High dimensional periodic sampling and cubature on Smolyak grids based on B spline quasi interpolation

High-dimensional periodic sampling and cubature on Smolyak grids
based on B-spline quasi-interpolation
Dinh D˜
ung
Vietnam National University, Hanoi, Information Technology Institute
144 Xuan Thuy, Cau Giay, Hanoi, Vietnam


August 03, 2014 -- Version 1.3

Abstract
We investigate linear algorithms of sampling recovery and cubature formulas on Smolyak
grids of periodic d-variate functions having Lipschitz-H¨older mixed smoothness based on Bspline quasi-interpolation, and their optimality when the number d of variables and the number
n of sampled function values may be very large. We establish upper and lower estimates of the
error of the optimal sampling recovery and the optimal integration on Smolyak grids, explicit
in d and n.
Keywords and Phrases Linear sampling algorithm · Cubature formula · Smolyak grid ·
Lipschitz-H¨
older mixed smoothnesse space of mixed smoothness · Quasi-interpolation representations by B-spline series.
Mathematics Subject Classifications (2000) 41A15 · 41A05 · 41A25 · 41A58 · 41A63.

1

Introduction

The aim of this paper is to investigate linear algorithms of sampling recovery and cubature formulas
on Smolyak grids of periodic d-variate functions having Lipschitz-H¨older mixed smoothness based
on B-spline quasi-interpolation, and their optimality when the number d of variables and the
number n of sampled function values may be very large. We stress on explicitly estimating from
above and below the error of the optimal sampling recovery and the optimal integration on Smolyak
grids as a function of two variables d and n.

We consider only functions on Rd 1-periodic at each variable. It is convenient to consider them
as functions defined in the d-torus Td = [0, 1]d which is defined as the cross product of d copies of
the interval [0, 1] with the identification of the end points. To avoid confusion, we use the notation
Id to denote the standard unit d-cube [0, 1]d .
1


For m ∈ N, the well known periodic Smolyak grid of sampled points Gd (m) ⊂ Td is defined as
Gd (m) := {ξ = 2−k s : k ∈ Nd , |k|1 = m, s ∈ I d (k)},

(1.1)

where I d (k) := {s ∈ Zd+ : sj = 0, 1, ..., 2kj − 1, j ∈ [d]}. Here and in what follows, we use the
notations: xy := (x1 y1 , ..., xd yd ); 2x := (2x1 , ..., 2xd ); |x|1 := di=1 |xi | for x, y ∈ Rd ; [d] denotes the
set of all natural numbers from 1 to d; xi denotes the ith coordinate of x ∈ Rd , i.e., x := (x1 , ..., xd ).
The number |Gd (m)| of points in the grid Gd (m) is
 m d
m
s=0 s ,



|Gd (m)| = 2d ,
m=d



 m m−1
2 d−1 , m > d.

Notice the grid Gd (m) is full if and only if m ≥ d.
Sparse grids Gd (m) for sampling recovery and numerical integration were first considered
by Smolyak [36]. Temlyakov [37] – [39] and the author of the present paper [9] – [11] developed Smolyak’s construction for studying the sampling recovery for periodic Sobolev classes and
Nikol’skii classes having mixed smoothness. Recently, Sickel and Ullrich [34] have investigated the
sampling recovery for periodic Besov classes having mixed smoothness. For non-periodic functions
of mixed smoothness linear sampling algorithms on Smolyak grids have been recently studied by
Triebel [40] (d = 2), Dinh D˜
ung [14], Sickel and Ullrich [35], using the mixed tensor product hat
functions and more general, B-splines. In [16], we have constructed methods of approximation by
arbitrary linear combinations of translates of the Korobov kernel κr,d on Smolyak grids of functions
from the Korobov space K2r (Td ) which is a reproducing kernel Hilbert space with the associated
kernel κr,d .
In numerical applications for high-dimensional approximation problems, Smolyak grids was first
considered by Zenger [42] in parallel algorithms for numerical solving PDEs. Numerical integration
on Smolyak grids was investigated in [21]. For non-periodic functions of mixed smoothness of
integer order, linear sampling algorithms on Smolyak grids have been investigated by Bungartz and
Griebel [3] employing hierarchical Lagrangian polynomials multilevel basis. There is a very large
number of papers on Smolyak grids and their modifications in various problems of approximations,
sampling recovery and integration with applications in data mining, mathematical finance, learning
theory, numerical solving of PDE and stochastic PDE, etc. to mention all of them. The reader
can see the surveys in [3, 29, 22] and the references therein. For recent further developments and
results see in [25, 24, 26, 19, 2].
For univariate functions f on R, the rth difference operator ∆rh is defined by
r

∆rh (f, x) :=

(−1)r−j
j=0

r
f (x + jh).
j

If u is any subset of [d], for multivariate functions on Rd the mixed (r, e)th difference operator ∆r,u
h
2


is defined by
∆r,u
h :=

∆rhi , ∆r,∅
h = I,
i∈u

where the univariate operator ∆rhi is applied to the univariate function f by considering f as a
function of variable xi with the other variables held fixed.
Denote by C(Td ) the normed space of all bounded continuous functions on Td with the max
norm · C(Td ) . For a function f on Td , let
ωru (f, t) :=

∆r,u
h (f )

sup
|hi |
C(Td ) ,

t ∈ Id ,

be the mixed (r, e)th modulus of smoothness of f (in particular, ωr∅ (f, t) = f

C(Td ) ).

d
α (u) for functions f ∈ C(T ) by
If 0 < α ≤ r, we introduce the semi-norm |f |H∞
α (u) := sup
|f |H∞

t∈Id i∈u

u
t−α
i ωr (f, t),

α (∅) = f
(in particular, |f |H∞

(1.2)

C(Td ) ).

α of mixed smoothness α is defined as the set of functions
The Lipschitz-H¨
older space H∞
f ∈ C(Td ) for which the norm

f

α
H∞

α (u)
:= sup |f |H∞

u⊂[d]

α the unit ball in H α .
is finite. Denote by U∞
∞

For a family Φ = {ϕξ }ξ∈Gd (m) of functions, we define the linear sampling algorithm Sm (Φ, ·)
on Smolyak grids Gd (m) by
Sm (Φ, f ) =

f (ξ)ϕξ .

