View of Approximation on the regular hexagon

J. Numer. Anal. Approx. Theory, vol. 49 (2020) no. 2, pp. 138–154 ictp.acad.ro/jnaat

TRIGONOMETRIC APPROXIMATION ON THE HEXAGON

ALI GUVEN^†

Dedicated to the memory of Dr. Figen Kiraz

Abstract. The degree of trigonometric approximation of continuous functions, which are periodic with respect to the hexagon lattice, is estimated in uniform and H¨older norms. Approximating trigonometric polynomials are matrix means of hexagonal Fourier series.

2020 Mathematics Subject Classification. 41A25, 41A63, 42B08.

Keywords. Hexagonal Fourier series, H¨older class, matrix mean, regular hexagon.

1. INTRODUCTION

Approximation problems of functions of several variables defined on cubes of the Euclidean space are usually studied by assuming that the functions are periodic in each of their variables (see, for example [20, §§ 5.3, 6.3], [23, vol. II, ch. XVII], [22, part 2], [15], [16] and [17]). But, we need other defini- tions of periodicity to study approximation problems on non-tensor product domains, for example on hexagonal domains of R². The periodicity defined by lattices is the most useful one.

In the Euclidean plane R², besides the standard lattice Z² and the rect- angular domain [−¹₂,¹₂)², the simplest lattice is the hexagon lattice and the simplest spectral set is the regular hexagon. The hexagon lattice has impor- tance, since it offers the densest packing of the plane with unit circles. In this section we give basic information about hexagonal lattice and hexagonal Fourier series. More detailed information can be found in [11] and [21].

The generator matrix and the spectral set of the hexagonal latticeHZ² are given by

H =

√3 0

−1 2

!

and

ΩH =ⁿ(x1, x2)∈R²:−1≤x2,

√3

2 x1±¹₂x2 <1^o.

†Department of Mathematics, Balikesir University, 10145 Balikesir, Turkey, e-mail:

[email protected].

It is more convenient to use the homogeneous coordinates (t1, t2, t3) that satis- fiest₁+t₂+t₃ = 0. As it is pointed out in [21], using homogeneous coordinates reveals symmetry in various formulas. If we set

t1 :=−^x₂² +^√3x₂ ¹, t2 :=x2, t3:=−^x₂² −

√3x1

2 , the hexagon ΩH becomes

Ω =ⁿ(t₁, t₂, t₃)∈R³ :−1≤t₁, t₂,−t₃<1, t₁+t₂+t₃ = 0^o,

which is the intersection of the plane t₁+t₂+t₃= 0 with the cube [−1,1]³. We use bold letterst for homogeneous coordinates and we set

R³_H =ⁿt= (t₁, t₂, t₃)∈R³ :t₁+t₂+t₃ = 0^o, Z³_H :=Z³∩R³_H. A function f :R² →Cis called H-periodic if

f(x+Hk) =f(x)

for all k∈Z² and x∈R².If we define t≡s (mod 3) as t₁−s₁ ≡t₂−s₂≡t₃−s₃ (mod 3)

for t = (t1, t2, t3), s = (s1, s2, s3)∈ R³_H, it follows that the function f is H-periodic if and only if f(t) =f(t+s) whenever s≡0 (mod 3),and

Z

Ω

f(t+s)dt=^Z

Ω

f(t)dt s∈R³H

forH-periodic integrable functionf [21].

L²(Ω) becomes a Hilbert space with respect to the inner product hf, gi_H := _|Ω|^{1 Z}

Ω

f(t)g(t)dt, where|Ω|denotes the area of Ω.The functions

φ_j(t) :=e^{2πi3 hj,ti} (t∈R³H),

where hj,ti is the usual Euclidean inner product of j and t, are H-periodic, and by a theorem of B. Fuglede the set

nφ_j:j∈Z³H

o

becomes an orthonormal basis ofL²(Ω) [3] (see also [11]). For every natural number n,we define a subset of Z³_H by

Hn:=ⁿj= (j₁, j₂, j₃)∈Z³H :−n≤j₁, j₂, j₃ ≤n^o. The subspace

H_n:= spanφ_j:j∈Hn (n∈N)

has dimension #Hn = 3n²+ 3n+ 1, and its members are called hexagonal trigonometric polynomials of degree n.

