Bernstein and Kantorovich polynomials diminish the Λ -variation
aa r X i v : . [ m a t h . F A ] M a r BERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISHTHE Λ -VARIATION KLAUDIUSZ CZUDEK
Abstract.
We prove the Λ-variation diminishing property of the Bernsteinand Kantorovich polynomials. Next we apply this result to characterize thespace C Λ BV c as the closure of the space of polynomials in the k · k Λ norm. Anew proof of the separability of C Λ BV c is given. Introduction
It is a well-known fact that every continuous function defined on the interval[0 ,
1] can be approximated by polynomials. Several proofs of this theorem werepublished, the first one by Karl Weierstrass and later by Henri Lebesgue, MarshallH. Stone and Sergei Bernstein, among others. This one given by Bernstein isparticulary important in our paper. It establishes that for a continuous function f : [0 , → R polynomials of the form: B n f ( x ) = n X k =0 f (cid:18) kn (cid:19) (cid:18) nk (cid:19) x k (1 − x ) n − k tend to f uniformly on [0 , f , were intensively studied in the last century, also inthe case when the initial function f is not continuous. Basic facts related to thistopic may be found in the classic positions [9] or [14]. The Bernstein polynomialshave also appeared attractive in numerous applications. We would like to suggestthe recent paper [10] as an overview of these achievements. Observe that one maylook at the Bernstein polynomials as a family of linear operators f B n f what isthe reason why we use terms ”Bernstein operators” and ”Bernstein polynomials”interchangeably.In the twentieth century a lot of generalizations of the Bernstein operators ap-peared, so-called Bernstein-type operators. One of the most significant are theSz´asz-Mirakyan, the Baskakov and the Kantorovich operators. The last one has animportant place in our paper. For a function f , integrable on the interval [0 , n -th Kantorovich polynomial: K n f ( x ) = n X k =0 (cid:18) nk (cid:19) x k (1 − x ) n − k ( n + 1) Z k +1 n +1 kn +1 f ( t ) dt. In comparsion with the Bernstein polynomials, the value of f at the point kn isreplaced with the mean value of f on the interval [ kn +1 , k +1 n +1 ]. In 1930 Kantorovich Date : March 7, 2017.
Key words and phrases.
Λ-variation, Bernstein polynomials, functions continuous in Λ-variation. proved that k K n f − f k → f , where k · k denotesthe standard norm in the space of integrable functions L . In [9] we may findother approximation theorems related to Kantorovich polynomials, including L p and supremum norms.Among a number of properties of Bernstein polynomials we would like to distin-guish the variation diminishing property, proved by G.G. Lorentz in [13]. It statesthat for an arbitrary function f : [0 , → R of bounded variation the variation ofthe Bernstein polynomial of the function f is not greater than the variation of f ,i.e. V ( B n f ) ≤ V ( f ). Recall that the variation of a function f : [0 , → R is thequantity: V ( f ) = sup n − X i =1 | f ( x i +1 ) − f ( x i ) | where supremum is taken over all finite sequences 0 ≤ x < ... < x n ≤
1. Thisproperty was also proven in the case of other Bernstein-type operators, in particularin the case of Kantorovich polynomials. Additionaly Lorentz showed that k B n f − f k BV → f is absolutely continuous, where k · k BV denotes thestandard norm in the space BV of functions of bounded variation k u k BV = V ( u ) + | u (0) | . Proofs of all these facts and rates of approximation may be found in [4]. Letus mention here that the variation diminishing property is called in [4] the variationdetracting property, while the term ”variation diminishing” is used to express thestatement that a number of zeros of a Bernstein polynomial of a function f withmultiplicities counted is not greater than the number of sign changes of the function f . Our terminology follows from [2].The variation diminishing property of the Bernstein polynomials was also provenfor certain generalization of the regular variation, so-called ϕ -variation. Given aconvex, nondecreasing function on [0 , ∞ ) continuous in 0 and satisfying ϕ (0) = 0define the ϕ -variation of f : [0 , → R as the number: V ϕ ( f ) = sup n − X i =1 ϕ ( | f ( x i +1 ) − f ( x i ) | )where supremum is taken over all finite sequences 0 ≤ x < ... < x n ≤