(1.3)

ξ∈Gd (m)

We shall use the linear sampling algorithm Sm (Φ, ·) for approximate recovery of functions from
α.
U∞
Let us introduce the quantity of optimal sampling recovery sm (Fd )p on Smolyak grids Gd (m)
with respect to the function class Fd by

sm (Fd )p := inf sup
Φ

f − Sm (Φ, f ) p .

(1.4)

f ∈Fd

α we will consider its subsets U
˚α which consists of all functions f ∈ U α
Paralleling with U∞
∞
∞
such that f (x) = 0 if xj = 0 for some index j ∈ [d].

3


˚α , we use the linear sampling algorithm S
˚m (Φ, ·) on
For sampling recovery of functions from U
∞
d
˚
the grids G (m) by
˚m (Φ, f ) =
S

f (ξ)ϕξ ,

(1.5)

˚d (m)
ξ∈G

˚d (m) of the Smolyak grid Gd (m) by
where for m ∈ N, we define the subset G
˚d (m) := {ξ = 2−k s : k ∈ Nd , |k|1 = m, s ∈ ˚
G
I d (k)}

(1.6)

with ˚
I d (k) := {s ∈ Zd+ : sj = 1, ..., 2kj − 1, j ∈ [d]}.
˚d (m) of points in the grid G
˚d (m) is
Notice that for m ∈ N, the number G
d

˚d (m)| =
|G

(2kj − 1) ≤ 2m
k∈Nd :

|k|1 =m j=1

m−1

.
d−1

(1.7)

˚d (m)
Let us introduce the quantity of optimal sampling recovery ˚
sm (Fd )p on Smolyak grids G
with respect to the function class Fd by
˚
sm (Fd )p := inf sup
Φ

˚m (Φ, f ) p .
f −S

(1.8)

f ∈Fd

In previous papers on sampling recovery and integration on Smolyak grids of classes of
functions with mixed smoothness, a typical form of lower and upper bounds of sm (Fd )p is
C(d, α) 2−am m(d−1)b , where C(d, α) is a constant not explicitly computed in d. Specially, in the
recent papers [14, 18] it has been proven that for 1 ≤ p ≤ ∞ and α > 1/p,
˚α )p ≤ sm (U α )p ≤ C(d, α, p) 2−αm md−1 .
C (d, α, p) 2−αm md−1 ≤ ˚
sm (U
∞
∞

(1.9)

In high-dimensional approximation problems using function values information, the number m
of sampled values is the main parameter in the study of convergence rates of the approximation
error with respect to m going to infinity. However, the parameter d may hardly affect this rate
when d is large. In the present paper, inspired by the relation (1.9), we establish upper and
α ) and ˚
α ) explicitly in n and d as a function of two
˚∞
lower bounds for the quantities sm (U∞
sm (U
p
p
variables m, d, in particular, upper bounds for the constant C(d, α, p) and C (d, α, p) in (1.9). To
obtain these upper and lower bounds we construct linear sampling algorithms on Smolyak grids
of the form (1.5) based on a B-spline quasi-interpolation series specially on Faber-Schauder series
related to the well-known hat functions. As consequences we obtain upper and lower bounds for
α ) and Int
˚ m (U
˚α )p of optimal cubature formula on Smolyak grids explicit
the quantities Intm (U∞
p
∞
in n and d. Related to the problems investigated in the present paper, is problems of hyperbolic
cross approximation of functions having mixed smoothness in high-dimensional setting in terms of
n-widths and ε-dimensions which have been invesigated in [4, 17].
The paper is organized as follows. In Section 2, we establish upper and lower bounds we
α)
˚∞
construct linear sampling algorithms on Smolyak grids based on Faber-Schauder series for ˚

sm (U
p
4


α ) for 0 < α ≤ 2. As consequence, we derive upper and lower bounds for the error
and sm (U∞
p
α)
˚ sm (U
˚∞
of cubature formulas on Smolyak grids and of optimal integration on Smolyak grids Int
p
s
s
α ) . In Section 3, we extend the results for s (U α ) and Int (U α ) in Section 2 to
and Intm (U∞
p
m
∞ p
m
∞ p
the case of large mixed smoothness α > 2, based on B-spine quasi-interpolation representations
α . In Section 4, we give some example of polynomials inducing generating
for function from H∞
quasi-interpolation operators.

2

Sampling recovery based on Faber-Schauder series

2.1

Faber-Schauder series

Let M2 (x) = (1 − |x − 1|)+ , x ∈ I, be the piece-wise linear B-spine with knot at 0, 1, 2. Since the
support of functions M2 (2k+1 ·) for k ∈ Z+ is the interval [0, 2−k ] we can extend these functions to
an 1-periodic function on the whole R. Denote this periodic extension by ϕk .
The univariate periodic Faber-Schauder system of the hat functions is defined by
F := {ϕk,s : s ∈ Z(k), k ∈ Z+ },
where Z(0) := {0} and Z(k) := {0, 1, ..., 2k−1 − 1} for k > 0,
ϕ0,0 (x) := 1, x ∈ T,
and for k > 0 and s ∈ Z(k)
ϕk,s (x) := ϕk (x − 2s), x ∈ T.
Let
1
λk,s (f ) := − ∆22−k (f, 2−k+1 s), k > 0, and λ0,0 (f ) := f (0).
2
Put Z d (k) :=

d
i=1 Z(ki ).