The hexagonal Fourier series of anH-periodic function f ∈L¹(Ω) is

(1) f(t)∼ ^X

j∈Z³H

fb_jφ_j(t), where

fbj = _|Ω|^{1 Z}

Ω

f(t)φj(t)dt (j∈Z³_H).

The nth hexagonal partial sum of the series (1) is defined by S_n(f) (t) := ^X

j∈Hn

fb_jφ_j(t) (n∈N). It is clear that

Sn(f) (t) = _|Ω|^{1 Z}

Ω

f(t−u)Dn(u)du, whereDn is the Dirichlet kernel, defined by

Dn(t) := ^X

j∈Hn

φj(t).

It is known that the Dirichlet kernel can be expressed as (2) D_n(t) = Θn(t)−Θn−1(t) (n≥1), where

(3) Θn(t) := ^{sin(n+1)(t31−t2)πsin(n+1)(t32−t3)πsin(n+1)(t33−t1)π}

sin^(t1−t₃^2)πsin^(t2−t₃^3)πsin^(t3−t₃^1)π

fort= (t1, t2, t3)∈R³H [11].

The degree of approximation ofH-periodic continuous functions by Ces`aro, Riesz and N¨orlund means of their hexagonal Fourier series was investigated by us in [4], [5], [6], [7] and [8]. In this paper, we studied the degree of approxi- mation by matrix means of hexagonal Fourier and we obtained generalizations of previous results.

2. MAIN RESULTS

Let C_H(Ω) be the Banach space of complex valued H-periodic continuous functions defined on R³_H,whose norm is the uniform norm:

kfk_C

H(^Ω) := sup

n|f(t)|:t∈Ω^o.

The modulus of continuity of the functionf ∈CH(Ω) is defined by ω_H(f, δ) := sup

0<ktk≤δ

kf −f(·+t)k_C

H(Ω), where

ktk:= max{|t₁|,|t₂|,|t₃|}

fort= (t1, t2, t3)∈R³_H. ω_H(f,·) is a nonnegative and nondecreasing function, and satisfies

(4) ωH(f, λδ)≤(1 +λ)ωH(f, δ) forλ >0 [21].

A functionf ∈CH(Ω) is said to belong to the H¨older spaceH^α(Ω) (0< α≤1) if

Λ^α(f) := sup

t6=s

|f(t)−f(s)|

kt−sk^α <∞.

H^α(Ω) becomes a Banach space with respect to the H¨older norm kfk_Hα(^Ω) :=kfk_C

H(^Ω) + Λ^α(f).

LetA= (a_n,k) (n, k= 0,1, . . .) be a lower triangular infinite matrix of real numbers. TheA-transform of the sequence (S_n(f)) of partial sums the series (1) is defined by

T_n^(A)(f) (t) :=

n

X

k=0

a_n,kS_k(f) (t) (n∈N).

We shall assume that the lower triangular matrix A = (an,k) satisfies the conditions

(5) a_n,k ≥0 (n= 0,1, . . . ,0≤k≤n), (6) an,k+1≥an,k (n= 0,1, . . . ,0≤k≤n−1), and

(7)

n

X

k=0

a_n,k = 1 (n= 0,1, . . .). Also we use the notations

A^∗_n,k:=

n

X

ν=k

a_n,ν (0≤k≤n), A^∗_n(u) :=A^∗_n,n−[u] ,a^∗_n(u) :=a_n,n−[u] (u >0), where [u] is the integer part of u.

Hereafter, the relation x . y will mean that there exists an absolute con- stant c >0 such thatx≤cy holds for quantitiesx and y.