1. Wesay that a function f is of bounded ϕ -variation if V ϕ ( f ) < ∞ . The space of allfunctions of bounded ϕ -variation is denoted by ϕBV . Basic facts about ϕ -variationmay be found in [15]. J. A. Adell and J. de la Cal proved in [2], using probabilisticrepresentation of the Bernstein operators, the ϕ -variation diminishing property, i.e.that V ϕ ( B n f ) ≤ V ϕ ( f ) for f ∈ ϕBV . Let us mention that Lorentz’s techniquefrom [13] could not be applied here, since he used the fact that every function ofbounded variation is a difference of two monotone functions, what does not remainvalid in the case of the ϕ -variation.There exists also other generalization of the regular variation, the Λ-variation,introduced by Daniel Waterman in [23] for the sake of its applicability to the studyof the uniform convergence of Fourier series. We say that a sequence Λ = ( λ n ) n of positive reals is a Λ-sequence if Λ is nondecreasing and P n λ n = ∞ . Givenan interval I = [ a, b ] and a function f denote | f ( I ) | = | f ( b ) − f ( a ) | . A finite orinfinite sequence ( I n ) n of closed intervals is called nonoverlapping if for every k = n intervals I n , I k intersect at most at the endpoints. We define the Λ-variation of a ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 3 function f : [0 , → R as the number: V Λ ( f ) = sup X n | f ( I n ) | λ n , where supremum is taken over all sequences (finite or infinite) of nonoverlappingintervals ( I n ) n contained in [0 , f is of bounded Λ-variation if V Λ ( f ) < ∞ . Notice that for Λ = (1 , , , .... ) we get that V Λ ( f ) = V ( f ).Given a finite sequence of nonoverlapping closed intervals I = h I , ..., I m i con-tained in [0 , f : [0 , → R define: σ Λ ( f, I ) = m X k =1 | f ( I k ) | λ k It is readily seen that we may define the Λ-variation of the function f : [0 , → R equivalently as the number: V Λ ( f ) = sup σ Λ ( f, I )where supremum is taken over all finite sequences I of nonoverlapping closed in-tervals contained in [0 , σ Λ was used in the papers ofPrus-Wi´sniowski but with a slightly different meaning (cf. [17], [18], [19]).For a fixed Λ-sequence Λ, the space of functions of bounded Λ-variation, denotedby Λ BV , is a linear space, which may be equipped with the norm k f k Λ = V Λ ( f ) + | f (0) | . Note that the Λ-variation is only a seminorm. Moreover, the normed space(Λ BV, k · k Λ ) is a Banach space (Section 3. in [22]). Obviously, every function fromΛ BV is bounded. Using simple arguments one can show that convergence in the k · k Λ norm implies convergence in the supremum norm. Notice that we may deducefrom this fact that the space C Λ BV of continuous functions of bounded Λ-variationis a closed subspace of Λ BV .Further, if Λ is a proper Λ-sequence, i.e. λ n → ∞ , then a function f issaid to be continuous in Λ-variation if lim m →∞ V Λ ( m ) ( f ) →
0, where Λ ( m ) de-notes a Λ-sequence obtained from Λ by omission of the first m terms: Λ ( m ) =( λ m +1 , λ m +2 , ... ). We denote by Λ BV c and C Λ BV c the space of functions con-tinuous in Λ-variation and the space of continuous functions that are continuousin Λ-variation, respectively. The concept of continuity in Λ-variation was intro-duced by Waterman in [24] in connection with ( C, β )-summability of Fourier series.It gained more importance later, since it turned out that Fourier series of thesefunctions have much more interesting properties (cf. [3]).The main purpose of this paper is to generalize Lorentz’s results from [13] to theΛ-variation. In the first section we prove that the Bernstein polynomials diminishthe Λ-variation. In the proof we use idea from [2] to use the probabilistic represen-tation of the Bernstein operators. We conclude from this fact that the Kantorovichoperators also diminish the Λ-variation (as far as we know, there does not exista probabilistic representation of the Kantorovich operators satisfying the desiredcondition). In the third section we deal with related notions of Λ-variation. Finally,in the fourth section we apply the Λ-variation diminishing property to characterize C Λ BV c as the closure of the space of polynomials in the k · k Λ norm and thereforewe partially answer the question asked by Waterman in [22] about characterizationof the space Λ BV c . Among others, this question has been already answered byPrus-Wi´sniowski but we will look at the problem from different point of view. A KLAUDIUSZ CZUDEK new proof of the separability of C Λ BV c and a new characterization of compactnessin this space are also given.2. The Λ -variation diminishing property Let (Ω , M , P ) be a probability space, ( X k ) k a sequence of independent randomvariables defined on Ω and uniformly distributed on the interval [0 , k and x ∈ [0 ,
