5} (ℓ_{n−5} + ℓ_{n}) + f_{n−4} (ℓ_{n−4} − ℓ_{n}) + f_{n−3} (ℓ_{n−3} + ℓ_{n})
+ f_{n−2} (ℓ_{n−2} − ℓ_{n}) + f_{n−1} (ℓ_{n−1} + ℓ_{n})}.

The definitions for l ⩾ 4 are analogous:

Q_{nl} (f, x) ≡ Q_{nl} (f) (1.5)

= {f_{1}(ℓ_{1} + ℓ_{2l}) + f_{2}(ℓ_{2} − ℓ_{2l}) + ⋯+ f_{2l−1} (ℓ_{2l−1} + ℓ_{2l})}+
+ {f_{2l+1} (ℓ_{2l+1} + ℓ_{4l}) + f_{2l+2} (ℓ_{2l+2} − ℓ_{4l}) + ⋯+ f_{4l−1} (ℓ_{4l−1} + ℓ_{4l})} + ⋯
+ {f_{n−(2l−1)} (ℓ_{n−(2l−1)} + ℓ_{n}) + ⋯+ f_{n−1} (ℓ_{n−1} + ℓ_{n})}.

You may consult with [1] or [2] (above f_{k} = f(x_{kn}) and ℓ_{k} ≡ ℓ_{kn} (T, x); moreover q is large enough).

If N = n + r, n = 2lq, 0 < r < 2l, the definition of Q_{Nl} is as follows (cf. [1] or [2])

Q_{Nl} (f) := Q_{nl} (f) + ∑_{k=n+1}^{N} f_{k}ℓ_{k}.

1.2. By the above definitions we have with e_{0}(x) ≡ 1

Q_{nl} (e_{0}, x) ≡ ∑_{k=1}^{n} ℓ_{kn} (T, x) ≡ 1, (1.6)

Q_{nl} (f, x_{kn}) = f(x_{kn}) if k ≠ 2l, 4l, …, 2lq, (1.7)

i.e. Q_{nl} interpolates at n − q = 2lq − q nodes. This number is “very close" to n if the (fixed) l is large enough while q (and n, too) tends to infinity, i.e., for large l our Q_{nl} is “very close" to the Lagrange interpolation L_{n}. However, Q_{nl} converges for every f ∈ C, when n → ∞ (cf. Proposition 1.1 and Theorem 2.1), which generally does not hold for L_{n}.

Later we use that (1.6) and (1.7) hold true forarbitrary point system.

ON A CONVERGENT PROCESS OF BERNSTEIN 235

1.3. In [1] Bernstein proved

Proposition 1.1. Let l be a fixed positive integer and f ∈ C. Then

lim_{n→∞} ∥f(x) − Q_{nl} (f, x)∥ = 0.

Above, ∥g(x)∥ = max_{|x|⩽1} |g(x)|, g ∈ C. Actually, he proved for N = n + r, too; the case when N = n + r demands only small technical changes in the proof.

1.4. The Bernstein process and its generalizations were exhaustively investigated by Kis (sometimes with coauthors). For more details we suggest the papers [6, 7, 8] and references therein.

2. The Bernstein process for Jacobi abscissas

2.1. The aim of this note is to prove a statement similar to Proposition 1.1 for Jacobi roots. Let the Jacobi polynomials P_{n}^{(α,β)} (x) be defined by

(1 − x)^{α} (1 + x)^{β}P_{n}^{(α,β)} (x) = \frac{(−1)^{n}}{2^{n}n!} \frac{d^{n}}{dx^{n}} \{(1 − x)^{α+n} (1 + x)^{β+n} \} (α, β > −1).

For the roots x_{kn}^{(α,β)} = cosϑ_{kn}^{(α,β)} , 0 < ϑ_{kn}^{(α,β)} < π, of P_{n}^{(α,β)} (x) we have

−1 < x_{nn}^{(α,β)} < x_{n−1,n}^{(α,β)} < ⋯< x_{1n}^{(α,β)} < 1.

