PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 161–171 DOI: 10.2298/PIM1409161T

ON THE CONVERSE THEOREM OF APPROXIMATION IN VARIOUS METRICS FOR NONPERIODIC FUNCTIONS

Miloš Tomić

Communicated by Gradimir Milovanović

Abstract. The modulus of smoothness in the norm of space L_{q} of nonperiodic functions of several variables is estimated by best approximations by entire functions of exponential type in the metric of space L_{p}, 1 ⩽ p ⩽ q < ∞.

1. Introduction and preliminaries

A converse theorem of approximation in various metrics for 2π periodic functions of several variables was proved in [5]. We prove the theorem of representation for the derivative of a function, and then the analogous converse theorem for nonperiodic functions defined on the space R^{n}. In this way we generalize and improve the results from [4, 6.4].

As usually we say that f(x_{1}, …, x_{n}) ∈ L_{p}(R^{n}), 1 ⩽ p < ∞ if

∥f∥_{p} = ( ∫_{R^{n}} |f|^{p}dx_{1} …dx_{n}) ^{1/p} = ( ∫_{R^{n}} |f|^{p}dx) ^{1/p} < ∞, x = (x_{1}, x_{2}, …, x_{n})

The notions of the best approximation and of the modulus of smoothness are given in [2] and [4].

Let g_{ν} = g_{ν_{1} ...ν_{n}} (x_{1}, …, x_{n}), ν = (ν_{1}, …, ν_{n}), (g_{ν} ∈ L_{p}) be an entire function of exponential type ν_{i} with respect to the variable x_{i} (i = 1, 2, …, n). The best approximation E_{ν_{1} ,...,ν_{n}} (f)_{p} of a function f ∈ L_{p}(R^{n}) by entire functions of exponential type is the quantity

E_{ν_{1} ,...,ν_{n}} (f)_{p} = inf_{g_{ν}} ∥f − g_{ν_{1} ...ν_{n}} ∥_{p}

2010 Mathematics Subject Classification: Primary 42B99.

Key words and phrases: modulus of smoothness, the converse theorem of approximation.

The modulus of smoothness of order k of a function f with respect to the variable x_{i} is

ω_{k}(f; δ_{i})_{p} = ω_{k}(f; 0, …, 0, δ_{i}, 0, …, 0)_{p} = sup_{|h_{i} |⩽δ_{i}} ∥Δ_{h_{i}}^{k} f∥_{p}

where

∥Δ_{h_{i}}^{k} f = ∑_{j=0}^{k} (−1)^{k−j} \binom{k}{j} f(x_{1}, …, x_{i−1} , x_{i} + jh_{i}, x_{i+1} , …, x_{n}).

The derivative of a function f is denoted by

f^{(ν_{1} ,...,ν_{n} )} = \frac{∂^{r_{1} +···+r_{n}} f}{∂x_{1}^{r_{1}} …∂x_{n}^{r_{n}}}

Lemma 1.1. If A_{i} ↓ 0 as i → ∞, then for λ = 1, 2, … and s ⩾ 1 the following inequalities hold

(1.1)  2^{(λ−1)s} A_{2^{λ}} ⩽ ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} A_{i}

(1.2)  2^{(λ+1)s} A_{2^{λ}} ⩽ 2^{2s} ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} A_{i}

Proof. We have

∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} = (2^{λ−1} + 1)^{s−1} + ⋯+ (2^{λ})^{s−1} ⩾ (2^{λ−1} + 1)^{s−1} · 2^{λ−1} ⩾ (2^{λ−1} )^{s}.

Therefore

(1.3)  2^{(λ−1)s} ⩽ ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} .