For k ∈ Zd+ , s ∈ Z d (k), define the tensor product hat functions

d

ϕk,s (x) :=

ϕki ,si (xi ),

i=1

and the d-variate periodic Faber-Schauder system Fd by
Fd := {ϕk,s : s ∈ Z d (k), k ∈ Zd+ }.
Let λk,s (f ) be defined in the manner of the definition (3.10) as
λk,s (f ) := λk1 ,s1 (λk2 ,s2 (...λkd ,sd (f ))).
5


Lemma 2.1 The d-variate periodic Faber-Schauder system Fd is a basis in C(Td ). Moreover,
function f ∈ C(Td ) can be represented by the series
λk,s (f )ϕk,s ,

f =
k∈Zd+

(2.1)

s∈Z d (k)

converging in the norm of C(Td ).
Proof. For the univariate case (d = 1), this lemma can be deduced from its well-known counterpart
for non-periodic functions on I (see, e.g., [30, Theorem 1, Chapter VI]). For the multivariate case
(d > 1), it can be proven by the tensor product argument.
Put Zd+ (u) := {k ∈ Zd+ : supp k = u}, where supp k denotes the support of k, i.e., the subset of
all j ∈ [d] such that kj = 0.
α can be represented by the
Theorem 2.1 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then a function f ∈ H∞
series (2.1) converging in the norm of C(Td ). Moreover, we have for every k ∈ Zd+ (u),

qk (f )

∞

≤

sup
s∈Z d (k)

α (u) ,
|λk,s (f )| ≤ 2−|u| 2−α|k|1 |f |H∞

(2.2)

and
qk (f )

p

α (u) , 0 < p < ∞.
≤ 2−|u| (p + 1)−|u|/p 2−α|k|1 |f |H∞

(2.3)

Proof. The first part of the lemma on representation and convergence is in Lemma 2.1. Let us
α we have for
prove the inequalities (3.42) and (2.3). By the definitions of ϕk,s , λk,s (f ) and f H∞
d
d
every k ∈ Z+ (u) and x ∈ T ,

|qk (f )(x)| ≤

sup

|λk,s (f )|

s∈Z d (k)

ϕk,s (x) ≤

and consequently,
qk (f )

∞

≤

sup

|λk,s (f )|

s∈Z d (k)

≤

sup
s∈Z d (k)

j∈u

sup
s∈Z d (k)

s∈Z d (k)

1
− ∆22−kj f (xk,s (u))
2

≤ 2−|u| ωru (f, 2−k )
α (u) .
≤ 2−|u| 2−α|k|1 |f |H∞

6

|λk,s (f )|,


The inequality (3.42) is proven. Let us prove (2.3). We first consider the case 1 ≤ p < ∞. We
have for every k ∈ Zd+ (u),
qk (f )

p
p

p

λk,s (f )ϕk,s (x) dx

=

Td

≤

s∈Z d (k)

ϕk,s (x) dx
Td

s∈Z d (k)

=

p

|λk,s (f )|p

sup

s∈Z d (k)

|λk,s (f )|p |Z d (k)|

sup

|ϕk,0 (x)|p dx
Td

s∈Z d (k)

2−kj

=

p

d

|λk,s (f )| |Z (k)|

sup
s∈Z d (k)

0

j∈u

α (u)
2−|u| 2−α|k|1 |f |H∞

≤

|2kj xj |p dxj

2

p

2−|u| 2|k|1 2|u| (p + 1)−|u| 2−|k|1

= 2−|u| (p + 1)−|u| 2−pα|k|1 |f |pH α (u) .
∞

This proves (2.3) for 1 ≤ p < ∞. The case 0 < p < 1 can be proven in a similar way starting from
the inequality
qk (f )

p
p

Td

s∈Z d (k)

2.2

|ϕk,s (x)|p dx.

|λk,s (f )|p

≤

Auxiliary lemmas

For m, n ∈ N with m ≥ n, we introduce the function Fm,n : (0, 1) → R by
∞

Fm,n (t) :=
s=0

m+s s
t .
n

From the definition it follows that Fm,n (t) =

m
n

+ tFm+1,n (t) and consequently,

tFm+1,n (t) < Fm,n (t) < Fm+1,n (t), t ∈ (0, 1).

(2.4)

We will need following equation proven in [3, (3.67), p.29].
1
Fm,n (t) =
1−t

n
s=0

m
s

t
1−t

n−s

.

(2.5)

7


For nonnegative integer n, we define the function

(1 − 2t)−1 ,
t < 1/2,




bn (t) := 2(n + 1),
t = 1/2,




(2t − 1)−1 [t/(1 − t)]n+1 , t > 1/2.

(2.6)

Lemma 2.2 Let m, n ∈ N, m ≥ n, and 0 < t < 1. Then we have
Fm,n (t) ≤

m

bn (t),
n∗

(2.7)

where n∗ := min{n, m/2 }.
Proof. We have
n
s=0

m
s

≤ (n + 1)

m
.
n∗

(2.8)

Hence, by (2.5) the case t = 1/2 of the inequality (2.7) is proven. Next, we have for x > 0 and
x = 1,
n
s=0

m s
x ≤
s

n

m
n∗

xs =
s=0

xn+1 − 1
,
x−1

m
n∗

(2.9)

and

(x − 1)−1 xn+1 ,

xn+1

−1
≤
(1 − x)−1 ,
x−1

x > 1,
(2.10)

x < 1.

By using the last two inequalities for x = (1 − t)/t from (2.5) we prove the cases t < 1/2 and
t > 1/2 of the inequality (2.7).
Lemma 2.3 Let m, n ∈ N, m ≥ n, and 0 < t < 1. Then we have
Fm,n (t) ≤

1
1−t

t
1−t

n

exp

1−t
t

mn

(2.11)

Proof. We have for x > 0,
n
s=0

m s
x =

s

n
s=0

m!
xs ≤
s!(m − s)!

n

ms
s=0

xs
≤ mn ex .
s!

Using this estimate from (2.5) we deduce the lemma.

8

(2.12)


2.3

Upper bounds

By the definition we can see that



α
α
˚
U∞ = f ∈ U∞
:f =




qk (f ) .

d

k∈N

˚m for f ∈ U
˚α by
For m ∈ Z+ , we define the operator R
∞
˚m (f ) :=
R

λk,s (f )ϕk,s .

qk (f ) =
k∈Nd : |k|

k∈Nd : |k|

1 ≤m

1 ≤m

s∈Z d (k)

˚m defines a linear sampling algorithm S
˚m (Ψ, f ) on the grid G
˚d (m) by
The operator R
˚m (f ) = S
˚m (Ψ, f ) =
R

f (ξ)ψξ ,
˚d (m)
ξ∈G

where Ψ∗ := {ψξ }ξ∈G
˚d (m) ,
d

ψ2−ki si (xi ), k ∈ Nd , s ∈ I d (k),

ψ2−k s (x) =
i=1

and the univariate functions ψk,s , k ∈ N, s ∈ I 1 (k), are defined by



1 − ϕ0,0 ,
k = 1, s = 0






ϕ0,0 ,
k = 1, s = 1
ψ2−k s =


− 12 (ϕk,j + ϕk,j−1 ), k > 1, s = 2j, 0 ≤ j < 2k−1 − 1,






ϕk,j ,
k > 1, s = 2j + 1, 0 < j < 2k−1 − 1.
In what follows, for notational convenience we write x0 = 1 for x ∈ [0, ∞]. For 0 < p ≤ ∞,
α > 0 and m ≥ l − 1, put
b(α, p) := 2(2α − 1)(p + 1)1/p ,
and
l−1
α

−l

β(α, l, m) := (2 − 1)

s=0

m
(2α − 1)s ,
s

β(α, 0, m) := 1.