Theorem 1. Let f ∈CH(Ω)and let A= (an,k) (n, k= 0,1, . . .) be a lower triangular infinite matrix of real numbers which satisfies(5),(6)and(7). Then the estimate

(8) f−T_n^(A)(f)

CH(^Ω) .log (n+ 1) (

ωH(f, an,n) +^Xn

k=1

ωH(^f,1k)

k A^∗_n,n−k )

holds.

Corollary 2. If f ∈ H^α(Ω) (0< α≤1) and A = (a_n,k) (n, k= 0,1, . . .) as in Theorem 1, then

(9) f −T_n^(A)(f)_C

H(^Ω) .log (n+ 1) (

a^α_n,n+

n

X

k=1 A^∗_n,n−k

k^1+α

) .

Theorem 3. Let 0 ≤ β < α ≤ 1, f ∈ H^α(Ω) and let A = (an,k) (n, k= 0,1, . . .) be a lower triangular infinite matrix of real numbers which satisfies (5), (6) and (7). Then

(10)

f −T_n^(A)(f)_Hβ(^Ω).log (n+ 1)

n

X

k=1 A^∗_n,n−k

k

!_α^β 



 a^α−β_n,n +

n

X

k=1 A^∗_n,n−k

k^1+α

!1−^β_α



 . Analogues of these results were obtained in [10], [13], [14] and [2] for matrix means of trigonometric Fourier series of 2π-periodic continuous functions.

3. PROOFS OF MAIN RESULTS

Proof of Theorem 1. It is clear that

f(t)−T_n^(A)(f) (t) ≤ _|Ω|¹ Z

Ω

|f(t)−f(t−u)|

n

X

k=0

an,kDk(u)

du

. _|Ω|¹ Z

Ω

ω_H(f,kuk)

n

X

k=0

a_n,kD_k(u)

du.

If we set Θ−1(u) := 0,by (2) we get Z

Ω

ωH(f,kuk)

n

X

k=0

an,kDk(u)

du=

=^Z

Ω

ω_H(f,kuk)

n

X

k=0

a_n,kΘk(u)−Θk−1(u)

du.

The function

t→ωH(f,ktk)

n

X

k=0

an,k(Θk(t)−Θk−1(t))

is symmetric with respect to variablest1, t2 and t3, where t= (t1, t2, t3)∈Ω. Hence it is sufficient to estimate the integral over the triangle

∆ :=ⁿt= (t1, t2, t3)∈R³H : 0≤t1, t2,−t₃ ≤1^o

={(t1, t2) :t1≥0, t2 ≥0, t1+t2 ≤1},

which is one of the six equilateral triangles in Ω.By considering the formula (3), we obtain

Z

∆

ω_H(f,ktk)

n

X

k=0

a_n,k(Θk(t)−Θk−1(t))

dt=

=^Z

∆

ω_H(f, t₁+t₂)

n

X

k=0

a_n,k

sin^(k+1)(t₃^1−t2)πsin^(k+1)(t₃^2−t3)πsin^(k+1)(t₃^3−t1)π sin^(t1−t₃^2)πsin^(t2−t₃^3)πsin^(t3−t₃^1)π

−^sin

k(t1−t2)π

3 sin^k(t2−t₃ ^3)πsin^k(t3−t₃ ^1)π sin^(t1−t₃^2)πsin^(t2−t₃^3)πsin^(t3−t₃^1)π

dt.

If we use the change of variables

(11) s₁ := ^t1−t₃ ³ = ^2t1₃^+t2, s₂ := ^t2−t₃ ³ = ^t1+2t₃ ², the integral becomes

3^Z

∆e

ωH(f, s1+s2)

n

X

k=0

an,k

sin((k+1)(s1−s2)π) sin((k+1)s2π) sin((k+1)(−s1π)) sin((s1−s2)π) sin(s2π) sin(−s1π)

− ^{sin(k(s1−s2)π) sin(ks2}π) sin(k(−s₁π)) sin((s1−s₂)π) sin(s2π) sin(−s₁π)

ds₁ds₂, where∆ is the image of ∆ in the plane, that is^e

∆ :=e {(s₁, s₂) : 0≤s₁ ≤2s₂, 0≤s₂ ≤2s₁, s₁+s₂ ≤1}.