Since the sequence A_{i} is monotonic, (1.1) follows from (1.3). Multiplying inequality (1.1) by 2^{2s}, we get inequality (1.2). □

Lemma 1.2. If A_{i} ↓ 0 as i → ∞, and s ⩾ 1, then the following inequality holds

(1.4)  ∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} A_{i} ⩽ 2^{2s−1} ∑_{i=2^{m−2} +1}^{2^{m−1}} i^{s−1} A_{i}, m = 2, 3, …

Proof. The following inequalities hold because the sequence A_{i} is monotonic

∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} A_{i} ⩽ A_{2^{m−1}} ∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} , (1.5)

∑_{i=2^{m−2} +1}^{2^{m−1}} i^{s−1} A_{i} ⩾ A_{2^{m−1}} ∑_{i=2^{m−2} +1}^{2^{m−1}} i^{s−1} , (1.6)

We have

∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} ⩽ (2^{m})^{s−1} · 2^{m−1} ,

∑_{i=2^{m−2} +1}^{2^{m−1}} i^{s−1} ⩾ (2^{m−2} )^{s−1} · 2^{m−2} = 2^{1−2s} · (2^{m})^{s−1} · 2^{m−1} .

From the above two inequalities it follows

(1.7)  ∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} ⩽ 2^{2s−1} ∑_{i=2^{m−2} +1}^{2^{m−1}} i^{s−1} .

Multiplying (1.7) by A_{2^{m−1}} and in view of (1.5) and (1.6), we get (1.2). □

Remark 1.1. Lemmas 1.1 and 1.2 are valid for 0 < s < 1 also, with different constants C = C(s). So inequality (1.1) becomes

2^{(λ−1)s} A_{2^{λ}} ⩽ 2^{s−1} ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} A_{i} (0 < s < 1).

2. Theorem of representation

Let g_{ν} = g_{ν_{1} ...ν_{n}} (x_{1}, …, x_{n}), ν = (ν_{1}, …, ν_{n}), be an entire L_{p} function of exponential type ν_{i} with respect to the variable x_{i} (i = 1, 2, …, n), by which the best approximation E_{ν_{1} ,...,ν_{n}} (f)_{p} is achieved, i.e., let

(2.1)  E_{ν_{1} ,...,ν_{n}} (f)_{p} = ∥f − g_{ν_{1} ...ν_{n}} ∥_{p}.

From these entire functions g_{ν_{1} ...ν_{n}} (x_{1}, …, x_{n}) we create entire functions

(2.2)  ξ_{λ} = g_{2^{(λ+1)l_{1}} ...2^{λ+1} ...2^{(λ+1)ln}} − g_{2^{λl_{1}} ...2^{λ} ...2^{λln}} , λ = 0, 1, 2, …

for given natural numbers l_{j} (j = 1, 2, …, n) where l_{i} = 1 for a chosen number i ∈ {1, 2, …, n}. The function ξ_{λ} is entire of exponential type 2^{(λ+1)l_{j}} with respect to x_{j}.

Theorem 2.1. Let f ∈ L_{p}(R^{n}) and r_{j} be nonnegative integers, and l_{j} (j = 1, …, n) be natural numbers, where l_{i} = 1 for some i ∈ {1, 2, …, n}. If the following inequality holds for the best approximation of the function

(2.3)  ∑_{λ=1}^{∞} λ^{qσ−1} E_{λ^{l_{1}} ...λ...λ^{ln}} (f)_{p} < ∞,

where

(2.4)  σ = ∑_{j=1}^{n} l_{j} ( r_{j} + \frac{1}{p} − \frac{1}{q}) , 1 ⩽ p ⩽ q < ∞,

then the function f has a derivative f^{(r_{1} ...r_{n} )} belonging to L_{q} and in the sense of L_{q} the equality

(2.5)  f^{(r_{1} ,...,r_{n} )} =^{(q)} g_{1...1}^{(ν_{1} ...ν_{n} )} + ∑_{λ=0}^{∞} ξ_{λ}^{r_{1} ,...,r_{n}}

holds.

Proof. For the sum

(2.6)  G_{m} = g_{1...1} + ∑_{λ=0}^{m} ξ_{λ}, m = 0, 1, 2 …

the equality

(2.7)  G_{m} = g_{2^{(m+1)l_{1}} ...2^{m+1} ...2^{(m+1)ln}}

holds. In view of (2.1) and (2.7) we conclude that

∥f − G_{m}∥_{p} = E_{2^{(m+1)l_{1}} ...2^{m+1} ...2^{(m+1)ln}} (f)_{p}

hence, it follows that

(2.8)  ∥f − G_{m}∥_{p} → 0 as m → ∞.