˚α and and every
Theorem 2.2 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U
∞
m ≥ d − 1,
˚m (f )
f −R

p

≤ [2(p + 1)1/p ]−d 2−αm β(α, d, m) ≤ exp(2α − 1) [b(α, p)]−d 2−αm md−1 . (2.13)
9


Moreover, if in addition, m ≥ 2(d − 1),
˚m (f )
f −R

p

≤ a◦ (α, p, d) 2−αm

m
,
d−1

(2.14)

where

[2(p + 1)1/p ]−d ,




a◦ (α, p, d) := |2α − 2|−| sgn(α−1)| × d [2(p + 1)1/p ]−d ,




[2(p + 1)1/p (2α − 1)]−d ,
Further, we have
˚d (f )
f −R

α > 1,
α = 1,
α < 1.

≤ [b(α, p)]−d .

p

(2.15)

(2.16)

Proof. Let us prove the lemma for 1 ≤ p ≤ ∞. It can be proven in a similar way with a slight
modification for 0 < p < 1. Put t = 2−α . From Theorem 2.1 and (2.5) it follows that for every
˚α and and every m ≥ d − 1,
f ∈U
∞
f − Rm (f )

p

≤

qk (f )

p

k∈Nd : |k|1 >m

2−d (p + 1)−d/p 2−α|k|1

≤
k∈Nd : |k|1 >m

= 2−d (p + 1)−d/p

2−α|k|1

(2.17)

k∈Nd : |k|1 >m
∞

= 2−d (p + 1)−d/p
j=1

m + j − 1 −α(m+j)
2
d−1

= 2−α(m+1) 2−d (p + 1)−d/p Fm,d−1 (t).
Hence, replacing t = 2−α we obtain the first inequality in (2.13) and the second one by applying
Lemma 2.2.
The inequality (2.14) can be derived from (2.17) by applying Lemma 2.3 for t = 2−α .
Let us prove (2.16) where m = d. From (2.17), and (2.5) we have
f − Rd (f )

p

≤ 2

−d(α+1)

(p + 1)

−d/p

1
1−t

= 2−d(α+1) (p + 1)−d/p

d−1
s=0

1
1−t

= [2(2α − 1)(p + 1)1/p ]−d

10

d

d−1
s

t
1−t

d−1−s

(2.18)

For m ∈ Z+ , we define the operator Rm by
Rm (f ) :=

qk (f ) =
k∈Zd+ : |k|1 ≤m

λk,s (f )ϕk,s .
k∈Zd+ : |k|1 ≤m s∈Z d (k)

For functions f on Td , Rm defines a linear sampling algorithm Sm (Ψ, f ) on the Smolyak grid
by

Gd (m)

f (ξ)ψξ ,

Rm (f ) = Sm (Ψ, f ) =
ξ∈Gd (m)

where Ψ := {ψξ }ξ∈Gd (m) .
Put
d

γ(α, d, p, m) :=
l=0

d −l
2 (p + 1)−l/p β(α, l, m).
l

α and every
Theorem 2.3 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U∞
m ≥ d − 1,

f − Rm (f )

p

≤ 2−αm γ(α, d, p, m) ≤ exp(2α − 1) [1 + 1/b(α, p)]d 2−αm md−1 .

(2.19)

Moreover, if in addition, m ≥ 2(d − 1),
f − Rm (f )

p

≤ a(α, p, d) 2−αm

m
,
d−1

(2.20)

where

[1 + 1/2(p + 1)1/p ]d ,





a(α, p, d) := |2α − 2|−| sgn(α−1)| × d [1 + 1/2(p + 1)1/p ]d ,




[1 + 1/2(p + 1)1/p (2α − 1)]d ,

α > 1,
α = 1,

(2.21)

α < 1.

Further, we have
f − Rd (f )

p

≤ [1 + 1/(2α − 1)]d [1 + 1/2(p + 1)1/p ]d .

(2.22)

Proof. Let us prove the lemma for 1 ≤ p ≤ ∞. It can be proven in a similar way with a slight
modification for 0 < p < 1. Put t = 2−α . From Theorem 2.1 and Lemma 2.2 it follows that for

11

α and and every m ≥ d − 1,
every f ∈ U∞

f − Rm (f )

p

≤

qk (f )

p

|u|≤d k∈Zd+ (u): |k|1 >m

2−|u| (p + 1)−|u|/p 2−α|k|1

≤
|u|≤d k∈Zd+ (u): |k|1 >m
d

=
l=0
d

=
l=0

d −l
2 (p + 1)−l/p
l
d −l
2 (p + 1)−l/p
l
d

= 2−α(m+1)
l=0

2−α|k|1
(2.23)

k∈Nl : |k|1 >m
∞
j=1

m + j − 1 −α(m+j)
2
l−1

d −l
2 (p + 1)−l/p Fm,l−1 (t)
l

Hence, replacing t = 2−α we proves the first inequality in (2.19). This inequality easily implies
(2.22) by considering m = d.
Next, applying Lemma 2.2 to Fm,l−1 (t) in (2.23) we get
d

f − Rm (f )

p

≤ 2−α(m+1)
l=0

≤ 2−α(m+1) exp

1−t
t

d

t−1 md−1
l=0
d

= 2−αm exp (2α − 1) md−1
l=0

1−t
t

d −l
2 (p + 1)−l/p
l

t

1−t

12

t
1−t

d −l
2 (p + 1)−l/p (2α − 1)−l
l

= exp(2α − 1) [1 + 1/b(α, p)]d 2−αm md−1 .
The second inequality in (2.19) is proven.

