Approximation by generalized bivariate Kantorovich ... - Springer Link

J Anal https://doi.org/10.1007/s41478-018-0085-6 ORIGINAL RESEARCH PAPER

Approximation by generalized bivariate Kantorovich sampling type series Sathish Kumar Angamuthu1 • Devaraj Ponnaian2

Received: 23 December 2017 / Accepted: 12 May 2018 Ó Forum D’Analystes, Chennai 2018

Abstract The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators ðKwu f Þw [ 0 : First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family ðKwu f Þw [ 0 : Finally, we give some examples of kernels such as box spline kernels and Bochner–Riesz kernel to which the theory can be applied. Keywords Bivariate Kantorovich sampling operators Voronovskaja type formula Rate of convergence Modulus of smoothness Mathematics subject classification 94A20 41A25 26A15

1 Introduction In recent years, theory of generalized sampling series is an attractive topic in approximation theory due to its wide range of applications, especially in signal and image processing. The theory of generalized sampling series was first initiated by & Sathish Kumar Angamuthu [email protected] Devaraj Ponnaian [email protected] 1

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, Nagpur 440010, India

2

School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Maruthamala, Vithura, Thiruvananthapuram 695551, India

123

S. K. Angamuthu, D. Ponnaian

Butzer and his school [13, 14]. After this many researchers have studied in this direction and obtained the convergence properties of these operators (cf. [7, 28, 29, 31] etc.). Also, the Kantorovich type generalizations of the generalized sampling operators were introduced by Butzer and his school (see e.g. [14, 17, 23, 34] etc). In [10], the authors introduced the sampling Kantorovich operators and studied their rate of convergence in the general settings of Orlicz spaces. Danilo and Vinti [21] obtained the rate of approximation for the family of sampling Kantorovich operators in the uniform norm for uniformly continuous bounded functions belonging to Lipschitz classes and for functions in Orlicz spaces. Also, the nonlinear version of sampling Kantorovich operators have been studied in [20, 35]. Let u : R2 ! R be a suitable kernel function. Then, the two dimensional generalized sampling series of a function f : R2 ! R is defined by 1 1 X X k j k j u ; u w x ðTw f Þðx; yÞ ¼ ;w y f ; w w w w k¼1 j¼1 where w 2 N and ðx; yÞ 2 R2 : These operators have great importance in the development of models for signal recovery. These type of operators have been studied by many authors (cf. [9, 15, 16, 25] etc.). Altomare and Leonessa [4] considered a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. Such operators include the Kantorovich operators as a particular case. Later, in order to obtain an approximation process for spaces of locally integrable functions on unbounded intervals, Altomare et. al. introduced and studied the generalized Sza´sz–Mirakjan–Kantorovich operators in [5]. Also in [19], the authors obtained some qualitative properties and an asymptotic formula for such a sequence of operators. Motivated by the above works, we consider the bivariate generalized Kantorovich sampling series. Let fak gk2Z ; fbk gk2Z ; fcj gj2Z and fdj gj2Z be sequences of real numbers such that for every k 2 Z, ak \bk and for every j 2 Z; cj \dj : In this paper, we analyse the approximation properties of the following type of bivariate generalized Kantorovich sampling series: For f 2 CðR2 Þ (The class of all uniformly continuous and bounded functions on R2 ), ðKwu f Þðx; yÞ

1 1 X X

w2 ¼ uðwx k; wy jÞ D D k¼1 j¼1 ak cj

Z

kþbk w kþak w

Z

jþdj w

jþcj w

f ðu; vÞdvdu;

where Dak ¼ bk ak and Dcj ¼ dj cj : The purpose of this study is to obtain some approximation properties of the Kantorovich type bivariate generalization of sampling operators ðKwu f Þw [ 0 : First, we give the pointwise convergence theorem and Voronovskaja type asymptotic theorem and then we obtain the degree of approximation and quantitative version of Voronovskaja type theorem of these operators. At the end we give some examples

123

Approximation by generalized bivariate\ldots

of kernels such as Box spline kernels and Bochner–Riesz kernel to which the theory can be applied.