1] the indicator function I ( X k ≤ x ) is a random variable on Ωand therefore: S n ( x ) = n X k =0 I ( X k ≤ x ) is a random variable. If x ∈ [0 ,
1] and k is a natural number, then the indicatorfunction I ( X k ≤ x ) takes two values: 1 with probability x and 0 with probability 1 − x .It is quite clear, that S n ( x ) has the binomial distribution with parameters n, x i.e.takes value k , 0 ≤ k ≤ n , with probability (cid:0) nk (cid:1) x k (1 − x ) n − k . Finally, define therandom variable: Z xn = S n ( x ) n Notice that Z xn takes values kn , for 0 ≤ k ≤ n with probability (cid:0) nk (cid:1) x k (1 − x ) n − k .Take any Borel measurable function f : [0 , → R . The random variable f ( Z xn )is a simple, measurable function which takes values f ( kn ), for 0 ≤ k ≤ n withprobability (cid:0) nk (cid:1) x k (1 − x ) n − k and therefore:(1) E f ( Z xn ) = n X k =0 f (cid:18) kn (cid:19) (cid:18) nk (cid:19) x k (1 − x ) n − k = B n f ( x )where B n f denotes the n -th Bernstein polynomial of the function f . Inequality x ≤ x implies that the set ( X k ≤ x ) is contained in the set ( X k ≤ x ) for everynatural k , hence I ( X k ≤ x ) ≤ I ( X k ≤ x ) and eventually:(2) Z x n ≤ Z x n for x ≤ x , what is a crucial property in our paper. The probabilistic representa-tion was, according to [2], firstly used by Lindvall in [12] to deduce some propertiesof the Bernstein polynomials. As an example let us take a look at the followingproposition which will be used in the next section. Proposition 1. If f is nondecreasing (nonincreasing), then B n f is nondecreasing(nonincreasing).Proof. Trivial from (1) and (2). (cid:3)
More theorems and proofs in this spirit, as the preservation of convexity, may befound in [1] or [12].We are ready to show the Λ-variation diminishing property of the Bernsteinpolynomials. The proof of the theorem below follows the lines of the proof of theanalogous result in [2]. In fact, the authors of the paper [2] proved this statementfor a wide class of Bernstein-type operators with a probabilistic representations, i.e.represented as the mean value of f composed with some double-indexed stochasticprocess { Z xn : n ∈ N , x ∈ [0 , } satisfying (2). Moreover, they do not restrict onlyto operators acting on spaces of functions defined on the interval [0 , ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 5
Note that: V Λ ( f ) = sup m − X i =1 | f ( x j +1 ) − f ( x j ) | λ β ( j ) where supremum is taken over all finite partitions 0 ≤ x ≤ x ≤ ... ≤ x m ≤ ,
1] and permutations β of the set { , , ..., m − } . This observation isused in several papers to simplify a notation of proofs (for example [17], [18], [19],the proof of this fact in [7]). Theorem 1. If Λ is a Λ -sequence, f ∈ Λ BV , then V Λ ( B n f ) ≤ V Λ ( f ) .Proof. Function f has one-sided limits in every point of its domain (see Theorem 4in [16]), what implies that it is a Borel function and hence representation (1) holdsfor B n f . Take 0 ≤ x ≤ x ≤ ... ≤ x m ≤
1. From (2) we have that Z x n ≤ ... ≤ Z x m n .Let β be any permutation of a set { , , ..., m − } . Then: m − X j =1 | B n f ( x j +1 ) − B n f ( x j ) | λ β ( j ) = m − X j =1 | E f ( Z x j +1 n ) − E f ( Z x j n ) | λ β ( j ) ≤≤ E m − X j =1 | f ( Z x j +1 n ) − f ( Z x j n ) | λ β ( j ) Now, Z x n ( ω ) ≤ ... ≤ Z x m n ( ω ) are numbers from interval [0 ,
1] for all ω ∈ Ω andhence for every ω ∈ Ω: m − X j =1 | f ( Z x j +1 n ( ω )) − f ( Z x j n ( ω )) | λ β ( j ) ≤ V Λ ( f )and therefore: E m − X j =1 | f ( Z x j +1 n ) − f ( Z x j n ) | λ β ( j ) ≤ Z Ω V Λ ( f ) dP = V Λ ( f ) . Finally: m − X j =1 | B n f ( x j +1 ) − B n f ( x j ) | λ β ( j ) ≤ V Λ ( f )for arbitrary 0 ≤ x ≤ ... ≤ x m ≤ β , what implies that V Λ ( B n f ) ≤ V Λ ( f ). (cid:3) Now we are going to conclude from Theorem 1 the Λ-variation diminishing prop-erty of the Kantorovich polynomials. Recall that if f ∈ Λ BV , then f is a boundedBorel function, therefore it is integrable, hence the Kantorovich polynomials of thisfunction may be considered. We start with an easy observation: Proposition 2. If f ∈ L [0 , , [ a, b ] ⊆ [0 , then there exist x , x ∈ ( a, b ) suchthat: b − a Z ba f ( t ) dt ≤ f ( x ) , and b − a Z ba f ( t ) dt ≥ f ( x ) . KLAUDIUSZ CZUDEK
We say that a point x ∈ (0 ,