This means that the equality

(2.9)  f =^{(p)} g_{1...1} + ∑_{λ=0}^{∞} ξ_{λ}

holds in L_{p}.

In the next step we prove (2.9) holds in L_{q}. For ξ_{λ} we have

(2.10)  ∥ξ_{λ}∥_{p} ⩽ 2E_{2^{λl_{1}} ...2^{λ} ...2^{λln}} (f)_{p}.

Applying the inequality of various metrics of Nikolsky [2, 3.3.5] we obtain

∥ξ_{λ}∥_{q} ⩽ 2^{n}( ∏_{j=1}^{n} 2^{(λ+1)l_{j}} ) ^{1/p−1/q} ∥ξ_{λ}∥_{p}

hence, in view of (2.10), it follows

(2.11)  ∥ξ_{λ}∥_{q} ≪ 2^{n}( ∏_{j=1}^{n} 2^{(λ+1)l_{j}} ) ^{1/p−1/q} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}} (f)_{p}.

We will estimate the sum

(2.12)  G_{t} − G_{m} = ∑_{λ=m+1}^{t} ξ_{λ}, m < t,

in the norm L_{q}. With the aim of estimating the quantity A = ∥G_{t} − G_{m}∥_{q}^{q} we will apply a method which has been used in several papers. For example, the method was applied in [3] and [1] (see the estimate of A in Lemma 1). The method was also applied in [6] to estimate quantity A from (2.6) to (2.45). Therefore, taking into account (2.11), from (2.12), we get

(2.13) ∥G_{t} − G_{m}∥_{q} ≪ \{ ∑_{λ=m+1}^{t} exp_{2} ( λq( \frac{1}{p} − \frac{1}{q}) ∑_{j=1}^{n} l_{j} ) E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}\}^{1/q}.

Following the proof in [6] and starting from equality (2.12), we will now prove inequality (2.13). Denote

(2.14)  A = ∥G_{t} − G_{m}∥_{q}^{q} = ∥ ∑_{λ=m+1}^{t} ξ_{λ} ∥_{q}^{q}, m < t.

For a given number q denote [q] + 1 = k. This means that k ∈ {2, 3, …} and that q/k < 1. From (2.14) it follows that (2.15)

A = ∫ | ∑_{λ=m+1}^{t} ξ_{λ} |^{q}dx = ∫ | ∑_{λ=m+1}^{t} ξ_{λ} |^{\frac{q}{k} k}dx ⩽ ∫ ( ∑_{λ=m+1}^{t} |ξ_{λ}|^{\frac{q}{k}} ) ^{k}dx, ∫ = ∫_{R^{n}} .

Denote

(2.16)  δ_{λ} = |ξ_{λ}|^{q/k}.

We get

(2.17)  A ⩽ ∫ ( ∑_{λ=m+1}^{t} δ_{λ}) ^{k}dx.

As k = k(q) is an integer, then

(2.18)  ( ∑_{λ=m+1}^{t} δ_{λ}) ^{k} ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} ∏_{j=1}^{k} δ_{λ_{j}} .

Now from (2.17), based on (2.18), we get

(2.19)  A ⩽ ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} ∫ ∏_{j=1}^{k} δ_{λ_{j}} dx.

Using the equality

(2.20)  ∏_{j=1}^{k} D_{j} = ( ∏_{r,s=1, r<s}^{k} D_{r}D_{s}) ^{1/(k−1)}

for D_{j} = δ_{λ_{j}} from (2.19) we obtain

(2.21)  A ⩽ ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} ∫ ( ∏_{r,s=1, r<s}^{k} δ_{λ_{r}} δ_{λ_{s}} ) ^{1/(k−1)} dx.

Applying Hölder’s integral inequality to a product of \frac{1}{2}k(k − 1) factors, from (2.21) we get that

(2.22)  A ⩽ ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} ∏_{r,s=1, r<s}^{k} [ ∫ (δ_{λ_{r}} δ_{λ_{s}} )^{k/2} dx]^{2/k(k−1)} .

Based on (2.16) we get

(2.23)  Γ_{rs} = ∫ (δ_{λ_{r}} δ_{λ_{s}} )^{k/2} dx = ∫ (|ξ_{λ_{r}} |^{q/2} |ξ_{λ_{s}} |^{q/2})dx.

