Theorem 2.1 . Assume that f ∈ W (0, ∞ ) is a **function** of **bounded** **variation** on every ﬁnite subinterval of (0, ∞ ). Furthermore, let α ≥ 1, λ > 2, and x ∈ (0, ∞ ) be given. Then, for each r ∈ N , there exists a constant M(f , α, r , x) such that for suﬃciently large n, the Bézier–type Baskakov-Durrmeyer operators V n,α satisfy the estimate

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Abstract. In this paper, using rest **bounded** **variation** sequences and head **bounded** **variation** sequences, some new results on approximation of **function**- s (signals) by almost generalized N¨ orlund means of their Fourier series are obtained. To our best knowledge this the first time to use such classes of sequences on approximations of the type treated in this paper. In addition, several corollaries are derived from our results as well as those obtained pre- viously by others.

RV ϕ I C = f ∈ RV ϕ I X f I ∈ C (20) Theorem 3.1. Let ( X , ⋅ ) be a real normed space, ( Y , ⋅ ) a real Banach space, C ⊂ X a convex cone, I ⊂ an arbitrary interval and let ϕ ψ ∈ , . Suppose that set-valued **function** h I : × → C clb Y ( ) is such that, for any t ∈ I the **function** h t ( ) , : ⋅ C → clb Y ( ) is continuous with respect to the second variable. If the composition operator H generated by the set-valued **function** h maps RW ϕ ( I C , ) into RW ψ ( I clb Y , ( ) ) , and satisfies the inequality

Abstract. For 0 < α < 1 let V (α) denote the supremum of the numbers v such that every α-H¨ older continuous **function** is of **bounded** **variation** on a set of Hausdorff dimension v. Kahane and Katznelson (2009) proved the estimate 1/2 ≤ V (α) ≤ 1/(2−α) and asked whether the upper bound is sharp. We show that in fact V (α) = max{1/2, α}. Let dim H and dim M denote the Hausdorff

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In the paper we introduce the class of linear combinations of functions which are subordinated to a convex **function**. Some relationships between this class and the class of real-valued functions with **bounded** **variation** on [0, 2 π ] are obtained. Next, we deﬁne classes of functions associated with **bounded** Mocanu **variation**. By using the properties of multivalent prestarlike functions, we obtain various inclusion relationships between deﬁned classes of functions. Some applications of the main results are also considered.

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Abstract. Some inequalities for functions of **bounded** **variation** that provide reverses for the inequality between the integral mean and the p−norm for p ∈ [1, ∞] are established. Applications related to the celebrated Landau inequality between the norms of the derivatives of a **function** are also pointed out.

Inequalities that estimate deviation of a **function** from its mean value using diﬀerent characteristics of the **function** are usually called Ostrowski type inequalities. Such inequal- ities have many applications, in particular in the area of numerical methods, and are heav- ily studied. See [] and the references therein for results connected with Ostrowski type inequalities for univariate functions of **bounded** **variation** and their applications.

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Recently, Guessab and Schmeisser [23], in the effort of incorporating together the mid-point and trapezoid inequality, have proved amongst others, the following companion of Ostrowski’s inequality. Theorem 1.3. Assume that the **function** f : [a, b] → R is of H − r−H¨ older type with r ∈ (0, 1], i.e., |f (t) − f (s)| ≤ H |t − s| r f oranyt, s ∈ [a, b] . (1.3) Then, for each x ∈

Definition 3. (Hardy-Krause). The **function** f (x, y) is said tobe of **bounded** **variation** if it satisfies the condition of Definition 1 and if in addition f (x, y) is of **bounded** **variation** in y (i.e. φ(x) is finite) for at least one x and f(x, y) is of **bounded** **variation** in y (i.e. ψ(y) is finite) for at least one y. Definition 4. (Arzel` a). Let (x i , y i ) (i = 0, 1, 2, ..., m) be any set of points satisfiying the conditions

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In recent years, there has been an increasing interest in the study of various mathematical problems with variable exponents. With the emergency of nonlinear problems in applied sciences, standard Lebesgue and Sobolev spaces demonstrated their limitations in applications. The class of nonlinear problems with exponent growth is a new research field and it reflects a new kind of physical phenomena. In 2000, the field began to expand even further. Motivated by problems in the study of electrorheological fluids, Diening [10] raised the question of when the Hardy-Littlewood maximal operator and other classical operators in harmonic analysis were **bounded** on the variable Lebesgue spaces. These and related problems are the subject of active research to this day. These problems were interesting in applications (see [11]-[14]) and gave rise to a revival of the interest in Lebesgue and Sobolev spaces with variable exponent, the origins of which could be traced back to the work of Orlicz in the 1930’s [15]. In the 1950’s, this study was carried on by Nakano [16] [17] who made the first systematic study of spaces with variable exponent. Later, Polish and Czechoslovak mathematicians investigated the modular **function** spaces (see for the example Musielak [18] [19], Kovacik and Rakosnik [20]). We refer to books [14] for the detailed information on the theoretical approach to the Lebesgue and Sobolev spaces with variable exponents. In [21], Castillo, Merentes and Rafeiro studied a new space of functions of generalized **bounded** **variation**. There, the authors introduced the notion of **bounded** **variation** in the Wiener sense with the exponent p(⋅)-variable.

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As we have mentioned before, for Schr¨ odinger operators in more than one dimension the question of self-adjointness and boundary conditions is much more subtle. We will focus on the case when V (x) is a **function** of |x| alone and quote a result from Reed–Simon [50, Appendix to Section X.1, Example 4]. This will be a decomposition theorem which, under certain conditions, reduces spherically symmetric Schr¨ odinger operators to a direct sum of half-line Schr¨ odinger operators.