1) is a point of varying monotonicity of a function f : [0 , → R if there is no neighbourhood of x on which f is strictly monotone orconstant. The points 0 and 1 are said to be points of varying monotonicity of f if f is non-constant on every neighbourhood of 0 or 1, respectively. The set of allpoints of varying monotonicity of a function f will be denoted by M f . Accordingto [17], this concept was introduced in [5]. Theorem 2. If Λ is a Λ -sequence, f ∈ Λ BV , then V Λ ( K n f ) ≤ V Λ ( f ) .Proof. Let us define a function g as a piecewise linear function such that: g (cid:18) kn (cid:19) = ( n + 1) Z k +1 n +1 kn +1 f ( t ) dt, k = 0 , , ..., n It is clear that K n f = B n g and therefore it sufficies to prove that V Λ ( g ) ≤ V Λ ( f ),since V Λ ( K n f ) = V Λ ( B n g ) ≤ V Λ ( g ) from Theorem 1.It is trivial if g is constant. Take any partition 0 = x < ... < x m = 1 of[0 ,
1] such that | g ( x i +1 ) − g ( x i ) | > i = 1 , ..., m −
1. We may assume that x k ∈ M g for k = 1 , ..., m (see Proposition 1.1 in [17]) and therefore that x k = i k n for certain natural i k ≤ n and every k = 1 , ..., m , since only such points are pointsof varying monotonicity of g . Moreover, we may assume that for k = 1 , ..., m − g ( i k n ) , g ( i k +1 n ) , g ( i k +2 n ) is not monotone. Indeed, if it is monotone thenfor an arbitrary permutation β of the set { , ..., m − } : | g ( i k +1 n ) − g ( i k n ) | λ β ( k ) + | g ( i k +2 n ) − g ( i k +1 n ) | λ β ( k +1) ≤ | g ( i k +2 n ) − g ( i k n ) | min { λ β ( k +1) , λ β ( k ) } . Now, take k = 1 , ..., m . If g ( i k n ) is greater than values of g at neighbouring pointsof our partition, then using Proposition 2 we define t k as a point from ( i k n +1 , i k +1 n +1 )such that: f ( t k ) ≥ ( n + 1) Z ik +1 n +1 ikn +1 f ( t ) dt = g (cid:18) i k n (cid:19) . Similarly, if g ( i k n ) is less than values of g at neighbouring points of our partition,then we define t k as a point from ( i k n +1 , i k +1 n +1 ) such that: f ( t k ) ≤ ( n + 1) Z ik +1 n +1 ikn +1 f ( t ) dt = g (cid:18) i k n (cid:19) . Obviously, 0 ≤ t ≤ t ≤ ... ≤ t m ≤ m − X k =1 | g ( i k +1 n ) − g ( i k n ) | λ β ( k ) ≤ m − X k =1 | f ( t k +1 ) − f ( t k ) | λ β ( k ) . The initial partition and the permutation β were arbitrary, hence V Λ ( g ) ≤ V Λ ( f ). (cid:3) Other notions of Λ -variation Apart from the regular Λ- and ϕ -variation presented in the previous sectionsthere are considered also other, related concepts of variation. Some of them will bedescribed below. ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 7 If I is a finite sequence of intervals contained in [0 , kIk the length of the longest interval in this sequence. Further, define for a function f : [0 , → R : V Λ ,δ ( f ) = sup σ Λ ( f, I )where the supremum is taken over all finite sequences I of nonoverlapping closedintervals with kIk ≤ δ , contained in [0 , f : [0 , → R : W Λ ( f ) = lim δ → V Λ ,δ ( f ) . Similarly, define: V ϕ,δ ( f ) = sup n − X i =1 ϕ ( | f ( x i +1 ) − f ( x i ) | )where supremum is taken over all finite sequences 0 ≤ x < ... < x n ≤ x i +1 − x i ≤ δ . The value: V ∗ ϕ ( f ) = lim δ → V ϕ,δ ( f ) . is called the fine ϕ -variation of f .In [2] the authors prove under some assumptions on ϕ and f that: V ∗ ϕ ( B n f ) ≤ V ∗ ϕ ( f )This observation becomes trivial in the case of the Λ-variation of a function definedon the interval [0 , Proposition 3. If Λ is a proper Λ -sequence, f : [0 , → R is a Lipschitz functionwith Lipschitz constant L , then W Λ ( f ) = 0 .Proof. Take ε >
0. Let m be such that Lλ m < ε and δ > mLδλ < ε .Let I = h I , ..., I k i be any finite sequence of intervals such that kIk ≤ δ . If k > m then: k X j =1 | f ( I j ) | λ j = m X j =1 | f ( I j ) | λ j + k X j = m +1 | f ( I j ) | λ j ≤≤ mLδλ + Lλ m k X j = m +1 | I j | ≤ mLδλ + Lλ m < ε. Similarly if k ≤ m . (cid:3) One may ask if it is true that V δ, Λ ( B n f ) ≤ V δ, Λ ( f ) for every δ > n . Suprisingly, we can give a counterexample to this claim. Let: f ( x ) = . x ≤ x ≤ . < x ≤ . x − . < x ≤ f is a piecewise linear function such that f (0) = 0 , f ( ) = 0 . , f ( ) =0 . , f (1) = 1. We shall prove that V δ, Λ ( B n f ) > V δ, Λ ( f ) for every natural n , Λ suchthat λ < λ and 1 > δ > .First, we are going to prove that the partition I = h [0 , δ ] , [ δ, i has the prop-erty that σ Λ ( f, I ) = V Λ ,δ ( f ). Let ω be the modulus of continuity of f , ω ( ε ) =sup {| f ( t ) − f ( t ) | : | t − t | ≤ ε } for ε >
0. Observe that from the monotonicity and
KLAUDIUSZ CZUDEK the continuity of f , there exists x ≤ − δ < such that ω ( δ ) = f ( x + δ ) − f ( x ).Obviously, x + δ ≥ and therefore: ω ( δ ) = f ( x + δ ) − f ( x ) = 1 . x + δ ) − . − . x = 1 . δ − . | f ( δ ) − f (0) | . Observation 1. If I ′ = h I , I , ..., I m i is an arbitrary sequence of nonoverlapping,closed intervals with kI ′ k ≤ δ , then: σ Λ ( f, I ′ ) ≤ σ Λ ( f, I ) = | f ( δ ) − f (0) | λ + | f (1) − f ( δ ) | λ . Proof.