For α = \frac{p+q}{p} , α′ = \frac{p+q}{q} , we have \frac{1}{α} + \frac{1}{α′} = 1. Therefore by applying Hölder’s inequality, we get

(2.24)  Γ_{rs} ⩽ (∥ξ_{λ_{r}} ∥_{qα/2} )^{q/2}(∥ξ_{λ_{s}} ∥_{qα′/2} )^{q/2}.

The function ξ_{λ} is entire of exponential type 2^{(λ+1)l_{j}} with respect to x_{j}, j = 1, 2, …, n. Therefore applying the inequality of Nikolsky [2, 3.3.5] we get

(2.25)  (∥ξ_{λ_{r}} ∥_{qα/2} )^{q/2} ≪ (∥ξ_{λ_{r}} ∥_{p})^{q/2} exp_{2} ( ( ∑_{j=1}^{n} λ_{r}l_{j} ) ( \frac{q}{2p} − \frac{1}{α}) ) .

(2.26)  (∥ξ_{λ_{s}} ∥_{qα′/2} )^{q/2} ≪ (∥ξ_{λ_{s}} ∥_{p})^{q/2} exp_{2} ( ( ∑_{j=1}^{n} λ_{s}l_{j} ) ( \frac{q}{2p} − \frac{1}{α′}) ) .

Using the equality

(2.27)  \frac{q}{2p} − \frac{1}{β} = \frac{q}{2}( \frac{1}{p} − \frac{1}{q}) + \frac{1}{2} − \frac{1}{β}, β ∈ \{α, α′\},

from (2.24), based on (2.25), (2.26) and (2.10), we get

(2.28) Γ_{rs} ≪ exp_{2} ( [λ_{r}( \frac{1}{2} − \frac{1}{α}) + λ_{s}( \frac{1}{2} − \frac{1}{α}′) ] ∑_{j=1}^{n} l_{j} )

×\{exp_{2}( [(λ_{r}+λ_{s})q( \frac{1}{p}−\frac{1}{q}) ] ∑_{j=1}^{n} l_{j}) E_{2^{λr l_{1}} ...2^{λr} ...2^{λr ln}}^{q} (f)_{p}E_{2^{λs l_{1}} ...2^{λs} ...2^{λs ln}}^{q} (f)_{p}\}^{1/2}.

Denote

(2.29)  H_{i} = exp_{2} ( iq( \frac{1}{p} − \frac{1}{q}) ∑_{j=1}^{n} l_{j} ) E_{2^{il_{1}} ...2^{i} ...2^{iln}}^{q} (f)_{p}.

Then

(2.30)  Γ_{rs} ≪ exp_{2} ( [λ_{r}( \frac{1}{2} − \frac{1}{α}) + λ_{s}( \frac{1}{2} − \frac{1}{α′}) ] ∑_{j=1}^{n} l_{j} ) H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} .

Since \frac{1}{α′} = 1 − \frac{1}{α} , it holds that

λ_{r}( \frac{1}{2} − \frac{1}{α}) + λ_{s}( \frac{1}{2} − \frac{1}{α′}) = −(λ_{s} − λ_{r})( \frac{1}{2} − \frac{1}{α}) .

CONVERSE THEOREM OF APPROXIMATION 167

Therefore from (2.30) it follows

(2.31)  Γ_{rs} ≪ exp_{2} ( − (λ_{s} − λ_{r})( \frac{1}{2} − \frac{1}{α}) ∑_{j=1}^{n} l_{j} ) H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} .

If we apply Hölder’s inequality so that α′ relates to the first factor, and α to the second one, then in the same way we conclude that

(2.32)  Γ_{rs} ≪ exp_{2} ( − (λ_{r} − λ_{s})( \frac{1}{2} − \frac{1}{α}) ∑_{j=1}^{n} l_{j} ) H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} .

Based on (2.31) and (2.32) we conclude that

(2.33)  Γ_{rs} ≪ exp_{2} ( − |λ_{r} − λ_{s}|( \frac{1}{2} − \frac{1}{α}) ∑_{j=1}^{n} l_{j} ) H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} .

