Infinite Dimensional Widths and Optimal Recovery of a Function Class in Orlicz Spaces in L(R) Metric

In this paper, we study the infnite dimensional widths and optimal recovery of Wiener–Sobolev smooth function classes W M ,1 ( P r ( D )) determined by the r -th diferential operator P r ( D ) in Orlicz spaces with L(R) metric. Using tools such as the H ¨o lder inequality, we give the exact values of the infnite dimensional Kolmogorov width and linear width of W M ,1 ( P r ( D )) in L(R) metric. We also study the related optimal recovery problem.


Introduction
In [1], the infnite dimensional widths problem and the optimal recovery problem of the Wiener-Sobolev class determined by diferential operators in L p spaces in L(R) metric are studied (where L(R) metric means the L 1 metric on R).In this paper, we study the infnite dimensional widths problem and the optimal recovery problem of the Wiener-Sobolev class W M,1 (P r (D)) determined by diferential operators in Orlicz spaces.
It is well known that Orlicz spaces are extensions and refnements of L p spaces.In particular, the Orlicz spaces generated by N-functions that do not satisfy the ∆ 2 -condition are a substantial generalization and promotion of the L p spaces.Terefore, the study of approximation problems in Orlicz spaces has potential application value and development prospect, such as in references [2][3][4].In recent years, the research on widths problem in Orlicz spaces has made some progress, such as references [5][6][7][8].
In this paper, let M(u) and N(v) be complementary N-functions, the defnition and properties of N-function are as follows.
Defnition 1.A real valued function M(u) defned on R is called an N-function, if it has the following properties: (1) M(u) is an even continuous convex function and M(0) � 0 (2) M(u) > 0 for u > 0 (3) lim u ⟶ 0 (M(u)/u) � 0, lim u ⟶ ∞ (M(u)/u) � ∞ Te complementary N-function is given by N(v) �  v 0 (M ′ ) − 1 (u)du.Te properties of N-functions are discussed in [9].Te Orlicz norm is defned by the following expression: All measurable functions u(x) { } with fnite Orlicz norms constitute the Orlicz space L * M (I) associated with the N-function M(u), where ρ(v; N) �  I N(v(x))dx expresses the modulus of v(x) with respect to N(v).According to the reference [9], the Orlicz norm can also be defned as follows: In this paper, ‖•‖ M [a,b] represents the Orlicz norm taken on the corresponding interval [a, b], and ‖•‖ M represents the Orlicz norm taken on the interval of the defnition domain involved in the conditions of the relevant conclusions.C is used to represent a constant, and in diferent places, its value can be diferent. Set where Z is the set of integers.As in references [10,11], defne a function space by reference [1], L * M,1 (R) is a Banach space.Given a natural number r, let C r 0 (R) represent the set of smooth functions 1) are absolutely continuous functions in an arbitrary finite interval  . (5) be a polynomial with only real roots, and P r (D)(D � d/dt) is the induced diferential operator of P r (t).Defne the Wiener-Sobolev space and the Wiener-Sobolev class in the Orlicz spaces as follows:

Preliminaries
For arbitrary λ > 0, let be a standard function with period (2/λ) defned by the diferential operator P r (D).Specially, if P r (D) � D r , then the function Φ r,λ (x) has the following form: . Ten, according to reference [1], we know is a Banach space with metric ‖•‖ ∞,∞ .Let be the corresponding Wiener-Sobolev space and Wiener-Sobolev class.
and do not have any other restrictions.If [α, β] is the interval that contains the point η 0 and Φ r,λ (x) is monotonous on [α, β], then the following statements are true: (1) If Φ r,λ (x) increases monotonically on [α, β], the following inequalities are true: (2) If Φ r,λ (x) decreases monotonically on [α, β], the following inequalities are true: and [a, b] is an interval such that g has only two zeros a and b on where Φ r+1,λ (x) represents the standard function defned by DP r (D).
Proof.From Lemma 2, similar to the proof of Teorem 5.7.1 in reference [12], Lemma 3 is easy to be proved.