Obviously | f ( I ) | ≤ ω ( δ ) = | f ( δ ) − f (0) | and therefore: | f ( δ ) − f (0) | − | f ( I ) | λ ≥ | f ( δ ) − f (0) | − | f ( I ) | λ . From the monotonicity of f we get | f ( I ) | + | f ( I ) | + ... + | f ( I m ) | ≤ | f ( δ ) − f (0) | + | f (1) − f ( δ ) | = 1. Using both this facts: | f ( δ ) − f (0) | λ + | f (1) − f ( δ ) | λ == | f ( I ) | λ + | f (1) − f ( δ ) | λ + | f ( δ ) − f (0) | − | f ( I ) | λ ≥≥ | f ( I ) | λ + | f (1) − f ( δ ) | λ + | f ( δ ) − f (0) | − | f ( I ) | λ == | f ( I ) | λ + 1 − | f ( I ) | λ ≥≥ | f ( I ) | λ + | f ( I ) | + | f ( I ) | + ... + | f ( I m ) | λ ≥ σ Λ ( f, I ′ ) . (cid:3) Observation 1 allows us to claim that:(3) V δ, Λ ( f ) = | f ( δ ) − f (0) | λ + | f (1) − f ( δ ) | λ Observation 2. B n f ( δ ) > f ( δ ) . Proof.
Let us notice that the Bernstein operators are positive, i. e. nonnegativity ofa function h implies nonnegativity of B n h and therefore if we put g ( x ) = 1 . x − . f − g ≥ B n ( f − g ) ≥
0. From linearity of the Bernstein operator B n f ≥ B n g = g . The last equality follows from the fact that B n h = h for an arbitrarylinear function h (see the beginning of the chapter 3.2 in [10]). In particular wehave B n f ( δ ) ≥ f ( δ ) = g ( δ ), since δ > .Now it sufficies to show that the assumption B n f ( δ ) = f ( δ ) leads to a con-tradiction. Note that this equality implies that B n f ( δ ) = g ( δ ) = B n g ( δ ). Usingthe monotonicity of f − g and the preservation of monotonicity by the Bernsteinpolynomials (Proposition 1) we know that B n ( f − g )( δ ) ≥ B n ( f − g )( x ) ≥ x ≥ δ , but then B n ( f − g )( x ) = 0 for x ≥ δ , since B n ( f − g )( δ ) = 0, andthereby B n ( f − g )( x ) = 0 for all x ∈ [0 , f (0) > g (0), hence B n f (0) > B n g (0) directly from the definition of the Bernstein polynomials, whatis a desired contradiction. (cid:3) ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 9
From Observation 2:(4) | B n f ( δ ) − B n f (0) | − | f ( δ ) − f (0) | = (cid:0) B n f ( δ ) − B n f (0) (cid:1) − (cid:0) f ( δ ) − f (0) (cid:1) > f and the preservation of monotonicity by B n :(5) | f (1) − f ( δ ) | + | f ( δ ) − f (0) | = 1 | B n f (1) − B n f ( δ ) | + | B n f ( δ ) − B n f (0) | = 1Now: (cid:18) | B n f ( δ ) − B n f (0) | λ + | B n f (1) − B n f ( δ ) | λ (cid:19) − (cid:18) | f ( δ ) − f (0) | λ + | f (1) − f ( δ ) | λ (cid:19) == | B n f ( δ ) − B n f (0) | − | f ( δ ) − f (0) | λ − | f (1) − f ( δ ) | − | B n f (1) − B n f ( δ ) | λ >> | B n f ( δ ) − B n f (0) | − | f ( δ ) − f (0) | λ − | f (1) − f ( δ ) | − | B n f (1) − B n f ( δ ) | λ = 0 . In the inequality above we needed the assumption that λ < λ and (4). The lastequality follows from (5). Finally V Λ ,δ ( B n f ) > V Λ ,δ ( f ) from (3). Remark.