Denote

(2.34)  a(λ_{s}, λ_{r}) = exp_{2} ( − |λ_{r} − λ_{s}|( \frac{1}{2} − \frac{1}{α}) ∑_{j=1}^{n} l_{j} ) ,
(2.35)  Q = ∏_{r,s=1, r<s}^{k} {a(λ_{s}, λ_{r})H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} }^{2/k(k−1)} .

From (2.22), based on (2.23), (2.33), (2.34) and (2.35), it follows

(2.36)  A ⩽ ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} Q.

We will now estimate the product Q. Based on (2.20) it holds that

∏_{r,s=1, r<s}^{k} {H_{λ_{r}}^{1/2} H_{λ_{s}}^{1/2} }^{1/(k−1)} = ∏_{j=1}^{k} H_{λ_{j}}^{1/2}

and then, using (2.35), we get

(2.37)  Q = ∏_{j=1}^{k} H_{λ_{j}}^{1/k} ∏_{r,s=1, r<s}^{k} {a(λ_{s}, λ_{r})}^{2/k(k−1)} .

It holds a(λ_{s}, λ_{r}) = a(λ_{r}, λ_{s}) and a(λ_{r}, λ_{r}) = 1. Therefore

(2.38) ∏_{r,s=1, r<s}^{k} a(λ_{r}, λ_{s}) = ∏_{r=1}^{k} ∏_{s=1}^{k} a^{1/2} (λ_{r}, λ_{s}).

From (2.37) based on (2.38) it follows

(2.39)  Q = ∏_{r=1}^{k} H_{λ_{r}}^{1/k} { ∏_{s=1}^{k} [a(λ_{s}, λ_{r})]^{1/(k−1)} }^{1/k} .

Now from (2.36) based on (2.39) we get

(2.40)  A ≪ ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} ∏_{r=1}^{k} H_{λ_{r}}^{1/k} { ∏_{s=1}^{k} [a(λ_{r}, λ_{s})]^{1/(k−1)} }^{1/k} .

In the inequality (2.40) the product has k factors

L_{r} = H_{λ_{r}}^{1/k} { ∏_{s=1}^{k} [a(λ_{r}, λ_{s})]^{1/(k−1)} }^{1/k}

with the exponent 1/k. The sum of these exponents is 1. Therefore we can apply Hölder’s inequality and get

(2.41)  A ≪ ∏_{r=1}^{k} { ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} H_{λ_{r}} ∏_{s=1}^{k} [a(λ_{r}, λ_{s})]^{1/(k−1)} }^{1/k} .

Denote

(2.42)  M_{r} = ∑_{λ_{1} =m+1}^{t} ⋯ ∑_{λ_{k} =m+1}^{t} H_{λ_{r}} ∏_{s=1}^{k} [a(λ_{r}, λ_{s})]^{1/(k−1)} , r = 1, …, k.

Since λ_{r} = m + 1, …, t for every r = 1, …, k, then

(2.43)  M_{1} = M_{2} = ⋯= M_{k} = M.

We will estimate, for example, M_{1} = M. Since a(λ_{1}, λ_{1}) = 1, then from (2.42) after some calculation we get (2.44)

M = M_{1} = ∑_{λ_{1} =m+1}^{t} H_{λ_{1}} ∑_{λ_{2} =m+1}^{t} [a(λ_{1}, λ_{2})]^{1/(k−1)} ⋯ ∑_{λ_{k} =m+1}^{t} [a(λ_{1}, λ_{k})]^{1/(k−1)} .

Based on (2.34) we conclude that

(2.45)  ∑_{λ_{r} =m+1}^{t} [a(λ_{1}, λ_{r})]^{1/(k−1)} ⩽ C(p, q), r = 2, 3, …, k.

Now from (2.44) based on (2.45) it follows

(2.46)  M ≪ ∑_{λ_{1} =m+1}^{t} H_{λ_{1}} .

From (2.41), using (2.42), (2.43) and (2.46), we get

(2.47)  A ≪ ∏_{r=1}^{k} M^{1/k} = M ≪ ∑_{i=m+1}^{t} H_{i}.

