On Various Moduli of Smoothness and K -Functionals
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ON VARIOUS MODULI OF SMOOTHNESS AND K-FUNCTIONALS R. M. Trigub
UDC 517.5
We present a survey of the results on the exact rates of approximation of functions by linear means of Fourier series and Fourier integrals. The corresponding K-functionals are expressed via the special moduli of smoothness.
1. Introduction For a continuous 2⇡-periodic function f : R ! C, f 2 C(T), T = [−⇡, ⇡], the modulus of smoothness of order r 2 N with step h > 0 is defined as follows:
where
� �� !r (f ; h) = sup �∆rδ f (x)� : x 2 T, δ 2 (0, h] , ∆rδ f (x)
=
r ✓ ◆ X r ⌫=0
⌫
(−1)⌫ f (x + ⌫δ)
is the difference of function of order r with step δ > 0. The modulus of continuity ! = !1 was used by Lebesgue. For r ≥ 2, the modulus of continuity was introduced by Bernstein in 1912. Its main properties were investigated by Marchaud in 1927. It is known that !r (f ; h) = O(h↵ ),
0 < ↵ r,
h ! 0,
� � � � if and only if f [↵] 2 Lip ↵ − [↵] (for noninteger ↵), !2 f (↵−1) ; h = O(h) (for integer ↵ < r ), and finally, f (r−1) 2 Lip 1 (for ↵ = r). In a similar way, it is possible to define the modulus of smoothness !r (f ; h)p of a function in the metric of Lp ; moreover, R can be replaced with a compact set in C. For the general properties of the moduli of smoothness, see [1, 3–5]. The theorem on extension of a function with preservation of the order of decrease in !r as h ! 0 was proved by Besov [2]. The general theorem on extension was proved by Brudnyi (see [7]). Some other properties (e.g., of the modulus of smoothness of the product of functions) were established by the author. In [8], Bernstein applied the modulus of smoothness of a more general form defined by the “difference” of order r ≥ 2 as follows: f (x) −
r X
ak f (x + bk δ),
k=1
r X
ak = 1,
k=1
1 s r − 1,
r X
ak bsk = 0,
k=1
1 = b1 < b2 < . . . < br .
Donetsk National University, Donetsk, Ukraine; e-mail: [email protected]. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 7, pp. 971–996, July, 2020. Ukrainian DOI: 10.37863/umzh.v72i7.2384. Original article submitted February 24, 2020. 0041-5995/20/7207–1131
© 2020
Springer Science+Business Media, LLC
1131
R. M. T RIGUB
1132
This is a special case of the following modulus (f is a bounded and continuous function on the straight line and µ is a finite Borel measure): � 1 � � Z � � � � sup sup � f (x − δu)dµ(u)�� . x 0 0): � K(f ; h) = K(f ; h, E1 , E2 ) = inf kf1 kE1 + hkf2 kE2 , f = f1 + f2
(see [12] and [4]). In the case where at least one space is a space of smooth functions, the K-functional, regarded as a modulus of smoothness, specifies an intermediate smoothness as h ! 0. In [13], Ciesielski posed the problem of formulas for the K-functionals for periodic functions expressed in terms of the mean values of their Fourier series. In [14], these formulas were obtained for the spaces determined by the differential operators ∆r (∆ is the Laplace operator) and D2r =
d X @ 2r 2r , @x j j=1
2 N,
x = (x1 , . . . , xd ),
r 2 N.
For the Laplace operator ∆ (r = 1, ) the other form was pr
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