2 Preliminaries Let us denote by C 0 :¼ C 0 ðR2 Þ ¼ CðR2 Þ the space of all uniformly continuous and bounded functions f : R2 ! R with the usual supnorm kf k1 and that by Cn :¼ C n ðR2 Þ the space of all n-times continuously differentiable functions for which ojkj f 2 C0 ; oxk1 oyk2 for k ¼ ðk1 ; k2 Þ with jkj ¼ k1 þ k2 ¼ n 1. For d [ 0 and ðx; yÞ 2 R2 ; Ud ðx; yÞ denotes the ball of radius d centered at (x, y), that is Ud ðx; yÞ ¼ fðu; vÞ 2 R2 : ðx uÞ2 þ ðy vÞ2 d2 g: Let u 2 C 0 be fixed. For any m 2 N0 ¼ N [ f0g; h ¼ ðh1 ; h2 Þ 2 N20 with jhj ¼ h1 þ h2 ¼ m; we define the algebraic moments as mmðh1 ;h2 Þ ðuÞ :¼

1 1 X X

uðx k; y jÞðk xÞh1 ðj yÞh2

k¼1 j¼1

and the absolute moments by m Mðh ðuÞ :¼ sup 1 ;h2 Þ

1 1 X X

juðx k; y jÞj jk xjh1 jj yjh2

2

ðx;yÞ2R k¼1 j¼1

and m Mm ðuÞ :¼ max Mðh ðuÞ: 1 ;h2 Þ jhj¼v

Note that for l; m 2 N0 with l\m; Mm ðuÞ\ þ 1 implies Ml ðuÞ\ þ 1: Indeed for l\m; we have 1 1 X X juðx k; y jÞjjk xjh1 jj yjh2 k¼1 j¼1

¼

X

X

juðx k; y jÞjjk xjh1 jj yjh2

ðk;jÞ2U1 ðx;yÞ

þ

X

X

juðx k; y jÞjjk xjh1 jj yjh2 ¼ I1 þ I2 :

ðk;jÞ62U1 ðx;yÞ

It is easy to see that I1 4kuk1 : Now, we obtain I2 : Since ðk; jÞ 62 U1 ðx; yÞ; we have jk xj [ 1=2 or jj yj [ 1=2: Now, we get

123


J2

X

X

juðx k; y jÞj

jkxj [ 1=2

þ

X

X

jk xjmh2 jj yjh2 jk xjml

juðx k; y jÞj

jjyj [ 1=2

jk xjh1 jj yjmh1 2mlþ1 Mm ðuÞ: jj yjml

When u has compact support, we immediately have that Mm ðuÞ\ þ 1 for every m 2 N0 : We suppose that the following assumptions hold: (i)

for every ðx; yÞ 2 R2 ; we have 1 1 X X

uðx k; y jÞ ¼ 1;

k¼1 j¼1

(ii)

M2 ðuÞ\ þ 1 and there holds X X lim juðx k; y jÞjððk xÞ2 þ ðj yÞ2 ÞÞ ¼ 0 R!1

(iii)

ðk;jÞ62UR ðx;yÞ

uniformly with respect to ðx; yÞ 2 R2 ; for every ðx; yÞ 2 R2 ; m1ð1;0Þ ðu; x; yÞ :¼ m1ð0;1Þ ðu; x; yÞ :¼ m2ð2;0Þ ðu; x; yÞ :¼ m2ð0;2Þ ðu; x; yÞ :¼ m2ð1;1Þ ðu; x; yÞ :¼

1 1 X X k¼1 j¼1 1 1 X X k¼1 j¼1 1 1 X X k¼1 j¼1 1 1 X X k¼1 j¼1 1 1 X X

uðx k; y jÞðk xÞ ¼ 0 uðx k; y jÞðj yÞ ¼ 0 uðx k; y jÞðk xÞ2 uðx k; y jÞðj yÞ2 uðx k; y jÞðk xÞðj yÞ:

k¼1 j¼1

3 Main results First, we give a pointwise convergence theorem at continuity points of the function f for the bivariate operators. Theorem 3.1 For f 2 L1 ðR2 Þ; limw!1 ðKwu f Þðx; yÞ ¼ f ðx; yÞ at every point (x, y) of continuity of f. Moreover, if the function is uniformly continuous and bounded on R2 ; then limw!1 kKwu f f k ¼ 0:

123


Let [ 0 be fixed. By the continuity of f at the point (x, y), there exists qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d [ 0 such that jf ðu; vÞ f ðx; yÞj ; whenever ðu xÞ2 þ ðv yÞ2 d: Now, we have jðKwu f Þðx; yÞ f ðx; yÞj

Proof

X

X ðwk ;wj Þ2Ud ðx;yÞ 2

þ

X

w2 juðwx k; wy jÞj Dak Dcj

X ðwk ;wj Þ62Ud ðx;yÞ

Z

kþbk w

Z

jþcj w

kþak w


Z

jþdj w

kþbk w kþak w

Z

jf ðu; vÞ f ðx; yÞjdvdu jþdj w

jþcj w

jf ðu; vÞ f ðx; yÞjdvdu

2

¼ I1 þ I2 : By the continuity of f at the point (x, y), we get X X Il juðwx k; wy jÞj :M0 ðuÞ; ðwk ;wj Þ2Ud ðx;yÞ 2

for sufficiently large w [ 0: Let d [ 0 be fixed. If ðwk ; wj Þ 62 Ud ðx; yÞ then 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 wd ðwx kÞ þ ðwy jÞ [ 2 : Hence, we get I2

8kf k1 X X juðwx k; wy jÞjðwx kÞ2 þ ðwy jÞ2 ! 0 as w ! 1 w 2 d2 k j ð ; Þ62U ðx;yÞ ww

d

and hence the first part of the theorem follows. The second part can be proved similarly. h Remark 3.1 Since functions in L1 ðR2 Þ are locally Lebesgue integrable, the generalized Kantorovich sampling series used in the above theorem are well defined for f 2 L1 ðR2 Þ: Next, we obtain a Voronovskaja type theorem for the bivariate Kantorovich sampling operators. of of Theorem 3.2 Let f 2 L1 ðR2 Þ be such that ox ; oy exist at (x, y) and let fak gk2Z ; fbk gk2Z ; fcj gj2Z and fdj gj2Z be bounded sequences such that ak þ bk ¼ a; cj þ dj ¼ b; bk ak l0 [ 0; cj dj s0 [ 0 and supðk;jÞ fjak j; jbk j; jck j; jdk jg K for some K [ 0: Then, we have 1 of of u lim w½ðKw f Þðx; yÞ f ðx; yÞ ¼ a ðx; yÞ þ b ðx; yÞ : w!1 2 ox oy

123


Let K ¼ supfjak j; jbk j; jcj j; jdj jg: Let ðx; yÞ 2 R2 : By Taylor’s theorem, we

Proof

ðk;jÞ

have of of ðx; yÞðu xÞ þ ðx; yÞðv yÞ ox oy þ hðu x; v yÞÞððu xÞ þ ðv yÞÞ;

f ðu; vÞ ¼ f ðx; yÞ þ

for some bounded function h such that hðu; vÞ ! 0 as ðu; vÞ ! ð0; 0Þ: Thus, we have ðKwu f Þðx; yÞ f ðx; yÞ Z kZ j 1 1 X X w w of w2 ¼ ðx; yÞ uðwx k; wy jÞ kþa ðu xÞdvdu jþcj k oy D D k¼1 j¼1 ak cj w w jþd

kþb

Z kZ j 1 1 X X w w of w2 þ ðx; yÞ uðwx k; wy jÞ kþa ðv yÞdvdu jþcj k oy D D k¼1 j¼1 ak cj w w kþb

þ

jþd

1 1 X X

w2 uðwx k; wy jÞ D D k¼1 j¼1 ak cj Z

kþbk w

kþak w

Z

jþdj w

jþcj w

hðu x; v yÞððu xÞ þ ðv yÞÞdvdu

¼ I1 þ I2 þ I3 ; ð say Þ: First, we obtain I1 : Z kZ j 1 1 X X w w of w2 I1 ¼ ðx; yÞ uðwx k; wy jÞ kþa ðu xÞdvdu jþc j k ox D D k¼1 j¼1 ak cj w w 2 2 1 1 X X of w k þ bk k þ ak ¼ ðx; yÞ x x uðwx k; wy jÞ ox 2Dak w w k¼1 j¼1 kþb