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According to Lemma 3, we have the following expression.
Proof.Without losing generality, suppose c � 0, and we just consider this case.
From Lemma 2, we have the following expression: Hence, according to inequations (24) and (25) and the inequation above, the lemma can be proved.

Infinite Dimensional Widths Problem
Let T � t j   j∈Z be a real sequence and satisfy

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For each natural number m ≥ r, let P m (t) be an algebraic polynomial of degree m with only real roots, and P r (t) be its factor.Defne where where α � α j   j∈Z is a real sequence, N j,r (x) is the standardized B-spline where M j,r (x) is B-spline, its detailed defnition refer to reference [13], there is no need to go into details here.

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Terefore, to prove (35), we just need to prove inf Since  R N j,r (x)f (r) (x)dx � 0 and ‖f (r) ‖ N ≤ 1, using the H€ older inequality in the Orlicz spaces, we have the following expression: where the H€ older inequality in the Orlicz spaces is See the reference [9].Terefore, Also, for h ∈ E N , defne According to reference [14], we have We notice that N j,r (x) has compact support [t j , t j+r ] and the orthogonal condition  I N N j,r (x)h(x)dx � 0 satisfes for j ≤ − N − r and j ≥ N. So, the orthogonal condition is necessary only if − N − r < j < N.
According to the duality theorem of the best approximation of a function in a fnite dimensional subspace, we have sup where α * j,N � 0 if j ≤ − N − r and j ≥ N. Te right-hand side of (47) defnes a monotonically increasing bounded sequence, and we need to prove lim Journal of Mathematics Assume (49) is not true, then there exists ε > 0, such that when N is sufciently large, we have According to [14], there exists a constant C such that for every j ∈ Z, such that (51) Tus, using the diagonal rule we can fnd a sequence of positive integers N n   +∞ n�1 that satisfes for every j ∈ Z, we have the following expression: Set For ∀x ∈ R, we have x ∈ I N n if n is sufciently large.Since the support of N j,r (x) is compact,  α * j,N n N j,r (x) contains a fnite number of terms which is not zero.According to (53) and (54), we have Lemma, we have the following expression: So, we draw a contradiction.Terefore, inequation (49) holds.When the limit of (47) on the right side is taken, we can get the opposite inequation of (44).( 1) is proved.□ As seen in the proof of Lemma 5(1), for f ∈ L * M (R), we have the following expression: where Also, a function space according to reference [1], we know L * N,∞ (R) is also a Banach space.
Lemma 6 (see [10,15]). (61) is called the best approximation of g by S n (P m (D)), and is called the best approximation of W M,1 (P r (D)) by S n (P m (D)) in L(R) metric.
Let n ≥ 0 be a fxed number (not necessarily an integer), F n represent the set of all linear subspaces F on L(R), such that for every F ∈ F n , we have the following expression: where F| [− a,a] indicates the limit of F on [− a, a] and dim(F ) is the dimension of the linear space F| [− a,a] .Quantity Ten, F * is the optimal subspace that reaches d n .
Theorem .Let P r (t), P m (t) be defned as above.If n is a natural number, then where m ≥ r, and P m (0) � 0, lim t ⟶ 0 (P m (t)/P r (t)) � 0 for m > r.
Proof.From (58) and integrating by parts, we can obtain the following expression: From Lemma 6, we have the following expression: Terefore, from Lemma 2, Lemma 4, and Lemma 6, we have the following expression: Journal of Mathematics where u(x) satisfes  (j/n) (j− 1)/n M(u(x))dx ≤ 1 for any j � 1, . . ., n, and So, the theorem is proved.
Proof.First of all, we have (73) To prove the opposite inequality, let d n (A, X) represent the n − K width of A in the usual sense of X (X is the space of functions defned on a fnite interval).For every fnite interval where ‖•‖ represents the norm on X.
For every where χ I represents the characteristic function on the interval I. Terefore, we have the following expression: For f ∈  W M (P r (D), I N ), let F(t) � f(Nt/π), t ∈ I π , then we have the following expression: where By ( 78) and (79), we have the following expression: Let N ⟶ + ∞ on both sides of above-given inequation, then Hence, we obtain 8