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The class of nonlinear problems with exponent growth is a new research field and it reflects a new kind of physical phenomena. In 2000 the field began to expand even further. Motivated by problems in the study of electrorheological fluids, Diening [9] raised the question of when the Hardy-Littlewood maximal operator and other classical operators in harmonic analysis are **bounded** on variable Lebesgue spaces. These and related problems are the subject of active research to this day. These problems are interesting in applications (see [10] [11] [12] [13]) and gave rise to a revival of the interest in Lebesgue and Sobolev spaces with variable exponent, the origins of which can be traced back to the work of Orlicz [14] in the 1930’s. In the 1950’s, this study was carried on by Nakano [15] [16] who made the first systematic study of spaces with variable exponent. Later, Polish and Czechoslovak mathematicians investigated the modular **function** spaces (see for example Musielak [17] [18], Kovacik and Rakosnik [19] and Kozlowski [20]). We refer to the book [13] for detailed information on the theoretical approach for the Lebesgue and Sobolev spaces with variable exponents. Recently, in [21] Castillo, Merentes and Rafeiro studied a new space of functions of generalized **bounded** **variation**. They introduced the notion of **bounded** **variation** in the Wiener sense with variable exponent p ( ) ⋅ on [ ] a b , and study some of its properties.

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Kahane (1961) [3] gave a result which was concerned with how to obtain the variable knot of **function** in **bounded** **variation** space. Zhang (2009) [4] gave the corresponding numerical experiments by using the piece- wise constants **function**. Moreover, the advantages of Zhang’s method over other denoising methods, such as Visushrink [5] and SureShrinkage [6] were also analyzed [4].

a b The class of all functions of **bounded** **variation** on , is denoted as BV a b , . The renowned Jordan’s theorem ([1]) states that a **function** f : , a b is of **bounded** **variation** on a b , if, and only if, it is the diference of two monotone functions. In particular, every **function** in BV a b , has left limit f x at every point x a b , and right limit f x + at every point

its most compelling aspect is its vertical **variation**, that is, the sum of the vertical distances between its adjacent terms. Denoted by varw, the vertical **variation** of the sequence in (1.1) is var w = 2 + 1 + 0 + 2 + 1 = 6. Our purpose here is to compute the mean and vari- ance of var on four classical sets of combinatorial sequences.

In this paper, by use of some integral identities and inequalities developed in [4](see also [5]), we point out some approximations of the Fourier transform in terms of the complex exponential mean E (z, w) (see Section 2) and study the error of approximation for different classes of mappings of **bounded** **variation** defined on finite intervals.

Stella and Stock (2013) is in its treatment of the latent states. Following most of the existing literature, Stella and Stock (2013) model trend in‡ation and the NAIRU as driftless random walks. Modeling trend in‡ation as a ran- dom walk is a component of many macroeconomic models (e.g., among many others, Smets and Wouters, 2003, Ireland, 2007, Stock and Watson, 2007 and Cogley, Primiceri and Sargent, 2010), despite the fact that there are many reasons for thinking that trend in‡ation should not wander in an unbounded random-walk fashion. For instance, the existence of explicit or implicit in‡a- tion targets by central banks means that trend or underlying in‡ation will be kept within bounds and not allowed to grow in an unbounded fashion. For NAIRU bounding the counterfactual implications of unrestricted movements in the random walk is perhaps more important. With the exception of abrupt changes in employment law or unemployment bene…ts, which are observable events, one would expect the forces determining the NAIRU to be slow mov- ing and not lead to declines of unemployment to levels close to zero or levels above previous peaks in the unemployment rate driven by recessions. This is particularly the case because the unemployment rate by construction is a **bounded** variable and one would expect long-run equilibrium in the labor market to produce strong restrictions on its movement within this **bounded** interval. Further, by imposing the unrestricted random walk speci…cation on NAIRU researchers by construction add excess uncertainty to the possible location of NAIRU in the past, present and future. Building on our previ- ous work, Chan, Koop and Potter (2013), we embed this **bounded** model of NAIRU within a structure where the central bank keeps trend in‡ation well-contained, allowing us to better discriminate between cyclical and trend movements in in‡ation.

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Let f be a **function** defined on an infinite interval a, ∞, One can suppose that f is defined on a, ∞ assuming that f∞ 0. Thus, f is Henstock-Kurzweil integrable on a, ∞ if f extended on a, ∞ is HK-integrable. For functions defined over intervals −∞, a and −∞, ∞ One can makes similar considerations.

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Abstract. Let X be a Banach space and A ⊂ X an absolutely convex, closed, and **bounded** set. We give some suﬃcient and necessary conditions in order that A lies in the range of a measure valuedin the bidual space X ∗∗ andhaving boundedvariation. Among other results, we prove that X ∗ is a G. T.-space if andonly if A lies inside the range of some X ∗∗ -valuedmeasure with boundedvariation whenever X A is isomorphic to a Hilbert space.

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The Skorokhod reflection problem has a long history. Skorokhod [10] introduced it as a method for representing a diffusion process with a reflecting boundary at zero. Given a continuous **function** X : [0, ∞) → R , the standard Skorokhod reflection problem seeks to find (Q(t), t ≥ 0) and a continuous, nondecreasing **function** Y : [0, ∞) → R + with Y (0) = 0,

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