Based on (2.47) and (2.29) we conclude that

(2.48)  A ≪ ∑_{i=m+1}^{t} exp_{2} ( iq( \frac{1}{p} − \frac{1}{q}) ∑_{j−1}^{n} l_{j} ) E_{2^{il_{1}} ...2^{i} ...2^{iln}}^{q} (f)_{p}.

Finally, from (2.48), based on (2.14), the inequality (2.13) follows. If r_{j} = 0, then σ = ( \frac{1}{p} − \frac{1}{q}) ∑ _{j=1}^{n} l_{j}, therefore in view of (2.3) and (2.13) we deduce that the sequence {G_{m}} is a Cauchy sequence in the space L_{q} and therefore it tends to a function f in L_{q} [2, 1.3.9]. Thus, we have

(2.49)  f =^{(q)} g_{1...1} + ∑_{λ=0}^{∞} ξ_{λ}.

In the next step we prove equality (2.5). To do it we estimate the quantity

(2.50)  B = ∥G_{t}^{(r_{1} ,...,r_{n} )} − G_{m}^{(r_{1} ,...,r_{n} )} ∥_{q}^{q} = ∥ ∑_{λ=m+1}^{t} ξ_{λ}^{(r_{1} ,...,r_{n} )} ∥_{q}^{q}.

Applying the inequality of the Bernstein type [2, 3.2.2], we get

∥ξ_{λ}^{(r_{1} ,...,r_{n} )} ∥_{q} ⩽ ( ∏_{j=1}^{n} 2^{l_{j} r_{j}} ) 2^{λ(l_{1} r_{1} +···+l_{n} r_{n} )} ∥ξ_{λ}∥_{q}

hence, in view of (2.11), it follows

(2.51)  ∥ξ_{λ}^{(r_{1} ,...,r_{n} )} ∥ ≪ 2^{λσ} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}} (f)_{p}.

Now, using for B the same procedure by which we estimated A, we get (see the estimation of B in [6, (2.50)–(2.65)]

(2.52)  ∥G_{t}^{(r_{1} ,...,r_{n} )} − G_{m}^{(r_{1} ,...,r_{n} )} ∥_{q} ≪ { ∑_{λ=m+1}^{t} 2^{λqσ} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}}^{1/q}.

In view of condition (2.3) and inequality (2.52) we conclude that the sequence {G_{m}^{(r_{1} ,...,r_{n} )} } is a Cauchy sequence in L_{q}. If we denote G_{m}^{(r_{1} ,...,r_{n} )} → h, m → ∞, then we conclude (see [2, 4.4.7] or [4, 6.3.31]) that h = f^{(r_{1} ,...,r_{n} )} . This means that equality (2.5) holds. □

3. The converse theorem of approximation

Now we are going to prove a converse theorem of approximation, analogously to the result in [5] and give some consequences.

Theorem 3.1. Let the conditions of Theorem 2.1 be satisfied (the condition (2.3) where σ is given by (2.4)), and let k and m_{i} be given natural numbers. Then the inequality

ω_{k}( f^{(r_{1} ,...,r_{n} )} ; 0, …, 0, \frac{1}{m_{i}} , 0, …, 0)_{q} (3.1)

⩽ C\{ \frac{1}{m_{i}^{k}} [∥f∥_{p}^{q} + ∑_{λ=1}^{m_{i}} λ^{q(σ+k)−1} E_{λ^{l_{1}} ...λ...λ^{ln}}^{q} (f)_{p}]^{1/q}

+[ ∑_{λ=m_{i} +1}^{∞} λ^{qσ−1} E_{λ^{l_{1}} ...λ...λ^{ln}}^{q} (f)_{p}]^{1/q} }

holds, where the constant C does not depend either on f or m_{i} = 1, 2, . . ..

Proof. For the modulus of smoothness ω_{k} of the derivative f^{(r_{1} ,...,r_{n} )} of the function f we have

ω_{k}(f^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} ⩽ ω_{k}(f^{(r_{1} ,...,r_{n} )} − G_{m}^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} (3.2)

+ω_{k}(G_{m}^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} = I_{1} + I_{2}.