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Te theorem is proved.
as the infnite dimensional linear n-width of W M,1 (P r (D)) in L(R) metric, where A runs over the set of all linear operators such that A(D) ⊂ F for some F ∈ F n , where D denotes the linear closure of W M,1 (P r (D)) in L(R).If there is a linear operator (85) Ten, A * is called the optimal linear operator.
Lemma 9 (See [1]).If f ∈ L * M,1 (P r (D)), then there exists an unique s r (f) ∈ S n (P r (D)) that satisfes where G(x, t) satisfes Theorem 10.Let n be a natural number, then where s r : and α n is a fxed constant.
Proof.According to Lemma 9, we have the following expression: So, we just need to prove By ( 66) and (84), we have the following expression: Now we prove From Lemma 9, Lemma 6, and equation (88), we have the following expression: Terefore, the theorem can be proved by the above-given expressions and Lemma 4.

Optimal Recovery Problem
Let Θ n be the set of all sequence ξ � ξ j   j∈Z satisfying where is the mapping from I ξ (W M,1 (P r (D))) to L(R).Sometimes A is called an algorithm.Now, we discuss the following optimal recovery problem: where A takes the mapping set from I ξ (W M,1 (P r (D))) to L(R).T r M (∆) 0 ≔ f: f (r− 1) is absolutely continuous on I, f ξ j   � 0, ∀ξ j ∈∆, According to reference [1], we know S * r− 1 (∆) is the splines space corresponding to P r (− D) and with simple nodes on ∆.

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Lemma 12 (see [15,17]).Let n be a natural number, then Theorem 13.Let n be a natural number, then Furthermore, ξ * � (j + α n )/n   j∈Z is the set of optimal sampling points, and the basic interpolation operator s r defned in Lemma 9 is the optimal algorithm.
Proof.First, we give the lower estimate of By ( 74)-(77), we obtain the following expression: where From the properties of Kolmogorov n-width and Lemma 11(2), we obtain the following expression: According to reference [15] and the Teorem 7.2− 4 in reference [17], and make the appropriate calculations, we obtain the following expression: (120) Hence, according to (119) and (120), the theorem is proved.

Data Availability
Te data are not available as no new data were created or analyzed in this study.
If A only traverses the set of linear maps, E n (W M,1 (P r (D)), S, L) is replaced by E L n (W M,1 (P r (D)), S, L), and E L n (W M,1 (P r (D)), S, L) is called the n-th fundamental error.If S is the identity operator, D n (W M,1 (P r (D)), S, L) and E n (W M,1 (P r (D)), S, L) are replaced by D n (W M,1 (P r (D)), L) and E n (W M,1 (P r (D)), L), respectively.Let I � [a, b], ξ ≔ ξ j   j∈Z ∈ Θ n , ∆ ≔ ξ ∩ I, S * r− 1 (∆) ≔ s ∈ C r− 2 (I): P r (− D)s(x) � 0, x ∈ ξ j , ξ j+1  , ∀j, such that ξ j , ξ j+1   ∩ I ≠ ∅  ; sup ‖f‖ L I N ( ) : f ∈ T r M ∆ N  0   � E T r, * ∞ I N , S * r− 1 I N   N I N ( ) ≥ d N n +r T r, * ∞ I N , L * N I N   ≥ d N n +r  TTo make the distinction, we replace  T r, * ∞ (I N ) by  W ∞ (P r (− D), I N ), by appropriate variation, we obtain the following expression:d N n +r  W ∞ P r (− D), I N , L * N I N   � N π d N n +r  W ∞ P r − πD N  , I π  , L * N I π   ,(112)Journal of Mathematics where  W ∞ P r (− D), I N  ≔ f ∈ C r 0 (R): f is of period 1 and P r (− D)
* is the optimal sampling.For every ξ ∈ Θ n , let I ξ (W M,1 (P r (D))) represent the image set of I ξ on W M,1 (P r (D)), and card(ξ ∩ [− a, a]) is the number of elements in ξ ∩ [− a, a].For every ξ ∈ Θ n , let I ξ represent an information