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İbrahim Çanak - One of the best experts on this subject based on the ideXlab platform.
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TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS
Facta Universitatis Series: Mathematics and Informatics, 2020Co-Authors: Firat Ozsarac, İbrahim ÇanakAbstract:Let $q$ be a positive weight function on $\mathbf{R}_{+}:=[0, \infty)$ which is integrable in Lebesgue's sense over every finite interval $(0,x)$ for $0 0$ , $Q(0)=0$ and $Q(x) \rightarrow \infty $ as $x \to \infty $ . Given a real or complex-valued function $f \in L^{1}_{loc} (\mathbf{R}_{+})$ , we define $s(x):=\int_{0}^{x}f(t)dt$ and $$ \tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...), $$ provided that $Q(x)>0$ . We say that $\int_{0}^{\infty}f(x)dx$ is summable to $L$ by the $m$ - th iteration of weighted mean method determined by the function $q(x)$ , or for short, $(\overline{N},q,m)$ integrable to a finite number $L$ if $$ \Lim_{x\to \infty}\tau^{(m)}_q(x)=L. $$ In this case, we write $s(x)\rightarrow L(\overline{N},q,m)$ . It is known that if the Limit $\Lim _{x \to \infty} s(x)=L$ exists, then $\Lim _{x \to \infty} \tau^{(m)}_q(x)=L$ also exists. However, the converse of this implication is not always true. Some suitable conditions together with the existence of the Limit $\Lim _{x \to \infty} \tau^{(m)}_q(x)$ , which is so called Tauberian conditions, may imply convergence of $\Lim _{x \to \infty} s(x)$ . In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for $(\overline{N},q,m)$ summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces \` {a} ro summability $(C,1)$ and weighted mean method of summability $(\overline{N},q)$ have been extended and generalized.
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Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method
Proceedings of the National Academy of Sciences India Section A: Physical Sciences, 2020Co-Authors: Sefa Anıl Sezer, İbrahim ÇanakAbstract:Let $$(s_n)$$ ( s n ) be a sequence of real numbers. We say that $$(s_n)$$ ( s n ) is summable to $$\xi $$ ξ by the logarithmic power series method if $$\Lim _{x\rightarrow 1^-}f(x)=\xi, \quad {\text{where}}\quad f(x)=-\frac{1}{\log (1-x)}\sum_{n=0}^{\infty }\frac{s_n}{n+1}x^{n+1}. $$ Lim x → 1 - f ( x ) = ξ , where f ( x ) = - 1 log ( 1 - x ) ∑ n = 0 ∞ s n n + 1 x n + 1 . It is well known that if the Limit $$\Lim_{n \rightarrow \infty }s_n=\xi $$ Lim n → ∞ s n = ξ exists, then the Limit $$\Lim _{x \rightarrow 1^-} f(x)=\xi $$ Lim x → 1 - f ( x ) = ξ also exists. In this paper, we determine Tauberian conditions of slowly decreasing type to obtain ordinary convergence of $$(s_n)$$ ( s n ) from its summability by logarithmic power series method. As a consequence of our result, we give a short proof of an earlier Tauberian theorem due to Kwee (Can J Math 20:1324–1331, 1968 ).
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Tauberian Conditions Under Which Convergence Follows from Statistical Summability by Weighted Means
Advances in Summability and Approximation Theory, 2018Co-Authors: Zerrin Önder, İbrahim ÇanakAbstract:Let \((p_n)\) be a sequence of nonnegative numbers such that \(p_0>0\) and $$ P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty \,\,\,\,\text {as}\,\,\,\,n\rightarrow \infty . $$ Let \((s_n)\) be a sequence of real and complex numbers. The weighted mean of \((s_n)\) is defined by $$ t_n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k s_k\,\,\,\,\text {for}\,\,\,\,n =0,1,2,\ldots $$ We obtain some sufficient conditions, under which the existence of the Limit \(\Lim s_n=\mu \) follows from that of st-\(\Lim t_n=\mu \), where \(\mu \) is a finite number. If \((s_n)\) is a sequence of real numbers, then these Tauberian conditions are one-sided. If \((s_n)\) is a sequence of complex numbers, these Tauberian conditions are two-sided. These Tauberian conditions are satisfied if \((s_n)\) satisfies the one-sided condition of Landau type relative to \((P_n)\) in the case of real sequences or if \((s_n)\) satisfies the two-sided condition of Hardy type relative to \((P_n)\) in the case of complex numbers.
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Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method
2018Co-Authors: Sefa Anıl Sezer, İbrahim ÇanakAbstract:Let \((s_n)\) be a sequence of real numbers. We say that \((s_n)\) is summable to \(\xi \) by the logarithmic power series method if $$\Lim _{x\rightarrow 1^-}f(x)=\xi, \quad {\text{where}}\quad f(x)=-\frac{1}{\log (1-x)}\sum_{n=0}^{\infty }\frac{s_n}{n+1}x^{n+1}. $$ It is well known that if the Limit \(\Lim_{n \rightarrow \infty }s_n=\xi \) exists, then the Limit $$\Lim _{x \rightarrow 1^-} f(x)=\xi $$ also exists. In this paper, we determine Tauberian conditions of slowly decreasing type to obtain ordinary convergence of \((s_n)\) from its summability by logarithmic power series method. As a consequence of our result, we give a short proof of an earlier Tauberian theorem due to Kwee (Can J Math 20:1324–1331, 1968).
