“…Define [11] we defined θ 2k+2 (f, x, h) assuming the existence of D 2k f (x). Since the existence of D 2k f (x) implies the existence of D 2k a f (x) with D 2k a f (x) = D 2k f (x), this definition of θ 2k+2 (f, x, h) agrees with that in [11] whenever D 2k f (x) exists and so these definitions of the upper and lower derivates are more general than that in [11] in the sense that if D 2k f (x) exists then these definitions are the same.…”
Section: Definitions and Notationsmentioning
confidence: 99%
“…Higher order smoothness was defined in [11] and was studied in [7], which was again extended in [8] to cover a wide class of functions and studied various properties. Buczolich, Evans and Humke [2] obtained the Baire*1 property of the approximate symmetric d.l.V.P.…”
Section: Introductionmentioning
confidence: 99%
“…. If D n f 0 except on a countable set E ⊂ (a, b) and if f is n-smooth on E then f (n−2) exists, is continuous and convex in (a, b).This is a special case of Theorem 3.2 or Theorem 4.1 of[11].Theorem 12. Suppose that (i) f is continuous on [a, b], (ii) D f exists and D f ∈ D in (a, b) for = n − 2, n − 4, .…”
mentioning
confidence: 95%
“…Since D n f 0, a.e., by [11,Theorem 3.2] f (n−2) exists and is convex in (c, d) and so (c, d) ⊂ G which is a contradiction since (c, d) ∩ E 1 = ∅. If n is odd, we can arrive at a similar contradiction by suitably modifying (26) …”
It is proved that if f is continuous and the approximate symmetric d.l.V.P. derivatives D n−2 a f of f of order n − 2 exist in (a, b) then under a certain smoothness type condition on f , D n−2 a f is in Baire*1. Also Zahorski property and Denjoy property for the ordinary symmetric d.l.V.P. derivative D n f are established under certain suitable conditions.
“…Define [11] we defined θ 2k+2 (f, x, h) assuming the existence of D 2k f (x). Since the existence of D 2k f (x) implies the existence of D 2k a f (x) with D 2k a f (x) = D 2k f (x), this definition of θ 2k+2 (f, x, h) agrees with that in [11] whenever D 2k f (x) exists and so these definitions of the upper and lower derivates are more general than that in [11] in the sense that if D 2k f (x) exists then these definitions are the same.…”
Section: Definitions and Notationsmentioning
confidence: 99%
“…Higher order smoothness was defined in [11] and was studied in [7], which was again extended in [8] to cover a wide class of functions and studied various properties. Buczolich, Evans and Humke [2] obtained the Baire*1 property of the approximate symmetric d.l.V.P.…”
Section: Introductionmentioning
confidence: 99%
“…. If D n f 0 except on a countable set E ⊂ (a, b) and if f is n-smooth on E then f (n−2) exists, is continuous and convex in (a, b).This is a special case of Theorem 3.2 or Theorem 4.1 of[11].Theorem 12. Suppose that (i) f is continuous on [a, b], (ii) D f exists and D f ∈ D in (a, b) for = n − 2, n − 4, .…”
mentioning
confidence: 95%
“…Since D n f 0, a.e., by [11,Theorem 3.2] f (n−2) exists and is convex in (c, d) and so (c, d) ⊂ G which is a contradiction since (c, d) ∩ E 1 = ∅. If n is odd, we can arrive at a similar contradiction by suitably modifying (26) …”
It is proved that if f is continuous and the approximate symmetric d.l.V.P. derivatives D n−2 a f of f of order n − 2 exist in (a, b) then under a certain smoothness type condition on f , D n−2 a f is in Baire*1. Also Zahorski property and Denjoy property for the ordinary symmetric d.l.V.P. derivative D n f are established under certain suitable conditions.
“…The theory of the P n -integral is rather technical and reference should be made to the basic papers [4,5], as well as to [7]. It is sufficient to say this integral is defined in a Perron manner using major and minor functions in which the lower derivative of the major function is the lower symmetric de la Vallée Poussin derivative of order n, and the upper derivative of the minor function is the upper symmetric de la Vallée Poussin derivative of order n. The resulting P n -integral will then integrate finite symmetric de la Vallée Poussin derivatives of order n\ more general functions can also be integrated see for instance [7,Theorem 5.1]. Further, certain classes of summable trigonometric series are P n -Fourier series and a modified form of the classical Fourier formulae for the coefficients have been given; see [5, (4.5) and (4.6)] or [7, (8.4) and (8.5)].…”
It is shown that if a trigonometric series is (R, 3), respectively (R, 4), summable then its (R, 3) sum, respectively (R, 4) sum, is James P3—, respectively P4—, integrable and that such series are Fourier series with respect to these integrals.
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