l−1

exp

d −l
1
2 (p + 1)−l/p
l
1−t

ml−1

l

(2.24)

Next, let us verify (2.20). Applying Lemma 2.3 to Fm,l−1 (t) in (2.23) we have for m ≥ 2(d − 1),
d

f − Rm (f )

p

≤ 2−α(m+1)
l=0
d

l−1
s=0

m
s

t
1−t

l−1−s

d −l
2 (p + 1)−l/p β(α, l, m)
l

= 2−αm
l=0

d

= 2−α(m+1)
l=0

≤ 2−α(m+1)

d
l

m
d−1

= a(α, p, d) 2−αm

2.4

d −l
1
2 (p + 1)−l/p
l
1−t

m
2−l (p + 1)−l/p bl−1 (2−α )
l−1
d

(2.25)

d −l
2 (p + 1)−l/p bl−1 (2−α )
l

l=0

m
.
d−1

Lower bounds

˚α and and every
Theorem 2.4 Let 1 ≤ p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U
∞
m ≥ d − 1,
α
˚∞
˚
sm (U
)p ≥ 2−7d 2−αm β(α, d, m) ≥ (2α − 1)−1 2−7d 2−αm

m
,
d−1

(2.26)

and therefore,
˚α )p ≥ (2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 .

˚
sm (U
∞

(2.27)

Further, we have
α
˚∞
˚
sd (U
)p ≥ (2α − 1)−d 2−7d .

(2.28)

Proof. Let M4 be the cubic cardinal B-spline with support [0, 4] and knots at the points 0, 1, 2, 3, 4.
Since the support of functions M4 (2k+2 ·) for k ∈ Z+ is the interval [0, 2−k ], we can extend these
functions to an 1-periodic function on the whole R. Denote this periodic extension by µk . Define
the univariate nonnegative function gk,s and gk on T by
gk,s (x) := µk (x − 4s), k ∈ Z+ , s ∈ I 1 (k);

(2.29)

gk (x) :=

(2.30)

gk,s (x).
s∈I 1 (k)

13


(2)

From the identity M4 (x) = M2 (x) − 2M2 (x − 1) + M2 (x − 2), x ∈ R, it follows that
(2)

|M4 (x)| ≤ 2M2 (2−1 x),

x ∈ R,

and consequently,
(2)

|gk,s (x)| ≤ 22k+5 ϕk,s (x), x ∈ T.

(2.31)

One can also verify that
supp gk,s = Ik,s =: [2−k s, 2−k (s + 1)],
1
0 M4 (x) dx

and by the equation
1

int Ik,s ∩ int Ik,s = ∅, s = s ,

(2.32)

= 1,

gk,s (x) dx = 2−k−2 .

(2.33)

0

Let
d

gkj ,sj , k ∈ Zd+ , s ∈ I d (k).

gk,s :=
j=1

Put for u ⊂ [d],
Du2 :=

∂ 2|u|
2.
j∈u ∂xj

Then we have for k ∈ Zd+ , s, s ∈ I d (k) and u ⊂ [d],
|Du2 gk,s (x)| = 25|u| 22|ku |1

gkj ,sj (xj ), x ∈ Td ,

ϕkj ,sj (xj )

j∈u

(2.34)

j∈u

d

Ikj ,sj ,

supp gk,s = Ik,s =:

int Ik,s ∩ int Ik,s = ∅, s = s ,

j=1

and
gk,s (x) dx = 2−2d 2−|k|1 .
Td

Take m, n ∈ N with n > m and we define the function fm,n by
n

fm,n := 2−5d

2−αl
l=m

gk,s .
|k|1 =l

(2.35)

s∈I d (k)

Let us prove
α
˚∞
fm,n ∈ U
,

(2.36)
14


and
˚d (m).
fm,n (ξ) = 0, ξ ∈ G

(2.37)

Let us first prove (2.37). Let k, k ∈ Nd with |k|1 ≥ m and |k |1 = m. Then there is an index
j ∈ [d] such that kj ≥ kj . Hence, 2−kj sj ∈ int Ik1j ,sj for every s ∈ ˚
I d (k) and s ∈ ˚
I d (k ). This
means that gkj ,sj (2−kj sj ) = 0, and consequently, gk,s (2−k s ) = 0 for every s ∈ ˚
I d (k), s ∈ ˚
I d (k ).
This proves (2.37).
We now prove (2.46).

Observe that
n

fm,n = 2−5d

2−αl

gk ,
|k|1 =l

l=m

where
d

gkj , k ∈ Nd .

gk :=

(2.38)

j=1

To prove (2.46) it is sufficient to show that
∆2,u
h (fm,n , x) ≤

|hj |α , x ∈ Td , h ∈ Rd .

(2.39)

j∈u

Let us prove this inequality for u = [d] and h ∈ Rd+ , the general case of u can be proven in
a similar way with a slight modification. For h ∈ R and k ∈ N, let h(k) ∈ [0, 2−k ) is the number
defined by h = h(k) + s2−k for some s ∈ Z. For h ∈ Rd and k ∈ Nd , put
d
(kj )

h(k) :=

hj

.

j=1

Since gk is 2−kj -periodic in variable xj , we have
2,[d]

∆h

2,[d]

(gk , x) = ∆h(k) (gk , x), x ∈ Td , h ∈ Rd .

By using the formula
(x + y)[h−1 M2 (h−1 y)] dy

∆2h (f, x) = h2

R

for twice-differentiable function f on R (see, e.g., [7, p.45]), and (2.31) we have that
d
2,[d]

∆h

d

h2j

(gk,s , x) =
j=1

Rd

−1
h−1
j M2 (hj (yj − xj )) dy.

2
D[d]
gk,s (y)
j=1

15


2,[d]

∆h

2,[d]

∆h(k) (gk,s , x)

(gk , x) =
s∈I d (k)

d

d
(k)
(hj )2
s∈I d (k) j=1

=

(k)

Rd

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy.