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Tauberian conditions for the (C, \alpha ) integrability of functions
Positivity, 2016Co-Authors: Ümit Totur, İbrahim ÇanakAbstract:For a real-valued continuous function f(x) on \([0,\infty )\), we define $$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$ for \(x>0\). We say that \(\int _{0}^{\infty } f(u)du\) is \((C, \alpha )\) integrable to L for some \(\alpha >-1\) if the Limit \(\Lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L\) exists. It is known that \(\Lim _{x \rightarrow \infty } s(x) =L\) implies \(\Lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L\) for all \(\alpha >-1\). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the \((C, \alpha )\) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the \((C,\alpha )\) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.
Dapeng Zhan - One of the best experts on this subject based on the ideXlab platform.
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Two-Curve Green’s Function for 2-SLE: The Interior Case
Communications in Mathematical Physics, 2020Co-Authors: Dapeng ZhanAbstract:A 2- $$\hbox {SLE}_\kappa $$ SLE κ ( $$\kappa \in (0,8)$$ κ ∈ ( 0 , 8 ) ) is a pair of random curves $$(\eta _1,\eta _2)$$ ( η 1 , η 2 ) in a simply connected domain D connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal $$\hbox {SLE}_\kappa $$ SLE κ curve in a complement domain. In this paper we prove that for any $$z_0\in D$$ z 0 ∈ D , the Limit $$\Lim _{r\rightarrow 0^+}r^{-\alpha _0} {\mathbb {P}}[{{\,\mathrm{dist}\,}}(z_0,\eta _j)
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Two-Curve Green’s Function for 2-SLE: The Interior Case
Communications in Mathematical Physics, 2020Co-Authors: Dapeng ZhanAbstract:A 2- $$\hbox {SLE}_\kappa $$ SLE κ ( $$\kappa \in (0,8)$$ κ ∈ ( 0 , 8 ) ) is a pair of random curves $$(\eta _1,\eta _2)$$ ( η 1 , η 2 ) in a simply connected domain D connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal $$\hbox {SLE}_\kappa $$ SLE κ curve in a complement domain. In this paper we prove that for any $$z_0\in D$$ z 0 ∈ D , the Limit $$\Lim _{r\rightarrow 0^+}r^{-\alpha _0} {\mathbb {P}}[{{\,\mathrm{dist}\,}}(z_0,\eta _j)
Katarzyna Pietruska-pałuba - One of the best experts on this subject based on the ideXlab platform.
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Lifschitz tail for alloy-type models driven by the fractional Laplacian
Journal of Functional Analysis, 2020Co-Authors: Kamil Kaleta, Katarzyna Pietruska-pałubaAbstract:Abstract We establish precise asymptotics near zero of the integrated density of states for the random Schrodinger operators ( − Δ ) α / 2 + V ω in L 2 ( R d ) for the full range of α ∈ ( 0 , 2 ] and a fairly large class of random nonnegative alloy-type potentials V ω . The IDS exhibits the Lifschitz tail singularity. We prove the existence of the Limit Lim λ → 0 λ d / α ln N ( λ ) = − C ω d ( λ d ( α ) ) d / α , with C ∈ ( 0 , ∞ ] . The constant C is finite if and only if the common distribution of the lattice random variables charges {0}. In this case, the constant C is expressed explicitly in terms of this distribution. In the Limit formula, λ d ( α ) denotes the Dirichlet ground-state eigenvalue of the operator ( − Δ ) α / 2 in the unit ball in R d , and ω d is the volume of this ball.
Bo Cui - One of the best experts on this subject based on the ideXlab platform.
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on the inductive Limit of direct sums of simple tai algebras
arXiv: Operator Algebras, 2019Co-Authors: Bo Cui, Chunlan JiangAbstract:An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive Limit $\Lim\Limits_{n\rightarrow\infty}(A_{n}=\bigoplus\Limits_{i=1}A_{n}^{i},\phi_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).
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On the inductive Limit of direct sums of simple TAI algebras
2019Co-Authors: Bo Cui, Jiang Chunlan, Li LiangqingAbstract:An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive Limit $\Lim\Limits_{n\rightarrow\infty}(A_{n}=\bigoplus\Limits_{i=1}A_{n}^{i},\phi_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all ATAI algebras with the Hausdorffized algebraic $K_{1}$-group as an addition to the invariant used in [Jiang1]. The theorem is proved by reducing the class to the classification theorem of $\mathcal{AHD}$ algebras with ideal property which is done in [GJL1]. Our theorem generalizes the main theorem of [Fa] and [Jiang1] (see corollary 4.3).Comment: 24 pages. arXiv admin note: text overlap with arXiv:1607.0758
Kamil Kaleta - One of the best experts on this subject based on the ideXlab platform.
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Lifschitz tail for alloy-type models driven by the fractional Laplacian
Journal of Functional Analysis, 2020Co-Authors: Kamil Kaleta, Katarzyna Pietruska-pałubaAbstract:Abstract We establish precise asymptotics near zero of the integrated density of states for the random Schrodinger operators ( − Δ ) α / 2 + V ω in L 2 ( R d ) for the full range of α ∈ ( 0 , 2 ] and a fairly large class of random nonnegative alloy-type potentials V ω . The IDS exhibits the Lifschitz tail singularity. We prove the existence of the Limit Lim λ → 0 λ d / α ln N ( λ ) = − C ω d ( λ d ( α ) ) d / α , with C ∈ ( 0 , ∞ ] . The constant C is finite if and only if the common distribution of the lattice random variables charges {0}. In this case, the constant C is expressed explicitly in terms of this distribution. In the Limit formula, λ d ( α ) denotes the Dirichlet ground-state eigenvalue of the operator ( − Δ ) α / 2 in the unit ball in R d , and ω d is the volume of this ball.