jþd

¼

1 1 X of 1 X ðx; yÞ uðwx k; wy jÞ½ðbk þ ak Þ þ 2ðk wxÞ ox 2w k¼1 j¼1

¼

1 1 X of a X ðx; yÞ uðwx k; wy jÞ ox 2w k¼1 j¼1

þ ¼

1 1 X of 1 X ðx; yÞ uðwx k; wy jÞðk wxÞ ox w k¼1 j¼1

of a 1 of a of ðx; yÞ þ ðx; yÞm1ð1;0Þ ðu; wx; wyÞ ¼ ðx; yÞ: ox 2w w ox 2w ox

Similarly, we obtain I2 ¼

123

b of ðx; yÞ: 2w oy


Now, we estimate I3 : Let [ 0 be fixed. Then, there exists d [ 0 such that pffiffiffiffiffiffiffiffiffiffiffiffiffi jhðt; sÞj for every t2 þ s2 d: Now, we have jI3 j

X


w2 juðwx k; wy jÞ Dak Dcj

2

Z kþbk Z jþdj w w kþa hðu x; v yÞjðju xj þ jv yjÞdvdu k jþcj w

þ

w

X



2

Z

kþbk w

Z

jþdj w jþcj w

kþak w

jhðu x; v yÞjðju xj þ jv yjÞdvdu

¼ I30 þ I300 : First, we estimate I30 . For suitable large w, jI30 j

X



Z

kþbk w

Z

jþdj w

jþcj w

kþak w

ju xjdvdu

2

þ

X



Z

kþbk w kþak w

Z

jþdj w

jþcj w

jv yjdvdu

2

:¼ J1 þ J2 : X J1 2


X 2wl0

2

2 2 w k þ bk k þ ak x þ x juðwx k; wy jÞj Dak w w

X

juðwx k; wy jÞj

ðwk ;wj Þ2Ud ðx;yÞ 2

ða2k þ b2k Þ þ 2ðak þ bk Þjk wxj þ 2ðk wxÞ2 1 2 ðuÞ þ 2Mð2;0Þ ðuÞ : 2K 2 M0 ðuÞ þ 2jajMð1;0Þ 2wl0 Similarly, we get J2

1 2 ðuÞ þ 2Mð0;2Þ ðuÞ : 2K 2 M0 ðuÞ þ 2jbjMð0;1Þ 2ws0

Therefore, we have

123


jI30 j

1 1 aMð1;0Þ ðuÞ bMð0;1Þ ðuÞ 1 1 2 þ þ K M0 ðuÞ þ w l 0 s0 l0 s0 2 2 Mð2;0Þ ðuÞ Mð0;2Þ ðuÞ þ þ : l0 s0

Let M [ 0 be a constant such that jhðt; sÞj M: Let d [ 0 be fixed. If ðwk ; wj Þ 62 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 k j d x þ y [ and by assumption (ii), we have Ud ðx; yÞ; then 2 w w 2 ! X X ððk wxÞ2 þ ðj wyÞ2 Þ 00 jI3 j 4M juðwx k; wy jÞj dw2 k j ðw;wÞ62Ud ðx;yÞ 2

for sufficiently large w. Hence, we get lim wI3 ¼ 0: So the assertion follows. h w!1

Remark 3.2 The boundedness assumption on f in the previous theorem can be relaxed by assuming that there are two positive constants a, b such that jf ðx; yÞj a þ bðx2 þ y2 Þ; for every ðx; yÞ 2 R2 : Proof

One can easily verify that

jKwu f Þðx; yÞj

1 1 X X

w2 juðwx k; wy jÞj D D k¼1 j¼1 ak cj

Z

kþbk w

kþak w

Z

jþdj w

jþcj w

jf ðu; vÞjdvdu

Z kþbk Z jþdj w w w2 juðwx k; wy jÞj kþa ða þ bðu2 þ v2 ÞÞdvdu jþcj k D D k¼1 j¼1 ak cj w w b 2bK 2 2 2 M0 ðuÞ a þ bðx þ y Þ þ ðjajjxj þ jbjjyjÞ þ 2 w w 2b b 1 1 1 1 ðuÞ þ jyjMð0;1Þ ðuÞ þ 2 jajMð1;0Þ ðuÞ þ jbjMð0;1Þ ðuÞ jxjMð1;0Þ þ w w b 2 2 þ 2 Mð2;0Þ ðuÞ þ Mð0;2Þ ðuÞ w 1 1 X X

and hence the series Kwu f is absolutely convergent for every ðx; yÞ 2 R2 : Moreover, for a fixed ðx; yÞ 2 R2 ; P1 ðu; vÞ ¼ f ðx; yÞ þ

of of ðx; yÞðu xÞ þ ðx; yÞðv yÞ; ox oy

the Taylor’s polynomial of first order centered at the point (x, y) by the Taylor’s formula, we can write