2
gk,s (y)
D[d]

j=1

Hence,
d
2,[d]

|∆h

d
(k)

5d 2|k|1 ϕk,s (y)

(gk , x)| ≤
s∈I d (k)

Rd

(k)

j=1

j=1

d

≤ 25d 2|k|1
Rd

d

j=1

s∈I d (k)

d

(k)

(k)

j=1

d

(k)

(2−kj hj )2−α

(hj )α

ϕk,s (y)

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy

(hj )2

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy
j=1

d

≤ 25d 2|k|1

hαj

(k)

Rd

j=1

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy.

ϕk,s (y)

j=1

s∈I d (k)

Hence, by the inequalities (2.34) and
n

ϕk,s (y) ≤ 1, y ∈ Rd ,
l=m |k|1 =l s∈I d (k)

we derive that
n
2,[d]
|∆h (fm,n , x)|

≤ 2

−5d

2,[d]

2−αl

|∆h

(gk , x)|

|k|1 =l

l=m
n

d

≤ 2−5d

d

j=1

j=1

(k)

(k)

j=1

(k)

Rd j=1

d

hαj .

M2 (yj ) dy ≤
Rd

j=1

j=1

j=1

16

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy

(hj )−1 M2 ((hj )−1 (yj − xj )) dy

d

hαj

(k)

d

Rd l=m |k| =l
s∈I d (k)
1

d
j=1

s∈I d (k)

d

hαj

=

Rd

ϕk,s (y)

d

(k)

(hj )−1 M2 ((hj )−1 (yj − xj )) dy

ϕk,s (y)

n

hαj

≤

hαj
j=1

l=m |k|1 =l

≤

d

2−αl 25d 2|k|1


The inequality (2.39) is proven for u = [d] and h ∈ Rd+ .
From (2.38), (2.32), (2.33) we can also verify that if m is given then for arbitrary n ≥ m,

n

fm,n

p

≥

fm,n

= 2−5d

1

n

2−αl

gk

= 2−7d

1

|k|1 =l

l=m

2−αl
l=m

l−1
.
d−1

(2.40)

˚m (Φ, fm,n ) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary
By (2.37) we have S
n ≥ m,
n

˚α )p ≥
˚
sm (U
∞

˚m (Φ, fm,n )
fm,n − S

p

=

fm,n

p

≥ 2

−7d

2−αl
l=m

l−1
d−1

(2.41)

This means that for t = 2−α ,
∞
α
˚∞
˚
sm (U
)p ≥ 2−7d

2−αl
l=m

l−1
d−1

= 2−7d 2−αm Fm−1,d−1 (t).

(2.42)

m
(2α − 1)d−1

d−1
m
d−1

(2.43)

Therefore, applying (2.4) gives
˚α )p > 2−7d 2−αm tFm,d−1 (t)
˚
sm (U
∞
= 2−7d 2−αm β(α, d, m)
> 2−7d 2−αm (2α − 1)−d
= (2α − 1)−1 2−7d 2−αm

which proves (2.41). The inequality (2.28) can be deduced from the first inequality in (2.52) in a
way similar to (2.18).
Put
d

γ (α, d, m) :=
l=0

d −7l
2 (p + 1)−l/p β(α, l, m).
l

α and and every
Theorem 2.5 Let 1 ≤ p ≤ ∞ and 0 < α ≤ 2. Then we have for every f ∈ U∞
m ≥ d − 1,

α
sm (U∞
)p ≥ 2−αm γ (α, d, m) > [2−7 (2α − 1)]−1

2α −

127
128

d−1

2−αm

m
,
d−1

(2.44)

and
α
sd (U∞
)p > [2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 .

17

(2.45)


Proof. We take the univariate functions gk,s , gk as in (2.29), (2.30), and for every u ⊂ [d], define

the functions
u
gk,s
:=

gkj ,sj , k ∈ Zd+ , s ∈ I d (k),
j∈u

and
gku :=

gkj .
j∈u

If m, n ∈ N with n > m, and u ⊂ [d], we define the |u|-variate function fm,n (u) by
n
u
:= 2−5|u|
fm,n

2−αl

u
gk,s
,
|k|1 =l s∈I ( k)

l=m

and the d-variate function φm,n by

u
fm,n
.

φm,n :=
u⊂[d]

From the proof of Theorem 2.4 we can see that
α
φm,n ∈ U∞
,

(2.46)

φm,n (ξ) = 0, ξ ∈ Gd (m).

(2.47)

and

Observe that
n

2−5|u|

φm,n :=

2−αl
|k|1 =l

l=m

u⊂[d]

gku

From (2.38), (2.32), (2.33) we can also verify that if m is given then for arbitrary n > m,
φm,n

p

≥

φm,n

1
n

=

2

−5|u|

2−αl
n

2−7|u|

=

u⊂[d]

=
s=1

2−αl
l=m

d −7s
2
s

1

|k|1 =l

l=m

u⊂[d]

gku
l−1
|u| − 1

n

2−αl
l=m

(2.48)

l−1
s−1

18


By (2.37) we have Sm (Φ, φm,n ) = 0 for arbitrary Φ, and consequently, by (2.40) for arbitrary
n > m,
d
α
sm (U∞
)p ≥

φm,n

p

d −7s
2
s

≥
s=1

n

2−αl
l=m

l−1
.
s−1

(2.49)

This means that for t = 2−α ,
d
α
sm (U∞
)p

≥
s=1

d −7s
2
s

∞

2

−αl

l=m

d

l−1

s−1

=
s=1

d −7s
2
Fm−1,s−1 (t).
s

(2.50)

Therefore, applying (2.4) gives
d
α
sm (U∞
)p >
s=1
−αm

= 2

d −7s −αm
2
2
tFm,s−1 (t)
s
γ (α, d, m)
d

> 2−αm
s=1
d−1

= 2

−αm
s=0
d−1

= 2−αm
l=0

= [2

α

s−1
l=0

m
(2α − 1)l
l

d
2−7(s+1) (2α − 1)−(s+1)
s+1
m
(2α − 1)l
l

l

(2 − 1)]

−1 −αm

2

s
l=0

m
(2α − 1)l
l

d
2−7(s+1) (2α − 1)−(s+1)
s+1

s=0

m
(2α − 1)d−1
d−1

> 2−αm
−7

d −7s α

2
(2 − 1)−s
s

l
s=0

d − 1 −7(s+1) α
2
(2 − 1)−(s+1)
s

m
(2α − 1)d−1
d−1

d−1
s=0

d−1
[2−7 (2α − 1)]−s
s

= [2−7 (2α − 1)]−1 (2α − 1)d−1 [1 + 2−7 (2α − 1)−1 ]d−1 2−αm
= [2−7 (2α − 1)]−1

2α −

127
128

d−1

2−αm

which proves (2.44).

