Rate of convergence in $$L_{p}$$ L p approximation
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Rate of convergence in L p approximation ˙ Sakao˘glu C. Orhan · I.
Published online: 20 May 2014 © Akadémiai Kiadó, Budapest, Hungary 2014
Abstract In the present paper we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from L p [a, b] into itself using the concept of A-summation processes. We also study the rate of convergence of these operators. Keywords A-summation process · Positive linear operator · Korovkin type theorem · Second-order modulus of smoothness · Rate of convergence Mathematics Subject Classification
Primary 41A25 · 41A36 · Secondary 40A05
1 Introduction Korovkin theorems provide conditions for whether a given sequence of positive linear operators converges to the identity operator in the space of continuous functions on a compact interval (see, [9]). Some results concerning Korovkin type approximation theorems in the space L p [a, b] of Lebesgue integrable functions on a compact interval may be found in [5]. Quantitative Korovkin theorems for approximation by positive linear operators in L p [a, b] spaces have also been studied in [3,4,16,17]. In order to get stronger results it is useful to use certain summation processes. We first recall some basic concepts used throughout the paper. Let L p [a, b] , 1 ≤ p < ∞, denote the space of measurable real valued pth power 1/ p b Lebesgue integrable functions f on [a, b] with f p = a | f | p dμ .
C. Orhan · ˙I. Sakao˘glu Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara 06100, Turkey e-mail: [email protected] Present Address: ˙I. Sakao˘glu (B) Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara 06100, Turkey e-mail: [email protected]
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Rate of convergence
177
Let T : L p → L p be a linear operator. Recall that T is a positive operator if T f ≥ 0 whenever f ≥ 0. If T is a positive linear operator then f ≤ g implies that T f ≤ T g, and | f | ≤ g implies that |T f | ≤ T g. The operator norm T L p →L p is given by T L p →L p = sup f p =1 T f p . (n) Let A := A(n) = ak j be a sequence of infinite matrices with nonnegative real entries. A sequence T j of positive linear operators from L p [a, b] into itself is called a strong A- summation process in L p [a, b] if T j f is strongly A-summable to f for every f ∈ L p [a, b] , i.e., (n)
lim ak j T j f − f p = 0, uniformly in n. k
j
Some results concerning strong summation processes in L p [a, b] may be found in [14]. A sequence T j of positive linear from L p [a, b] into itself is called an A operators summation process in L p [a, b] if T j f is A-summable to f for every f ∈ L p [a, b] , i.e.,
(n)
= 0, uniformly in n, a T f − f lim
j k j
k
j p
where it is assumed that the series converges for each k, n and f. Recall that a sequence of (n) real numbers x j is said to be A-summable to L if limk j ak j x j = L , uniformly in n (see [2,15]). Results concerning summation processes on other spaces may be found in
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