For I_{1} we obtain

(3.3)  I_{1} ≪ ∥f^{(r_{1} ,...,r_{n} )} − G_{m}^{(r_{1} ,...,r_{n} )} ∥_{q} = ∥ ∑_{λ=m+1}^{∞} ξ_{λ}^{(r_{1} ,...,r_{n} )} ∥_{q}.

In the same way by which inequality (2.17) was established, in view of (3.3), we conclude that

(3.4)  I_{1} ≪ { ∑_{λ=m+1}^{∞} 2^{λqσ} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}}^{1/q}.

In virtue of the properties of the modulus of smoothness [2, 4.4.4(2)] we have

(3.5)  I_{2} = ω_{k}(G_{m}^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} ⩽ \frac{1}{m_{i}^{k}} ∥G_{m}^{(r_{1} ,...,r_{i} +k,...,r_{n} )} ∥_{q}.

In the same way by which the inequality (2.17) was established, putting r_{i} + k instead of r_{i}, and since l_{i} = 1, we get the estimate

(3.6)  ∥G_{m}^{(r_{1} ,...,r_{i} +k,...,r_{n} )} ∥_{q} ≪ { ∥f∥_{p}^{q} + ∑_{λ=0}^{∞} 2^{λq(σ+k)} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}}^{1/q}.

Now, in view of (3.2), (3.4), (3.5) and (3.6), we obtain

(3.7) ω_{k}(f^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} ≪ { ∑_{λ=m_{i} +1}^{∞} 2^{λqσ} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}}^{1/q}

+ \frac{1}{m_{i}^{k}} \{ ∥f∥_{p}^{q} + ∑_{λ=0}^{m_{i}} 2^{λq(σ+k)} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p}\}^{1/q}.

Let

(3.8)  q(σ + k) = s, E_{2^{λl_{1}} ...2^{λ} ...2^{λln}}^{q} (f)_{p} = A_{2^{λ}} .

Then, using inequality (1.1), (Lemma 1.1), we get

∑_{λ=0}^{m} 2^{λs} A_{2^{λ}} = A_{1} + 2^{s}A_{2} + 2^{s} ∑_{λ=2}^{m} 2^{(λ−1)s} A_{2^{λ}} ⩽ A_{1} + 2^{s}A_{2} + 2^{s} ∑_{λ=2}^{m} ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{s−1} A_{i}

= A_{1} + 2^{s}A_{2} + 2^{s}{ ∑_{i=3}^{2^{m−1}} i^{s−1} A_{i} + ∑_{i=2^{m−1} +1}^{2^{m}} i^{s−1} A_{i}} .

CONVERSE THEOREM OF APPROXIMATION 171

Using Lemma 1.2, from the previous inequality, it follows

(3.9)  ∑_{λ=0}^{m} 2^{λs} A_{2^{λ}} ≪ ∑_{i=1}^{2^{m−1}} i^{s−1} A_{i}.

Choosing m so that 2^{m−1} ⩽ m_{i} < 2^{m}, from (3.9) it follows ∑ _{λ=0}^{m} 2^{λs} A_{2^{λ}} ≪ ∑ _{i=1}^{m_{i}} i^{s−1} A_{i}, i.e.,

(3.10) ∑_{λ=0}^{m} 2^{λq(σ+k)} A_{2^{λ}} ≪ ∑_{i=1}^{m_{i}} i^{q(σ+k)−1} A_{i}.

To estimate the first sum in (3.7) we use (1.2), (Lemma 1.1.), and get

∑_{λ=m+1}^{∞} 2^{λqσ} A_{2^{λ}} = 2^{−qσ} ∑_{λ=m+1}^{∞} 2^{(λ+1)qσ} A_{2^{λ}} ⩽ 2^{−qσ} 2^{2qσ} ∑_{λ=m+1}^{∞} ∑_{i=2^{λ−1} +1}^{2^{λ}} i^{qσ−1} A_{i}
= 2^{qσ}{(2^{m} + 1)^{qσ−1} A_{2^{m} +1} + ⋯+ (2^{m+1} )^{qσ−1} A_{2^{m+1}} + ⋯ },

hence, using that m_{i} < 2^{m}, it follows

(3.11)  ∑_{λ=m+1}^{∞} 2^{λqσ} A_{2^{λ}} ⩽ 2^{qσ} ∑_{λ=m_{i} +1}^{∞} i^{qσ−1} A_{i}.