19

m
d−1

m
d−1

(2.51)


From the fifth inequality in (2.52) we have
d−1

d−1
(2α − 1)l
l

α
sd−1 (U∞
)p > 2−α(d−1)
l=0

d−1
−7

> [2

α

−1 −α(d−1)

(2 − 1)]

2

l=0
d−1

= [2−7 (2α − 1)]−1 2−α(d−1)
= [2

−7

α

−1 −α(d−1)

(2 − 1)]

2

l=0

α

l
s=0

d − 1 −7(s+1) α
2
(2 − 1)−(s+1)
s

d−1
(2α − 1)l
l

l
s=0

l
[2−7 (2α − 1)]−s
s
(2.52)

d−1
(2α − 1)l [1 + 2−7 (2α − 1)]l
l

[2 + 2−7 ]d−1

= [2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 .
This proves (2.45).

2.5

Cubature

We are interested in cubature formulas on Smolyak grid for approximately computing of the integral
I(f ) :=

f (x) dx.

(2.53)

[0,1]d
α.
for a f ∈ U∞

If Λm = (λξ )ξ∈Gd (m) , we consider the cubature formula Λsm (f ) := Λm (Gd (m), f ) on grids Gd (m)
given by
Λsm (f ) =

λξ f (ξ).
ξ∈Gd (m)

The quantity of optimal cubature Intsm (Fd ) on Smolyak grids Gd (m) is introduced by
Intsm (Fd ) := inf sup |f − Λsm (f )|.

(2.54)

Λm f ∈Fd

For a family Φ = {ϕξ }ξ∈Gd (m) of functions on Td , the linear sampling algorithm Sm (Φ, ·)
generates the cubature formula Λsm (f ) on Smolyak grid Gd (m) by
Λsm (f ) =

λξ f (ξ),

(2.55)

ξ∈Gd (m)

where the integration weights Λm = (λξ )ξ∈Gd (m) are given by
λξ =

ϕξ (x) dx.

(2.56)

Id

20


Hence, it is easy to see that
|I(f ) − Λsm (f )| ≤

f − Sm (Φ, f ) 1 ,

and, as a consequence of (1.8) and (2.54),
Intsm (Fd ) ≤ sm (Fd )1 .

(2.57)

˚α , based on the grids G
˚d (m), we can define the cubature formula ˚
For functions f ∈ U
Λsm (f ) :=
∞
s
d
d
˚
˚ (m), f ) on grids G
˚ (m), and the quantity of optimal cubature Int
˚ (Fd ) on Smolyak grids
Λ(G
m
d
˚
G (m). Moreover, thereholds true the inequality
˚ sm (Fd ) ≤ ˚
Int
sm (Fd )1 .

(2.58)

Theorem 2.6 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every every m ≥ 2d−1 ,
˚ s (U
˚α ) ≤ 4−d 2−αm β(α, d, m),
2−7d 2−αm β(α, d, m) ≤ Int
m

∞

(2.59)

and therefore,
˚ s (U
˚α ) ≤ exp(2α − 1) 4−d 2−αm md−1 . (2.60)
(2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 ≤ Int
m
∞
Moreover, if in addition, m ≥ 2(d − 1),
(2α − 1)−1 7−d 2−αm

m
d−1

α
˚ sm (U
˚∞
≤ Int
) ≤ a◦ (α, d) 2−αm

m
,
d−1

(2.61)

where
 −d


4 ,


a◦ (α, d) := |2α − 2|−| sgn(α−1)| × d 4−d ,




[4(2α − 1)]−d ,

α > 1,
(2.62)

α = 1,
α < 1,

and
α
˚ sd (U
˚∞
(2α − 1)−d 2−7d ≤ Int
) ≤ 4−d .

(2.63)

Proof. The upper bounds in (2.61)–(2.63) follow from (2.58) and Theorem 2.2.
α as in (2.35) with the property as
˚∞
To prove the lower bounds, we take the function fm,n ∈ U

in (2.37):

˚d (m).
fm,n (ξ) = 0, ξ ∈ G
Notice that fm,n is a nonnegative function. Hence, we have ˚
Λsm (Φ, fm,n ) = 0 for arbitrary Φ, and
consequently, by (2.40) for arbitrary n ≥ m,
α
˚ sm (U
˚∞
Int
) ≥ |fm,n − ˚
Λsm (Φ, fm,n )| =

n

fm,n

1

≥ 2−7d

2−αl
l=m

21

l−1
.
d−1

(2.64)


˚ s (U
˚α ) can be estimated from below as in the proof of
Comparing with (2.41), we can see that Int
m
∞
˚α ). This proves the lower bounds in in (2.61)–(2.63).
Theorem (2.4) for ˚
ssm (U
∞
Put
d

γ(α, d, m) :=
l=0

d −l
4 β(α, l, m).
l

In a similar way we can prove
Theorem 2.7 Let 0 < p ≤ ∞ and 0 < α ≤ 2. Then we have for every every m ≥ 2d−1 ,
α
) ≤ 2−αm γ(α, d, m),
2−αm γ (α, d, m) ≤ Intsm (U∞

(2.65)

and therefore,
α
(2α − 1)−1 2−7d (d − 1)−(d−1) 2−αm md−1 ≤ Intsm (U∞
) ≤ exp(2α − 1) 4−d 2−αm md−1 . (2.66)

Moreover, if in addition, m ≥ 2(d − 1),
[2−7 (2α − 1)]−1

2α −

127
128

d−1

2−αm

m
d−1

α
≤ Intsm (U∞
) ≤ a(α, d) 2−αm

m
, (2.67)
d−1

where


d

(5/4) ,


a(α, d) := |2α − 2|−| sgn(α−1)| × d (5/4)d ,




[1 + 1/4(2α − 1)]d ,

α > 1,
α = 1,

(2.68)

α < 1.