Let

ℓ_{kn}^{(α,β)} (x) = \frac{P_{n}^{(α,β)} (x)}{P_{n}^{(α,β) ′} (x_{kn})(x − x_{kn})} .

For a fixed positive integer l, we define Q_{nl}^{(α,β)} (f, x) according to (1.2)–(1.5); now ℓ_{k} and f_{k} stand for ℓ_{kn}^{(α,β)} (x) and f(x_{kn}^{(α,β)} ), respectively. As we noticed we have the properties analogous to (1.6) and (1.7) for Q_{nl}^{(α,β)} (f, x), too.

2.2. We prove (compare with Vértesi [3] dealing with Lagrange interpolation) Theorem 2.1. Let l be a fixed positive integer, n = 2lq (q = 1, 2, …) and

f ∈ C. Then

(2.1) lim_{n→∞} ∥f(x) − Q_{nl}^{(α,β)} (f, x)∥ = 0

for any processes Q_{nl}^{(α,β)} supposing −1 < α, β < 0.5.

Our statement follows from the next more informative pointwise estimations (compare with the result in Vértesi [4] on Lagrange interpolation).

Theorem 2.2. Let l be fixed natural number. Then for arbitrary fixed α, β > −1 and f ∈ C

(2.2)  |Q_{nl}^{(α,β)} (f, x) − f(x)| = O(1) ∑_{i=1}^{n} ω( f; \frac{\sqrt{1 − x^{2}}}{n} i + \frac{i^{2}}{n^{2}} ) \frac{1}{i^{γ}}

uniformly in n and x ∈ [−1, 1], where γ = min(2; 1.5 − α; 1.5 − β). (ω(f; t) is the modulus of continuity of f(x).)

2.3. It is easy to get (2.1) using Theorem 2.2. Indeed, let

ε_{n} = \{ \begin{matrix}\frac{1}{n} logn & if − 1 < α, β ⩽ −0.5 \\ n^{δ−0.5} & if max(α, β) =: δ > −0.5. \end{matrix}

We have by (2.2)

∥Q_{nl}^{(α,β)} (f, x) − f(x)∥ = O(1)ω(f; ε_{n})

if f ∈ C, whence we obtain (2.1).

2.4. Another consequence of Theorem 2.2 is the following

Corollary 2.1. If −1 < α, β ⩽ −0.5 and ω(f; t) ∼ t^{ϱ} (0 < ϱ < 0.5) then for f ∈ C

|Q_{nl}^{(α,β)} (f, x) − f(x)| = O(1)[( \frac{1}{n} \sqrt{1 − x^{2}} )^{ϱ} + \frac{1}{n^{2ϱ}} ]

uniformly for n and |x| ⩽ 1.

This formula of Timan type can be obtained by simple calculation.

Other estimations showing the connections between the parameters γ ∈ (0, 2] and ϱ ∈ (0, 1] are as follows

|Q_{nl}^{(α,β)} (f, x) − f(x)|

= O(1) \{ \begin{matrix}( \frac{1}{n} \sqrt{1 − x^{2}} )^{ϱ} & +\{ \begin{matrix}n^{−2ϱ} & if 0 < ϱ < \frac{1}{2}(γ − 1), \\ n^{−2ϱ} logn & if ϱ = \frac{1}{2}(γ − 1), \\ n^{−γ+1} & if \frac{1}{2}(γ − 1) < ϱ < γ − 1; \end{matrix} \\ ( \frac{1}{n} \sqrt{1 − x^{2}} )^{ϱ} logn & + \frac{1}{n^{γ−1}} if ϱ = γ − 1, \\ \frac{(\sqrt{1 − x^{2}} )^{ϱ}}{n^{γ−1}} & + \frac{1}{n^{γ−1}} if γ − 1 < ϱ ⩽ 1 \end{matrix}

uniformly for n and |x| ⩽ 1. These formulae can be obtained by simple calculation.