In [6] the authors defined so-called lower Λ-variation. For a real valuedfunction f on [0 ,
1] we define the lower Λ-variation of f as:var Λ ( f ) = inf { V Λ ( g ) : f = g a.e. } Let: Λ BV = { f ∈ L : var Λ ( f ) < ∞} . This space is a Banach space with the norm k f k Λ = k f k + var Λ ( f ). A function f ∈ Λ BV is called a good representative of its equivalence class in the space L if V Λ ( f ) = var Λ ( f ).Note that the Kantorovich operator is well-defined on this space. Theorem 2tells us that: V Λ ( K n f ) ≤ var Λ ( f )for f ∈ Λ BV . Moreover, Theorem 8 in [6] states that continuous functions aregood representatives and therefore:var Λ ( K n f ) ≤ var Λ ( f )i.e. the Kantorovich operators diminish the lower Λ-variation.4. Applications in Λ BV spaces We begin this section with several propositions generalizing Lorentz’s resultsdescribed in [14], Theorems 1.7.1 and 1.7.2. Let us define for nonempty K ⊆ [0 , f : [0 , → R : V Λ ( f, K ) = sup σ Λ ( f, I )where supremum is taken over all finite sequences of nonoverlapping closed intervalswhose endpoints are in K . The following statement is a straightforward general-ization of the analogous result in the case of the regular variation. The elementaryproof will be omitted. Proposition 4. If Λ is a Λ -sequence, K ⊆ [0 , is nonempty, f n ( x ) → f ( x ) forevery x ∈ K , then: V Λ ( f, K ) ≤ lim inf n →∞ V Λ ( f n , K ) . In particular, if f is not of bounded Λ -variation, then lim n →∞ V Λ ( f n ) = ∞ . Obviously, the Bernstein polynomials of a continuous function f tend to f pointwise and therefore we have the following proposition. Recall that k f k Λ = V Λ ( f ) + | f (0) | . Proposition 5. If Λ is a Λ -sequence, f ∈ C Λ BV , then: lim n →∞ k B n f k Λ = k f k Λ . Proof.
Theorem 1 implies that lim sup n →∞ V Λ ( B n f ) ≤ V Λ ( f ). Proposition 4, how-ever, yields that lim inf n →∞ V Λ ( B n f ) ≥ V Λ ( f ). The conclusion follows, since B n f (0) = f (0) for every natural n . (cid:3) Proposition 6. If Λ is a Λ -sequence, then: C Λ BV = { f ∈ C [0 ,
1] : sup n ∈ N k B n f k Λ < ∞} . Proof. If f ∈ Λ BV , then sup n k B n f k Λ ≤ k f k Λ . If f Λ BV , then lim n →∞ V Λ ( B n f ) = ∞ from Proposition 4. (cid:3) In Propositions 5 and 6 we had to restrict to continuous functions, since discon-tinuous functions in general do not have the property that B n f → f pointwise. Inthe case of discontinuous functions it is necessary to add some further assumptionson f .For arbitrary f : [0 , → R , let K f be the set of points of continuity of f ,and 0 and 1. Further, denote by A the set of all bounded functions with one-sided limits at every point of (0 ,
1) and such that min { f ( x − ) , f ( x +) } ≤ f ( x ) ≤ max { f ( x − ) , f ( x +) } . If f ∈ Λ BV for some Λ-sequence Λ, then f admits the firstone of these properties but not necessarily the second one (Theorem 4 in [16]).Observe that from the first property we may conclude that the set of points ofdiscontinuity of a function from A is at most countable. Indeed, if f ∈ A , then thisset is the union of the sets E n = { x ∈ [0 ,
1] : | f ( x +) − f ( x − ) | > n } for n = 1 , , ... and each E n is finite from the compactness of [0 ,
1] and the definintion of the family A . We are ready to formulate the next lemma. Lemma 1. If Λ is a Λ -sequence, f ∈ A , then V Λ ( f, K f ) = V Λ ( f ) .Proof. It is trivial when f is constant. Obviously, V Λ ( f, K f ) ≤ V Λ ( f ). Let 0 = x < ... < x m = 1 be an arbitrary partition of [0 ,
1] such that | f ( x k +1 ) − f ( x k ) | > k = 1 , ..., m −