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f ðu; vÞ P1 ðu; vÞ ¼ hðu x; v yÞ; ju xj þ jv yj where h is a function such that lim hðu; vÞ ¼ 0: Then h is bounded on a neighðu;vÞ!0

bourhood of (x, y), say Ud ðx; yÞ [ 0: For ðu; vÞ 62 Ud ðx; yÞ; we have of jf ðu; vÞj jf ðx; yÞj ju xj þ þ jhðu x; v yÞj ju xj þ jv yj ju xj þ jv yj ox ju xj þ jv yj of jv yj ; þ oy ju xj þ jv yj and all the terms on the right-hand side of the above inequality are bounded for ðu; vÞ 62 Ud : Hence, hð: x; : yÞ is bounded on R2 : The same Voronovskaya formula can be obtained by writing the proof along the lines of the proof of Theorem 3.2. h Remark 3.3 One can easily verify that the following set of sequences satisfy the 1 1 1 ; bk ¼ 2 jkjþ1 ; cj ¼ jjjþ1 and dj ¼ conditions of the above theorem: ak ¼ jkjþ1 1 1 þ jjjþ1 : Moreover, several sequences can be constructed in a similar way.

For f 2 CðR2 Þ; the modulus of continuity for the bivariate case is defined as follows:

xðf ; d1 ; d2 Þ :¼ sup jf ðt; sÞ f ðx; yÞj : ðt; sÞ; ðx; yÞ 2 R2 and jt xj d1 ; js yj d2 ;

where d1 [ 0 and d2 [ 0: Further, xðf ; d1 ; d2 Þ satisfies the following properties: (a) (b)

xðf ; d1 ; d2 Þ ! 0 if d1 ! 0 and d2 ! 0; jt xj js yj jf ðt; sÞ f ðx; yÞj xðf ; d1 ; d2 Þ 1 þ 1þ ; d1 d2

The details of the modulus of continuity for the bivariate case can be found in [6]. In what follows, we shall define the following K-functional:

og og 0 ð1Þ ð1Þ Kðf ; d; C ; C Þ :¼ inf kf gk1 þ d max ; ;g 2 C ; ox 1 oy 1 where f 2 C0 and d [ 0: For every f 2 C0 there holds 1 Kðf ; d=2; C0 ; C1 Þ ¼ xðf ; dÞ; 2

ð3:1Þ

where xðf ; :Þ denotes the least concave majorant of xðf ; :Þ (see e.g. [6]). Now, we give the estimate of the rate of convergence of the bivariate operators ðKwu f Þ:

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Theorem 3.3

Let f 2 CðR2 Þ and supfak ; bk ; cj ; dj g K for some K [ 0. k;j

Then, for all ðx; yÞ 2 R2 ; we have jðKwu f Þðx; yÞ

KM0 ðuÞ 1 1 K f ðx; yÞj xðf ; d1 ; d2 Þ M0 ðuÞ þ þ þ w d1 d2 wd1 d2 1 K 1 1 1 1 Mð1;0Þ þ Mð0;1Þ þ þ þ w d1 d2 d1 d2 w

for any w [ 0; and d1 [ 0; d2 [ 0: Proof Using the linearity and positivity of the operators and using the property (b) of Remark 3.3, we have jðKwu f Þðx; yÞ f ðx; yÞj

1 1 X X

w2 D D cj a k k¼1 j¼1 juðwx k; wy jÞj

xðf ; d1 ; d2 Þ

Z

kþbk w kþak w

Z

Z

kþbk w kþak w

Z

jþdj w jþcj w

jf ðu; vÞ f ðx; yÞjdvdu

1 1 X X

w2 juðwx k; wy jÞj D D k¼1 j¼1 ak cj jþdj w

jþcj w

1þ

ju xj d1

1þ

jv yj dvdu d2

:¼I1 þ I2 þ I3 þ I4 :