2.5. It is interesting to compare (2.2) to

|H_{n}^{(α,β)} (f; x) − f(x)| = O(1) ∑_{i=1}^{n} ω( f; \frac{1}{n} \sqrt{1 − x^{2}} i + \frac{i^{2}}{n^{2}} ) i^{2η−1} (x ∈ [−1, 1])

where H_{n}^{(α,β)} (f; x) is the Hermite–Fejér interpolatory polynomial of degree ⩽ 2n−1 defined by H_{n}^{(α,β)} (f; x_{kn}^{(α,β)} ) = f(x_{kn}^{(α,β)} ), H_{n}^{(α,β) ′} (f; x_{kn}^{(α,β)} ) = 0 (k = 1, 2, …, n), f ∈ C and η = max(−0.5, α, β) (see [5, 2.1]).

3. Proof of Theorem 2.2

We apply the main idea from [4]. Let x = cosϑ, x ∈ [−1, 1], ϑ ∈ [0, π] and define the index j = j(n) by min_{1⩽k⩽n} |x − x_{kn}^{(α,β)} | = |x − x_{jn}^{(α,β)} |.

3.1. First let l = 1. By (1.5) and (1.6) we can write

(3.1) Q_{nl}^{(α,β)} (f, x) − f(x) = ∑_{k=1}^{q} {( f(x_{2k−1}^{(α,β)} ) − f(x)) ( ℓ_{2k−1}^{(α,β)} (x) + ℓ_{2k}^{(α,β)} (x)) }

= ∑_{k=1}^{cq} ⋯+ ∑_{k>cq} ⋯= ∑ _{I} + ∑ _{II} .

Now we use Lemma 4.1 of [3], which says the following: Let −1 < α, β and ε, η > 0 be fixed. If k ⩾ M, ϑ_{kn}^{(α,β)} ⩽ π − ε, then for any x ∈ [−1 + η, 1] we have

(3.2)  |ℓ_{kn}^{(α,β)} (x) + ℓ_{k+1,n}^{(α,β)} (x)| = O(1)|ℓ_{kn}^{(α,β)} (x)|[ \frac{1}{k} + \frac{k}{(k + j)(|k − j| + 1)} ]

uniformly in x and k.

We note that instead of ℓ̃_{kn}^{(α,β)} (x) of [3] one can write ℓ_{kn}^{(α,β)} (x). Moreover (3.2) obviously holds true if 1 ⩽ k ⩽ M (maybe with another O(1)).

From (3.1) with obvious short notations we have

(3.3) ∑ _{I} = O(1) ∑_{k=1}^{cq} |f(x_{2k−1} ) − f(x)|

× ( |ℓ_{2k−1} (x)|[ \frac{1}{2k − 1} + \frac{2k − 1}{(2k − 1 + j)(|2k − 1 − j| + 1)} ])

if α, β > −1 and ε, η > 0 are fixed.

By (3.3) we get as in [3]: If γ = min(2; 1.5 − α; 1.5 − β), then

(3.4) ∑_{k=1}^{n−1} |f(x_{2k−1} ) − f(x)| |ℓ_{2k−1} (x) + ℓ_{2k}(x)| = O(1) ∑_{i=1}^{n} ω( f; \frac{sinϑ}{n} i + \frac{i^{2}}{n^{2}} ) \frac{1}{i^{γ}}

uniformly in x ∈ [−1, 1]; see [3, 4.10], where ∑ |f − f_{k}| |ℓ_{k}k^{−1} | (which by (3.2), is analogous to ∑ |f − f_{k}| |ℓ_{k} + ℓ_{k+1} |) is estimated.

Let us remark that getting (3.4) we have to define J = [ϑ_{j−1,n}^{(α,β)} , ϑ_{j+1,n}^{(α,β)} ) and for r = 1, 2, …

I_{r} = [ϑ_{j−2^{r} ,n}^{(α,β)} , ϑ_{j−2^{r−1} ,n}^{(α,β)} ), K_{r} = [ϑ_{j+2^{r−1} ,n}^{(α,β)} , ϑ_{j+2^{r} ,n}^{(α,β)} )

instead of the definition (4.2) of [4].

From the above formulas we obtain our theorem for l = 1.