Since the integrated function is symmetric with respect tos₁ands₂,estimating the integral over the triangle

∆^∗ :=ⁿ(s1, s2)∈∆ :^e s1 ≤s2

o={(s1, s2) :s1 ≤s2 ≤2s1, s1+s2 ≤1}, which is the half of∆^e,will be sufficient.The change of variables

(12) s₁ := ^u1−u₂ ², s₂ := ^u1+u₂ ² transforms the triangle ∆^∗ to the triangle

Γ :=(u1, u2) : 0≤u2 ≤ ^u₃¹, 0≤u1≤1 . Thus we have to estimate the integral

In:=^Z

Γ

ωH(f, u1)

n

X

k=0

an,kD_k^∗(u1, u2)

du1du2, where

D^∗_k(u₁, u₂) = sin((k+1)(u2)π) sin (k+1)^u1+₂^u2π

sin (k+1) ^u1−u₂ ²π

sin((u2)π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π

−^{sin(k(u2)π) sin k}

u1+u2 2 π

sin k ^u1−u₂ ²π

sin((u2)π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π .

By elementary trigonometric identities, we obtain

(13) D^∗_k(u1, u2) =D^∗_k,1(u1, u2) +D_k,2^∗ (u1, u2) +D_k,3^∗ (u1, u2), where

D^∗_k,1(u₁, u₂) := 2 cos(k+¹₂)u₂π^sin(¹2u2π)^{sin (k+1)u1+}2^u2π

sin (k+1)^u1−u₂ ²π

sin(u2π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π , D^∗_k,2(u₁, u₂) := 2 cos(k+¹₂)^u1+u₂ ²π^{sin(ku2π) sin}

1 2

u1+u2 2 π

sin (k+1)^u1−u₂ ²π

sin(u2π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π and

D_k,3^∗ (u1, u2) := 2 cos(k+¹₂)^u1−u₂ ²π^{sin(ku2π) sin k}

u1+u2 2 π

sin ¹₂^u1−u₂ ²π

sin(u2π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π . We partition the triangle Γ as Γ = Γ1∪Γ2∪Γ3,where

Γ1 :={(u₁, u₂)∈Γ :u₁ ≤a_n,n},

Γ2 :=(u1, u2)∈Γ :u1 ≥an,n, u2≤ ^an,n₃ , Γ3 :=(u1, u2)∈Γ :u1 ≥an,n, u2≥ ^an,n₃ . Hence In=In,1+In,2+In,3,where

I_n,j :=^Z

Γj

ω_H(f, u₁)

n

X

k=0

a_n,kD_k^∗(u₁, u₂)

du₁du₂ (j= 1,2,3). We need the inequalities

(14)

sinnt

sint

≤n, (n∈N), and

(15) sint≥ _π²t, 0≤t≤ ^π₂

to estimate integrals I_n,1, I_n,2 and I_n,3.

We divide Γ1 into three parts to estimateIn,1 as follows:

Γ⁰1:=ⁿ(u₁, u₂)∈Γ1 :u₁ ≤ _n+1^{1 o},

Γ⁰⁰1 :=ⁿ(u₁, u₂)∈Γ1 :u₁ ≥ _n+1¹ , u₂ ≤ _3(n+1)^{1 o}, Γ⁰⁰⁰₁ :=ⁿ(u1, u2)∈Γ1 :u1 ≥ _n+1¹ , u2 ≥ _3(n+1)^{1 o}. Hence we have

In,1 =





 Z

Γ⁰₁

+^Z

Γ⁰⁰₁

+^Z

Γ⁰⁰⁰₁





ωH(f, u1)

n

X

k=0

an,kD_k^∗(u1, u2)

du1du2.

