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# Roe L.I.T. F.I.T. VIPs

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Summability is a wide field of mathematics in functional analysis and has many applications, for instance, in numerical analysis to speed up the rate of convergence, in operator theory, the theory of orthogonal series, approximation theory, etc. Toeplitz  was the first to study summability methods as a class of transformations of complex sequences by complex infinite matrices. By w, we mean the space of all complex sequences. Any vector subspace of w is called a sequence space. The spaces of all bounded, convergent, and null sequences are denoted respectively by $$\ell_\infty$$, c, and $$c_0$$. We indicate the set of natural numbers including 0 by $$\mathbbN$$, and $$\mathcalG$$ denotes the collection of all finite subsets of $$\mathbbN$$. Let λ and η be two sequence spaces, and let $$A = (a_nk)$$ be an infinite matrix of real or complex numbers $$a_nk$$, where $$n, k \in\mathbbN$$. Then the matrix A defines the A-transformation from λ into η if, for every sequence $$x = (x_k) \in\lambda$$, the sequence $$Ax = \(Ax)_n\$$, the A-transform of x exists and is in η; where

Kızmaz  gave the concept of the spaces $$\ell_\infty(\Delta )$$, $$c(\Delta )$$, and $$c_0( \Delta )$$ by using difference operator, and it was additionally summed up by Et and Çolak . Let n, m be nonnegative integers, then for a given sequence space Z, we have

Taking $$n = 1$$, we get the spaces $$\ell_\infty(\Delta ^m)$$, $$c(\Delta ^m)$$, and $$c_0( \Delta ^m)$$ studied by Et and Çolak . Taking $$m = n = 1$$, we get the spaces $$\ell_\infty(\Delta )$$, $$c(\Delta )$$, and $$c_0( \Delta )$$ introduced and studied by Kızmaz .

To demonstrate that the spaces $$\mathcalN^t(\mathcalF,\Delta^m_n,\mu,q)$$ and $$\ell(q,\Delta^m_n)$$ are linearly isomorphic, we have to prove that there exists a linear bijection between these spaces. Define a linear transformation $$T:\mathcalN^t(\mathcalF,\Delta^m_n,\mu,q) \to\ell(q, \Delta^m_n)$$ by $$x \rightarrow y = Tx=\mathcalN^t(\mathcalF,\Delta^m_n, \mu,q)x$$ by using equation (2). So, linearity of T is trivial. Clearly, $$x = \theta$$ whenever $$Tx = \theta$$ and therefore T is injective.

Abstract:In this paper, the authors introduce the Orlicz spaces corresponding to the Young function and, by virtue of the equivalent theorem between the modified K-functional and modulus of smoothness, establish the direct, inverse, and equivalent theorems for linear combinations of modified summation operators of integral type in the Orlicz spaces.Keywords: approximation; linear combination; direct theorem; inverse theorem; equivalent theorem; Orlicz space; modified summation operators of integral type; K-functional; modulusMSC:41A17; 41A27; 41A35 153554b96e

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