Further, we have
α
) ≤ (5/4)d [1 + 1/(2α − 1)]d .
[2−7 (2α − 1)]−1 [1 + 2−7−α ]d−1 ≤ Intsd (U∞

3

3.1

(2.69)

Sampling recovery based on B-spine quasi-interpolation representations
Periodic B-spline quasi-interpolation representations

We introduce quasi-interpolation operators for functions on Rd . For a given natural number ,
denote by M the cardinal B-spline of order with support [0, ] and knots at the points 0, 1, ..., .
We introduce quasi-interpolation operators for functions on Rd . In what follows we fixed r ∈ N
and consider the cardinal B-spline M := M2r of even order 2r. Let Λ = {λ(s)}|j|≤µ be a given

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finite even sequence, i.e., λ(−j) = λ(j) for some µ ≥ r − 1. We define the linear operator Q for
functions f on R by
Q(f, x) :=

Λ(f, s)M (x − s),

(3.1)

λ(j)f (s − j + r).

(3.2)

s∈Z

where
Λ(f, s) :=
|j|≤µ

The operator Q is local and bounded in C(R) (see [5, p. 100–109]). An operator Q of the form
(3.1)–(3.2)is called a quasi-interpolation operator in C(R) if it reproduces P2r−1 , i.e., Q(f ) = f for
every f ∈ P2r−1 , where Pl denotes the set of d-variate polynomials of degree at most l − 1 in each
variable.
If Q is a quasi-interpolation operator of the form (3.1)–(3.2), for h > 0 and a function f on R,
we define the operator Q(·; h) by
Q(f ; h) := σh ◦ Q ◦ σ1/h (f ),
where σh (f, x) = f (x/h).
Let Q be a quasi-interpolation operator of the form (3.1)–(3.2) in C(R). If k ∈ Z+ , we introduce
the operator Qk by
Qk (f, x) := Q(f, x; h(k) ), x ∈ R,

h(k) := (2r)−1 2−k .

We define the integer translated dilation Mk,s of M by
Mk,s (x) := M (2r2k x − s), k ∈ Z+ , s ∈ Z,
where Z+ := {s ∈ Z : s ≥ 0}. Then we have for k ∈ Z+ ,
ak,s (f )Mk,s (x), ∀x ∈ R,

Qk (f )(x) =
s∈Z

where the coefficient functional ak,s is defined by
ak,s (f ) := Λ(f, s; h(k) ) =

λ(j)f (h(k) (s − j + r)).

(3.3)

|j|≤µ

Notice that Qk (f ) can be written in the form:
f (h(k) (s + r))Lk (x − s), ∀x ∈ R,

Qk (f )(x) =

(3.4)

s∈Z

where the function Lk is defined by
Lk := =

λ(j)Mk,j .

(3.5)

|j|≤µ

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From (3.4) and (3.5) we get for a function f on R,
Qk (f )

≤

C(R)

LΛ

C(R)

f

C(R)

≤

Λ

f

C(R) ,

(3.6)

where
λ(j)M (x − j − s),

LΛ (x) := =

|λ(j)|.

Λ =

s∈Z |j|≤µ

(3.7)

|j|≤µ

For k ∈ Zd+ , let the mixed operator Qk be defined by
d

Qk :=

Qki ,

(3.8)

i=1

where the univariate operator Qki is applied to the univariate function f by considering f as a
function of variable xi with the other variables held fixed.
We define the d-variable B-spline Mk,s by
d

Mki ,si (xi ), k ∈ Zd+ , s ∈ Zd ,

Mk,s (x) :=

(3.9)

i=1

where Zd+ := {s ∈ Zd : si ≥ 0, i ∈ [d]}. Then we have
Qk (f, x) =

ak,s (f )Mk,s (x),

∀x ∈ Rd ,

s∈Zd

where Mk,s is the mixed B-spline defined in (3.9), and


d

ak,s (f ) = 

akj ,sj  (f ),

(3.10)

j=1

and the univariate coefficient functional aki ,si is applied to the univariate function f by considering
f as a function of variable xi with the other variables held fixed.
Since M (2r2k x) = 0 for every k ∈ Z+ and x ∈
/ (0, 1), we can extend the univariate B-spline
M (2r.2k ·) to an 1-periodic function on the whole R. Denote this periodic extension by Nk and
define
Nk,s (x) := Nk (x − s), k ∈ Z+ , s ∈ I(k),
where I(k) := {0, 1, ..., 2r2k − 1}.
We define the d-variable B-spline Nk,s by
d

Nki ,si (xi ), k ∈ Zd+ , s ∈ I d (k),

Nk,s (x) :=
i=1

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(3.11)


d
i=1 I(ki ).

where I d (k) :=
Qk (f, x) =

Then we have for functions f on Td ,

ak,s (f )Nk,s (x),

∀x ∈ Td .

(3.12)

s∈I d (k)

Since the function LΛ defined in (3.7) is 1-periodic, from (3.6) it follows that for a function f
on T,
Qk (f )

C(T)

≤

LΛ

C(T)

f

≤

C(T)

Λ

f

C(T) ,

(3.13)

For k ∈ Zd+ , we write k → ∞ if ki → ∞ for i ∈ [d]). The following lemma can be proven in a
way similar to the proof of [15, Lemma 2.2].
Lemma 3.1 We have for every f ∈ C(Td ),
f − Qk (f )

C(Td )

u
ω2r

(f, 2−k ),

≤ C

(3.14)

u⊂[d], u=∅

and, consequently,
f − Qk (f )

C(Td )

→ 0, k → ∞.

(3.15)

For convenience we define the univariate operator Q−1 by putting Q−1 (f ) = 0 for all f on I.
Let the operators qk and qu,k be defined in the manner of the definition (3.8) by
d

(Qki − Qki −1 ) , k ∈ Zd+ .

qk :=

(3.16)

i=1

We have

Qk =

qk .

(3.17)

k ≤k

From (3.17) and (3.15) it is easy to see that a continuous function f has the decomposition
f =

qk (f )
k∈Zd+

with the convergence in the norm of C(Td ).
From the definition of (3.16) and the refinement equation for the B-spline M , in the univariate
case, we can represent the component functions qk (f ) as
qk (f ) =

ck,s (f )Nk,s ,

(3.18)

s∈I d (k)

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