By (14),

Z

Γ⁰₁

ωH(f, u1)

n

X

k=0

an,kD^∗_k(u1, u2)

du1du2 .

. Z

Γ⁰₁

ω_H(f, u₁)

n

X

k=0

(k+ 1)²a_n,k

!

du₁du₂

≤(n+ 1)²

1 3(n+1)

Z

0

1 n+1

Z

3u2

ωH(f, u1)du1du2

≤ω_Hf,_n+1¹ ≤ω_H(f, a_n,n). By (15),

Z

Γ⁰⁰₁

ω_H(f, u₁)

n

X

k=0

a_n,kD^∗_k,1(u₁, u₂)

du₁du₂ .

.

1 3(n+1)

Z

0 an,n

Z

1 n+1

ωH(f,u1)

u²₁ du₁du₂ ≤ω_H(f, a_n,n)

1 3(n+1)

Z

0 an,n

Z

1 n+1

1

u²₁du₁du₂ ≤ω_H(f, a_n,n). (14) and (15) gives forj= 1,2,

Z

Γ⁰⁰₁

ωH(f, u1)

n

X

k=0

a_n,kD_k,j^∗ (u1, u2)

du1du2.

. (n+ 1)

1 3(n+1)

Z

0 an,n

Z

1 n+1

ωH(f,u1)

u1 du₁du₂≤(n+ 1)ω_H(f, a_n,n)

1 3(n+1)

Z

0 an,n

Z

1 n+1

1

u1du₁du₂

≤ log (n+ 1)ω_H(f, an,n). Since

sin 2x+ sin 2y+ sin 2z=−4 sinxsinysinz forx+y+z= 0,we also get the expression

(16) D^∗_k(u₁, u₂) =H_k,1(u₁, u₂) +H_k,2(u₁, u₂) +H_k,3(u₁, u₂), where

H_k,1(u₁, u₂) := ¹₂ cos((2k+1)u2π) sin ^u1+₂^u2π

sin ^u1−u₂ ²π, H_k,2(u₁, u₂) :=−¹₂ ^{cos (2k+1)}

u1+u2 2 π

sin(u2π) sin ^u1−u₂ ²π,

H_k,3(u₁, u₂) := ¹₂ ^{cos (2k+1)u1}

−u2 2 π

sin(u2π) sin ^u1+₂^u2π. By the method used in [12, p. 179] we get

(17)

n

X

k=0

an,kcos (2k+ 1)t

.A^∗_n¹_t+_sin¹_ta^∗_n¹_t (0< t < π), and

(18)

n

X

k=0

an,kcos (2k+ 1)t

.A^∗_n¹_t 0< t≤ ^π₂. By aim of (18) and (15) we obtain

(19)

n

X

k=0

a_n,kH_k,1(u₁, u₂)

. _u¹2 1

A^∗_nπu¹

2

and (20)

n

X

k=0

a_n,kH_k,3(u₁, u₂)

. _u1¹_u2A^∗_nπu³

1

,

where both of for u1 and u2 are away from the origin. Also, for such u1 and u₂,it follows from (17),(15) and from the fact

sin ^u1₂^π.sin^(u1+u₂ ^2)π that

(21)

n

X

k=0

a_n,kH_k,2(u₁, u₂)

. _u1¹_u2A^∗_nπu³

1

.

Using (19) and the inequality (22) ω_H(f, δ₂)

δ2

≤2ω_H(f, δ₁)

δ1 (δ₁ < δ₂),

which is obtained from (4), and considering that the function A^∗_n is nonde- creasing yield

Z

Γ⁰⁰⁰₁

ωH(f, u1)

n

X

k=0

an,kHk,1(u1, u2)

du1du2.