Putting A_{i} = E_{i}^{q} (equality (3.8)), from (3.7) and (3.11), it follows (3.1). □

Corollary 3.1. For n = 1 it holds that l_{j} = 1, r_{j} = r, σ = r + \frac{1}{p} − \frac{1}{q} and we get the corresponding theorems and inequalities for a function of one variable.

Corollary 3.2. If l_{j} = 1, j = 1, 2, …, n and r_{j} = 0, j ≠ i, r_{i} = r, then σ = n(\frac{1}{p} − \frac{1}{q}) + r. Therefore, the condition

∑_{λ=1}^{∞} λ^{q[r+n(1/p−1/q)]−1} E_{λ...λ...λ}^{q} (f)_{p} < ∞

implies that the function f has a derivative ∂^{r}f/∂x^{r} with respect to any variable x_{i} belonging to L_{q}. For the modulus of smoothness the corresponding inequality holds.

Corollary 3.3. Applying the inequality (∑ a_{k})^{s} ⩽ ∑ (a_{k})^{s}, a_{k} ⩾ 0, 0 < s ⩽ 1, for s = 1/q, from (3.7) it follows

ω_{k}(f^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} ≪ ∑_{λ=m_{i} +1}^{∞} 2^{λσ} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}} (f)_{p}

+ \frac{1}{m_{i}^{k}} \{ ∥f∥_{p}^{q} + ∑_{λ=0}^{m_{i}} 2^{λ(σ+k)} E_{2^{λl_{1}} ...2^{λ} ...2^{λln}} (f)_{p}\} .

wherefrom

(3.12) ω_{k}(f^{(r_{1} ,...,r_{n} )} ; 1/m_{i})_{q} ≪ ∑_{λ=m_{i} +1}^{∞} λ^{σ−1} E_{λ^{l_{1}} ...λ...λ^{ln}} (f)_{p}

+ \frac{1}{m_{i}^{k}} \{ ∥f∥_{p} + ∑_{λ=1}^{m_{i}} λ^{σ+k−1} E_{λ^{l_{1}} ...λ...λ^{ln}} (f)_{p}\} .

For n = 1 inequality (3.12) implies inequality 6.4.1(3) in [4]. For r_{j} = 0, j ≠ i, r_{i} = r (j = 1, …, n) it holds that σ = r + (\frac{1}{p} − \frac{1}{q}) ∑ _{j=1}^{n} l_{j}, and from (3.12) it follows inequality 6.4.3(8) in [4].

Corollary 3.4. For p = q it holds that σ = ∑ _{j=1}^{n} l_{j} r_{j}, and from (3.12) we get the corresponding result in L_{p}.

Remark 3.1. Some results of this paper were presented at the First Mathematical Conference of the Republic of Srpska (Pale, May 21-22, 2011).

References

1. N. A. Il’yasov, An inverse theorem of approximation theory in different metrics, Mat. Zametki 50:6 (1991), 57–65, 158 (in Russian); translated in Math. Notes 50:5-6 (1991), 1253–1260

2. S. M. Nikol’skiĭ, Approximation of Functions of Several Variables and Imbedding Theorems, Second edition, revised and supplemented, Nauka, Moscow, 1977 (in Russian).

3. M. K. Potapov, Imbedding theorems in the mixed metric, Trudy Mat. inst. AN SSSR 156 (1980), 143–156 (in Russian)

4. A. F. Timan, Theory of Approximation of Functions of Real Variable, Gos. Izdat. FM, Moscow, 1960 (in Russian).

5. M. Tomić, On converse theorem of approximation in various metrics for periodic functions of several variables, Facta Univ., Ser. Math. Inform. 15 (2000), 49–56.

6. On representation of derivatives of functions in L_{p}, Mat. Vesn. 62(3) (2010), 235–250.

University of East Sarajevo (Received 12 08 2010) East Sarajevo (Revised 30 09 2013) Republic of Srpska