It is easy to see that I1 ¼ xðf ; d1 ; d2 ÞM0ðuÞ: k þ ak k þ bk Now, we estimate I2 : For every u 2 ; ; we have w w k þ ak k þ ak k maxfjak j; jbk jg þ x x þ ju xj u w w w w K k x þ : w w

Then, we obtain Z kZ j 1 1 X w w xðf ; d1 ; d2 Þ X w2 I2 juðwx k; wy jÞj kþa ju xjdvdu jþcj k d1 D D k¼1 j¼1 ak cj w w 1 1 X xðf ; d1 ; d2 Þ X juðwx k; wy jÞj jk wxj þ K d1 w k¼1 j¼1 xðf ; d1 ; d2 Þ 1 KM0 ðuÞ þ Mð0;1Þ ðuÞ : d1 w xðf ; d1 ; d2 Þ 0 1 Similarly, we obtain I3 ðuÞ þ Mð1;0Þ ðuÞ : KMð0;0Þ d1 w Now, kþb

123

jþd


1 1 X xðf ; d1 ; d2 Þ 1 X juðwx k; wy jÞj jk wxj þ K jj wyj þ K d1 d2 w2 k¼1 j¼1 xðf ; d1 ; d2 Þ 1 2 0 1 1 2 M ðuÞ þ K M ðuÞ þ M ðuÞ þ M ðuÞ : K ð1;0Þ ð0;1Þ ð1;1Þ d1 d2 w2

I4

Combining the estimates I1 I4 ; we get the desired result. Using an estimate of the remainder in two dimensional Taylor formula and a technique developed in [26] by Gonska et al., we obtain a quantitative version of the Voronovskaja formula for bivariate sampling series. We remark that quantitative Voronovskaja formulae have important links with the theory of semi-groups of operators (see [3, 11]). The strict connections between the two theories were described in [2] and recently developed in [1, 18, 30]. Now, we have the following quantitative version of the Voronovskaja formula for bivariate sampling series. h Lemma 3.1 Let f 2 C 1 and ðx; yÞ 2 R2 ; ðx0 ; y0 Þ 2 R2 : Then for the remainder R1 ðf Þ :¼ R1 ðf ; ðx0 ; y0 Þ; ðx; yÞÞ in the Taylor formula, we have of of ; jx x0 j þ jy y0 jx ; jy y0 j : jR1 ðf Þj jx x0 jx ox oy Proof The proof follows from the Taylor formula of the first order using the same analysis as of Lemma 1 in [9]. Theorem 3.4 Let f 2 C 1 ðR2 Þ and let ak ; bk ; cj ; dj satisfy the conditions of Theorem 3.2. Then, for any ðx; yÞ 2 R2 , we have 1 of of w½ðK u f Þðx; yÞ f ðx; yÞ a þb w 2 ox oy of of D B Ax ; þ Cx ; ; ox 1 Aw oy 1 Cw 1 2 ðuÞ þ 2Mð2;0Þ ðuÞ ; B ¼ K 2 M0 ðuÞ þ where A ¼ l10 2K 2 M0 ðuÞ þ 2aMð1;0Þ 1 2 1 2 1 2 aMð1;0Þ ðuÞ þ Mð2;0Þ ðuÞ ; C ¼ s0 2K M0 ðuÞ þ 2bMð0;1Þ ðuÞ þ 2Mð0;2Þ ðuÞ and 1 2 D ¼ K 2 M0 ðuÞ þ bMð0;1Þ ðuÞ þ Mð0;2Þ ðuÞ : Proof Let f 2 C 1 be fixed. Then, we can write u 1 w ðK f Þðx; yÞ f ðx; yÞ a of þ b of w 2 ox oy