.

an,n 3

Z

1 3(n+1)

an,n

Z

3u2

ωH(f,u1)

u²₁ A^∗_nπu¹

2

du1du2

≤2

an,n 3

Z

1 3(n+1)

an,n

Z

3u2

ωH(f,3u2)

3u1u2 A^∗_nπu¹

2

du₁du₂

= ²₃

an,n 3

Z

1 3(n+1)

ωH(f,3u2)

u2 log^a_3u^n,n₂A^∗_nπu¹

2

du₂

≤log ((n+ 1)a_n,n)

an,n

Z3

1 3(n+1)

ω_H(f,3u2)

u2 A^∗_nπu¹

2

du₂

≤log (n+ 1)

an,n

Z3

1 3(n+1)

ω_H(f,3u2)

u2 A^∗_nπu¹

2

du₂ = log (n+ 1)

3 π(n+1)

Z

3 π

ωH(f,_πt³)

t A^∗_n(t)dt

= log (n+ 1)

n

X

k=1









3 π(k+1)

Z

3 πk

ωH(^f,πt³)

t A^∗_n(t)dt









≤log (n+ 1)^Xn

k=1 ωH(f,_k¹)

k A^∗_n(k+ 1). Forj = 2,3,by (20) and (21)

Z

Γ⁰⁰⁰₁

ω_H(f, u₁)

n

X

k=0

a_n,kH_k,j(u₁, u₂)

du₁du₂ .

.

an,n

Z

1 n+1

u1

Z3

1 3(n+1)

ωH(f,u1)

u1u2 A^∗_nπu³

1

du2du1

=

an,n

Z

1 n+1

ωH(f,u1)

u1 log ((n+ 1)u1)A^∗_nπu³

1

du1

≤log ((n+ 1)a_n,n)

an,n

Z

1 n+1

ωH(f,u1)

u1 A^∗_nπu³

1

du₁ =

= log ((n+ 1)a_n,n)

3 π(n+1)

Z

3 π

ωH(^f,πt³)

t A^∗_n(t)dt

≤log (n+ 1)

n

X

k=1

ωH(^f,k¹)

k A^∗_n(k+ 1). Thus we get the estimate

(23) I_n,1 .log (n+ 1) (

ω_H(f, a_n,n) +

n

X

k=1

ωH(^f,k¹)

k A^∗_n(k+ 1) )

. We divide Γ2 into two domains as

Γ⁰₂:=ⁿ(u1, u2)∈Γ2 :u2 ≤ _3(n+1)^{an,n o}, Γ⁰⁰2 :=ⁿ(u1, u2)∈Γ2 :u2 ≥ _3(n+1)^{an,n o} to estimate In,2.By (15) and (22),

Z

Γ⁰₂

ωH(f, u1)

n

X

k=0

a_n,kD_k,1^∗ (u1, u2)

du1du2 .

.

an,n 3(n+1)

Z

0 1

Z

an,n

ωH(f,u1)

u²₁ du1du2≤2^ωH(f,a_an,n^n,n)

an,n 3(n+1)

Z

0 1

Z

an,n

1

u1du1du2

= 2^ωH(f,a_an,n^n,n)log_an,n¹ _3(n+1)^an,n ≤log (n+ 1)ωH(f, an,n). (14),(15) and (22) yield

Z

Γ⁰₂

ω_H(f, u1)

n

X

k=0

a_n,kD^∗_k,j(u1, u2)

du1du2 .

. (n+ 1)

an,n 3(n+1)

Z

0 1

Z

an,n

ωH(f,u1)

u1 du1du2≤2 (n+ 1)^ωH(f,a_an,n^n,n)

an,n 3(n+1)

Z

0 1

Z

an,n

du1du2

≤ ωH(f, an,n)

forj= 2,3.Since (15) implies|H_k,1(u₁, u₂)|. _u¹2

1 for (u₁, u₂)∈Γ⁰⁰₂,we get Z

Γ⁰⁰₂

ωH(f, u1)

n

X

k=0

an,kHk,1(u1, u2)

du1du2 .