1. Using the same argument as in the proof of Theorem 2 wemay assume that for k = 1 , ..., m − f ( x k ) , f ( x k +1 ) , f ( x k +2 ) is notmonotone. Put: δ = 12 min i =1 ,...,m − | x i +1 − x i | . Take ε > ε < min i =1 ,...,m − | f ( x i +1 ) − f ( x i ) | λ . We construct a new partition t < t < ... < t m in the following way: let t = x , t m = x m . If x k is a point of continuity of f , then put t k = x k , k = 2 , ..., m − x k for k = 2 , ..., m − t k be any point ofcontinuity such that | x k − t k | < δ and: ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 11 (i) f ( t k ) ≥ f ( x k ) − λ ε m − if f ( x k ) is greater than values of f at neighbouringpoints of the partition x < ... < x m (ii) f ( t k ) ≤ f ( x k ) + λ ε m − otherwiseIt is possible, since lim sup t → x k f ( t ) ≥ f ( x k ) , lim inf t → x k f ( t ) ≤ f ( x k ) for k =2 , ..., m − f is at most countable.For every k we have | x k − t k | < δ , hence t < t < ... < t m is a partition of [0 , λ εm − m − X i =1 λ β ( j ) ≤ λ εm − · m − λ = ε. Further, (6) implies that for k = 1 , ..., m − f ( t k ) , f ( t k +1 ) , f ( t k +2 ) isnot monotone and therefore for an arbitrary permutation β of the set { , ..., m − } we have: m − X j =1 | f ( t j +1 ) − f ( t j ) | λ β ( j ) ≥ m − X j =1 | f ( x j +1 ) − f ( x j ) | λ β ( j ) − λ εm − m − X i =1 λ β ( j ) ≥≥ m − X j =1 | f ( x j +1 ) − f ( x j ) | λ β ( j ) − ε. Since the inequality remains valid for any partition x < ... < x m and any ε > V Λ ( f, K f ) ≥ V Λ ( f ). (cid:3) Proofs of the next two propositions follow from Lemma 1, Proposition 4, Theorem1 and the fact that B n f ( x ) → f ( x ) at every point of continuity x of the function f (Theorem 1.1.1. in [14]). Proposition 7. If Λ is a Λ -sequence, f ∈ A , then: lim n →∞ k B n f k Λ = k f k Λ . Proposition 8. If Λ is a Λ -sequence, f ∈ A , then f is of bounded Λ -variation ifand only if sup n k B n f k Λ < ∞ . Let us now concentrate on the problem of a convergence of sequences ( B n f ) n and ( K n f ) n to f in the k · k Λ norm and thereby on the main result of this section.First, let us give the proof of the following proposition. Proposition 9. If Λ is a proper Λ -sequence, then Λ BV c and C Λ BV c are closedsubspaces of Λ BV in the k · k Λ norm.Proof. For every f ∈ Λ BV we have V Λ (1) ( f ) ≥ V Λ (2) ( f ) ≥ ... . We deduce from thisobservation that if f Λ BV c , then there exists ε > V Λ ( m ) ( f ) ≥ ε forevery natural m . Take an arbitrary g ∈ Λ BV such that k f − g k Λ < ε . Obviously, V Λ ( m ) ( f − g ) < ε for every natural m and therefore V Λ ( m ) ( g ) ≥ ε for every natural m from the reverse triangle inequality (recall that the Λ-variation is a seminorm).However, this implies that g Λ BV c , what proves that Λ BV c is a closed subspaceof Λ BV . The space C Λ BV c is a closed subspace as an intersection of two closedsubspaces. (cid:3) As it was mentioned in the introduction, the space C Λ BV c appeared in thetheory of the Λ-variation mainly due to good properties of Fourier series of functionsfrom this space. This led Waterman to ask about a characterization of the space Λ BV c (see [22]). He also expressed a suspicion (see [21]) that not every function ofbounded Λ-variation is continuous in Λ-variation. An example of such function wasgiven initially in [11] but in fact the most beautiful confirmation of Waterman’sconjecture was provided by F. Prus-Wi´sniowski in [18], where he defined the Shao-Sablin index S Λ of a Λ-sequence Λ as: S Λ = lim sup n →∞ P ni =1 1 λ i P ni =1 1 λ i and proved the theorem (Theorem 3.1 in [18]): Theorem 3 (Prus-Wi´sniowski) . If Λ is a proper Λ -sequence, then the followingstatements are equivalent:(i) The space C Λ BV is separable(ii) C Λ BV c = C Λ BV (iii) Λ BV c = Λ BV (iv) S Λ < . Several other characterizations of Λ BV c and C Λ BV c being an answer or par-tial answer to Waterman’s question from [22] have been published over last threedecades, for instance: Theorem 4 ([20]) . If Λ is a proper Λ -sequence, then f ∈ Λ BV c if and only if thereexists Λ -sequence Γ such that γ n λ n → and f ∈ Γ BV . Theorem 5 ([19], Theorem 2) . If Λ is a proper Λ -sequence, then f ∈ C Λ BV c ifand only if W Λ ( f ) = 0 . Theorem 6 ([19], Theorem 3) . If Λ is a proper Λ -sequence, then Λ BV c is theclosure of the set of all step functions of bounded Λ -variation in the k · k Λ -norm. For more theorems in this spirit involving also other notions of the Λ-variation see[19]. Now we give the proof of the main result of this section with three interestingcorollaries.