3.2. Now let l = 2. By (1.3) and (1.6) we get

(3.5) Q_{n2} (f) − f = {(f_{1} − f)(ℓ_{1} + ℓ_{4}) + (f_{2} − f)(ℓ_{2} − ℓ_{4}) + (f_{3} − f)(ℓ_{3} + ℓ_{4})}_{1}

+ {(f_{5} − f)(ℓ_{5} + ℓ_{8}) + (f_{6} − f)(ℓ_{6} − ℓ_{8}) + (f_{7} − f)(ℓ_{7} + ℓ_{8})}_{2} + · · ·.

In {⋯ }_{1},

|ℓ_{1} + ℓ_{4}| = |(ℓ_{1} + ℓ_{2}) − (ℓ_{2} + ℓ_{3}) + (ℓ_{3} + ℓ_{4})| ⩽ |ℓ_{1} + ℓ_{2}| + |ℓ_{2} + ℓ_{3}| + |ℓ_{3} + ℓ_{4}|

⩽ c ∑_{k=1}^{3} |ℓ_{k}(x)|[ \frac{1}{k} + \frac{k}{|k + j| (|k − j| + 1)} ]

⩽ c\{ |ℓ_{s}(x)|[ \frac{1}{s} + \frac{s}{|s + j| (|s − j| + 1)} ]\}_{s=k=1} .

Here we used (3.2) and that |ℓ_{k}(x) · ℓ_{k±m}^{−1} | ∼ 1 for any k whenever 0 ⩽ m ⩽ C.

Similar considerations are valid for the second term in {⋯ }_{1} by ℓ_{2} − ℓ_{4} = (ℓ_{2} + ℓ_{3}) − (ℓ_{3} + ℓ_{4}).

Taking into account that |f_{k} − f| ⩽ cω( \frac{sin ϑ}{n} i + \frac{i^{2}}{n^{2}} ) whenever i = |k − j| + m (see [4, (4.4)]; 0 ⩽ m ⩽ C), we get that

|\{⋯ \}_{1}| ⩽ cω( \frac{sinϑ}{n} i + \frac{i^{2}}{n^{2}} ) ( |ℓ_{s}(x)|[ \frac{1}{s} + \frac{s}{|s + j| (|s − j| + 1)} ])_{s=1} .

Using this last estimation and similar ones for {⋯ }_{2}, {⋯ }_{3}, …, we can get (3.4). If l > 2, the argument is similar. We may omit the further details. □

Acknowledgement. The authors thank the referee for the careful work. Some of the referee’s advices are in the paper.

References

1. S. Bernstein, Sur une modification de la formule d’interpolation de Lagrange (French), Communications Kharkow 5 (1932), 49–57. [Collected Works. Vol. II. The constructive theory of functions [1931–1953] (Russian), Izdat. Akad. Nauk SSSR, Moscow, 1954, 130–140.]

2. , Sur une formule d’interpolation (French), C. R.191 (1930), 635–637. [Collected Works. Vol. I. The constructive theory of functions [1905–1930] (Russian), Izdat. Akad. Nauk SSSR, Moscow, 1952, 520–522.]

3. P. Vértesi, One-sided convergence conditions for Lagrange interpolation based on the Jacobiroots, Acta Sci. Math. 45 (1983), 419–428.

4. , Lagrange interpolation for continuous functions of bounded variation, Acta Math. Acad. Sci. Hung. 35 (1980), 23–31.

5. , Notes on the Hermite–Fejér interpolation based on the Jacobi abscissas, Acta Math. Acad. Sci. Hung. 24 (1973), 233–239.

6. O. Kis, Ho Tho Kau, Investigation of a certain interpolation process. III, (Russian) Acta Math. Acad. Sci. Hung. 28 (1976), 157–176.

7. O. Kis, J. Szabados, On some de la Vallé-Poussin type discrete linear operators, Acta Math. Hung. 47 (1986), 239–260.

8. T. Hermann, O. Kis, P. Vértesi, On certain rational interpolatory operators, Acta Math. Hung. 49 (1987), 29–49.

Department of Numerical Analysis, Loránd Eötvös University

Budapest, Hungary

  szili@caesar.elte.hu

Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences

Budapest, Hungary

  veter@renyi.hu