123


Z kþbk Z jþdj 1 1 X X of w w w2 ¼ ðx; yÞ uðwx k; wy jÞ kþa ðu xÞdvdu jþcj k ox D D k¼1 j¼1 ak cj w w Z kZ j 1 1 X X w w of w2 þ ðx; yÞ uðwx k; wy jÞ kþa ðv yÞdvdu jþc j k oy D D k¼1 j¼1 ak cj w w jþd

kþb

þ

1 1 X X

w2 uðwx k; wy jÞ D D k¼1 j¼1 ak cj Z

kþbk w

Z

jþdj w

jþcj w

kþak w

hðu x; v yÞððu xÞ þ ðv yÞÞdudv

1 of of a þb 2 ox oy

1 1 X X


Z

kþbk w

Z

kþak w

jþdj w

jþcj w

jhðu x; v yÞjðju xj þ jv yjÞdvdu :¼ I1 :

Using the relation (3.1) and Lemma 3.1, we obtain 1 1 X X

w2 I1 juðwx k; wy jÞj D D k¼1 j¼1 ak cj

Z

1 1 X X

kþbk w

Z

jþcj w

kþak w

w2 þ juðwx k; wy jÞj D D k¼1 j¼1 ak cj

Z

jþdj w

kþbk w

Z

kþak w

of ; ju xj dvdu ju xjx ox

jþdj w

jþcj w

of ; jv yj dvdu jv yjx oy

:¼ I10 þ I100 : First, we estimate I10 : For g 2 C2 ; we have jI10 j 2

1 1 X X


Z

kþbk w kþak w

Z

jþdj w

jþcj w

ju xjK

of ju xj ; dvdu ox 2

1 1 X X 2w2 juðwx k; wy jÞj D D k¼1 j¼1 ak cj

Z kþbk Z jþdj oðf gÞ w w þ ju xj max og ; og kþa ju xj dvdu ox oy jþcj k ox 2 1 1 1 w w

123


1 1 X oðf gÞ X 2w2 juðwx k; wy jÞj ox 1 k¼1 j¼1 Dak Dcj Z

kþbk w

Z

jþdj w

1 1 X X


Z kþbk Z jþdj og og w w 2 ; kþa ðu xÞ max dvdu :¼ J1 þ J2 : ox oy jþcj k 1 1 w w

kþak w

jþcj w

ju xjdvdu þ

It is easy to see that 2 2 kþb 1 1 w oðfoxgÞ X X k þ a k k 1 x þ x jJ1 j juðwx k; wy jÞj Dak w w k¼1 j¼1 oðf gÞ 1 2 1 2 ox wl 2K M0 ðuÞ þ 2aMð1;0Þ ðuÞ þ 2Mð2;0Þ ðuÞ 1 0 Now, X 1 1 X og og w2 ; J2 max ox oy 1 1 k¼1 j¼1 Dak Dcj juðwx k; wy jÞj

Z

kþbk w kþak w

Z

jþdj w jþcj w

ðu xÞ2 dvdu

1 1 X og og w X ; juðwx k; wy jÞj max ox oy 1 1 3Dak k¼1 j¼1 3 3 ! k þ bk k þ ak x x w w og og 1 2 1 2 ; max M ðuÞ þ aM ðuÞ þ M ðuÞ : K 0 ð1;0Þ ð2;0Þ ox oy 2 1 1 w Thus, we have oðf gÞ þ 1 B max og ; og wI10 A : ox ox oy 1 wA 1 1 Taking the infimum over all g 2 C2 ; we obtain oðf gÞ B 0 wI1 Ax ; : ox 1 Aw Similarly, we obtain

123


wI100

oðf gÞ D Cx ; : oy 1 Cw

Combining the estimates I10 and I100 ; we get the required result.

h

4 Applications to special kernels In this section, we describe some particular examples of kernels u which illustrate the previous theory. In particular, we will examine the box splines kernel and Bochner–Riesz Kernel. 4.1 Separable box splines First, we consider the sampling Kantorovich operators based on box spline functions which are separable. Let bd1 and bd2 be two Cardinal central B-splines of degrees d1 and d2 respectively. That is bd1 ðxÞ :¼v½1;1 Hv½1;1 Hv½1;1 H Hv½1;1 ðd1 þ 1 times Þ 22

22

22

22

bd2 ðyÞ :¼v½1;1 Hv½1;1 Hv½1;1 H Hv½1;1 ðd2 þ 1 times Þ; 22

22

22

22

where v½1;1 ðxÞ ¼ 22

8