Theorem 7. If Λ is a proper Λ -sequence, then || B n f − f || Λ → if and only if f ∈ C Λ BV c .Proof. C Λ BV c is a closed subspace of Λ BV containing the space of all polynomials,what may be concluded from Theorem 5 and Proposition 3. It implies that if || B n f − f || Λ → f ∈ C Λ BV c .Assume that f ∈ C Λ BV c . Take ε >
0. There exists such m that V Λ ( m ) ( f ) ≤ ε . The Λ-variation diminishing property implies that V Λ ( m ) ( B n f ) ≤ ε for everynatural n . Moreover, there exists N ∈ N such that: || B n f − f || ∞ ≤ ελ m for n ≥ N .Let h I , ..., I k i be any sequence of nonoverlapping intervals contained in [0 , k > m , then: k X j =1 | ( B n f − f )( I j ) | λ j = m X j =1 | ( B n f − f )( I j ) | λ j + k X j = m +1 | ( B n f − f )( I j ) | λ j ≤ ERNSTEIN AND KANTOROVICH POLYNOMIALS DIMINISH THE Λ-VARIATION 13 ≤ λ m X j =1 | ( B n f − f )( I j ) | + k X j = m +1 | B n f ( I j ) | λ j + k X j = m +1 | f ( I j ) | λ j ≤≤ λ · m · ελ m + V Λ ( m ) ( B n f ) + V Λ ( m ) ( f ) ≤ ε for n ≥ N . Similarly if k ≤ m . Hence, using also fact that | ( B n f − f )(0) | = 0: || B n f − f || Λ ≤ ε for n ≥ N . Since ε was arbitrary, our theorem is proved. (cid:3) Remark.
In the proof of Theorem 7 only two properties of the Bernstein polynomialswere significant: that k B n f − f k ∞ → f and that theBernstein polynomials diminish the Λ-variation. The Kantorovich polynomials alsohave these properties. Indeed, the first one follows from chapter 10, paragraph 6 in[9] and the second one from Theorem 2. Therefore, we may prove in a very similarway that || K n f − f || Λ → f ∈ C Λ BV c . Corollary 1. If Λ is a proper Λ -sequence, then C Λ BV c is the clousure of the spaceof polynomials in the k · k Λ norm. This is a partial answer to Waterman’s question from [22].Before the next corollary let us recall that for every function f ∈ C [0 , V ( f ) = Z | f ′ ( x ) | dx The first proof of the next corollary was given by Prus-Wi´sniowski in [17] and wasrather technical and long. Observe that in our proof of this statement we use onlyTheorem 1 and Theorem 7.
Corollary 2. If Λ is a proper Λ -sequence, then C Λ BV c is separable.Proof. Obviously, now it sufficies to prove that the space of polynomials is separablein the k · k Λ norm. We may easily show using (7) that for every polynomial f and ε > g with rational coefficients such that V ( f − g ) < ε .For f, g ∈ BV we have that V Λ ( f − g ) ≤ λ − V ( f − g ) and the conclusion follows. (cid:3) The Bernstein operators are linear operators from Λ BV to Λ BV . Moreover,Theorem 1 tells us that these operators are also continuous and k B n k ≤
1, where k · k denotes the standard norm in the space of linear operators from Λ BV to Λ BV .For every constant function f we have B n f = f (the beginning of the chapter 3.2in [10]), what guarantees that B n is not a contraction and therefore k B n k = 1.The Bernstein operators are finite dimensional, continuous operators and hence arecompact.Now we use this fact to prove the characterization of compactness. At the end ofchapter 2.1 of [14] the author gives the criterion of compactness in L p spaces basedon the very similar idea with Kantorovich polynomials. Another characterizationwas provided by Prus-Wi´sniowski (cf. [17]). Some other results about compactessin Λ BV spaces were recently given also by Bugajewski et al (see [8]). Corollary 3. If Λ is a proper Λ -sequence, K ⊆ C Λ BV c is bounded and closed,then K is compact if and only if k B n f − f k Λ → uniformly for all f ∈ K . Proof.
Assume that k B n f − f k Λ → K is closed, it sufficies toshow that it is totally bounded. Set ε >
0. There exists n such that k B n f − f k Λ < ε for f ∈ K . Operator B n is compact, K is bounded and hence B n K is totallybounded. There exists f , f , ..., f m ∈ K such that B n K ⊆ S mi =1 B (cid:0) B n f i , ε (cid:1) . If f ∈ K , then there exists k such that k B n f − B n f k k Λ < ε , hence k B n f k − f k Λ < ε .Since ε was arbitrary, we have that K is totally bounded.Now, assume that K is compact. There exist f , ..., f k ∈ K such that K ⊆ S ki =1 B (cid:0) f i , ε (cid:1) . Let N be such that k B n f i − f i k Λ < ε for i = 1 , , ..., .k , n ≥ N .Let f ∈ K . There exists m such that f ∈ B (cid:0) f m , ε (cid:1) . Now for n ≥ N : k B n f − f k Λ ≤ k B n kk f − f m k Λ + k B n f m − f m k Λ + k f m − f k Λ < ε. (cid:3) Acknowledgments.
I am grateful to Jacek Gulgowski for suggesting the topic ofthis paper and much valuable advice. I also would like to thank the referee for hisconstructive comments.
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