A contribution to the theory of divergent sequences

From MaRDI portal
Publication:2647762

DOI10.1007/BF02393648zbMath0031.29501OpenAlexW2031685471MaRDI QIDQ2647762

George G. Lorentz

Publication date: 1948

Published in: Acta Mathematica (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf02393648




Related Items

The subspace of almost convergent sequencesOn the sublinear functional associated to a family of invariant meansApproximation of certain bivariate functions by almost Euler means of double Fourier seriesOn Browder's convergence theorem and Halpern iteration process for \(G\)-nonexpansive mappings in Hilbert spaces endowed with graphsDixmier traces and non-commutative analysisApproximation of signals (functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier seriesOn ergodic operator means in Banach spacesA note on \(F_ B\)-convergenceLebesgue constants of the Walsh system and Banach limitsAn abstract version of the Korovkin theorem via \(\mathcal{A}\)-summation processExtremal properties of the set of vector-valued Banach limitsFinite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjectureGeneralized limits and statistical convergenceSome estimations of Banach limitsSingular approximation in polydiscs by summability processSome \(\sigma\)-double sequence spaces defined by Orlicz functionApplication of Banach limits to the study of sets of integersOn the structure of invariant Banach limits\(\sigma\)-asymptotically lacunary statistical equivalent sequencesStrongly almost summable and strongly almost convergent sequencesOn a new \(I\)-convergent double-sequence spaceConvergence of positive linear approximation processesFourier-Stieltjes localization in neighborhoods of finite-dimensional irreducible representations of locally compact groupsSome spaces of double sequences obtained through invariant mean and related conceptsDomain of the double sequential band matrix \(B(\tilde{r}, \tilde{s})\) in the sequence space \(\ell(p)^\ast\)Uniform mean convergence theorems for hybrid mappings in Hilbert spacesAbstract Korovkin-type theorems in modular spaces and applicationsOn some new double sequence spaces defined by a modulusMatrix transformations between the spaces of Cesàro sequences and invariant meansDomain of the double sequential band matrix in the classical sequence spacesSome new sequence spaces derived by the domain of generalized difference matrixGeneralized limits with additional invariance properties and their applications to noncommutative geometryOn summability and positive linear operatorsAlmost \(\lambda_r\)-statistical and strongly almost \(\lambda_r\)-convergence of order \(\beta\) of sequences of fuzzy numbersBanach limits and traces on \(\mathcal{L}_{1, \infty}\)A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spacesWeak convergence theorems for nonexpansive mappings in Banach spacesApplications of double lacunary sequences to \(n\)-normRate of convergence in \(L_p\) approximationLack of Tauberian theorem for absolute almost convergenceStructure properties of the Banach limit setCharacteristic functions of Banach limitsQuantitative theorems on linear approximation processes of convolution operators in Banach spacesDurch Matrixverschiebungen erweiterte KonvergenzbegriffeA note on some strongly sequence spacesOn the Riesz almost convergent sequences spaceInvariant and absolute invariant means of double sequencesSome matrix transformations of convex and paranormed sequence spaces into the spaces of invariant meansOn some lacunary almost convergent double sequence spaces and Banach limitsOn infinite matrices and lacunary \(\sigma \)-convergenceSaturation of multiplier operators in Banach spacesOn \(\mathcal A\)-invariant mean and \(\mathcal A\)-almost convergenceAn application of almost convergence in approximation theorems\((A_\sigma)\)-double sequence spaces defined by Orlicz function and double statistical convergenceOn vector-valued Banach limitsA note on \(\tau \)-convergence, \(\tau \)-convergent algebra and applicationsLebesgue constants of the Walsh systemConcerning Fatou's theoremDual convergence theorems for the infinite products of resolvents in Banach spacesA Tauberian theorem related to almost convergenceApproximation theorems by positive linear operators in weighted spacesA unified approach to the asymptotic almost-equivalence of evolution systems without Lipschitz conditionsSome variations of the Bohman-Korovkin theoremOn convergence of iterates of positive contractions in L\(_p\) spacesOn \(\sigma \)-conservative matrices and compact operators on the space \(V_{\sigma }\)Almost convergence and generalized difference matrixOn almost \(p\)-bounded variation of lacunary sequencesOn asymptotic formulae via summabilityDixmier traces as singular symmetric functionals and applications to measurable operatorsOn some generalized sequence spaces of fuzzy numbers defined by a sequence of Orlicz functionsOn \(\sigma \)-conservative and boundedly \(\sigma \)-conservative four-dimensional matricesEffective metastability of Halpern iterates in \(CAT(0)\) spacesInvariant means on bounded vector-valued functionsQuantitative results on almost convergence of a sequence of positive linear operatorsInvariant Banach limits and applicationsAsymptotic almost-equivalence of Lipschitz evolution systems in Banach spacesOn the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak \(\omega\)-limit setStrong and absolute almost-convergence of vector functions and related Tauberian theoremsDouble sequence spaces characterized by lacunary sequencesMultidimensional matrix characterisations of the Banach and Pringsheim coreFourier-Haar coefficients and Banach limitsSummation process of positive linear operatorsRegularly \(\sigma \)-conservative and \(\sigma \)-coercive four dimensional matricesAlmost convergence and nonlinear ergodic theoremsOn some new sequence spaces and statistical convergence methods for double sequencesTraces and residues of pseudo-differential operators on the torusThe set of almost convergent sequences as intersections of summability fieldsAlmost convergence and two matrix limitation methodsOn generalized paranormed statistically convergent sequence spaces defined by Orlicz functionZur Limitierung der beschränkten FolgenOn the invariant mean and statistical convergenceAn application of medial limits to iterative functional equationsTauberian conditions for almost convergenceTauberian conditions for \(w\)-almost convergent double sequencesAlmost convergence and Euler totient matrixGeneralized almost statistical convergenceBound states and scattered states for contraction semigroupsA note on a mean ergodic theorem for nonlinear semigroupsMeans on commutative semigroups and nonlinear ergodic theoremsAsymptotic behavior of solutions to nonlinear Volterra integral equationsUnnamed ItemUnnamed ItemSummability of a Class of Fourier SeriesA note on a theorem of Maddox on strong almost convergenceUnnamed ItemSome Sequence Spaces and Absolute Almost ConvergenceA Sequence Space and Uniform $$(A,\varphi )$$—Statistical ConvergenceMultiple disjointness and invariant measures on minimal distal flowsA nonlinear ergodic theorem for asymptotically nonexpansive mappingsKorovkin Type Approximation Theorem for Almost and Statistical ConvergenceInvariant mean and a Korovkin-type approximation theoremOn some new matrix transformationsAsymptotic behavior of commutative semigroups of nonexpansive mappings in Banach spacesOn some difference sequence spaces defined by a sequence of Orlicz functionsErgodic sequences of averages of group representationsOn existence of finite invariant measuresInäquivalenzsätze bei LimitierungsverfahrenDirect theorems on methods of summability. III. Absolute summability functionsAn ergodic theorem for asymptotically nonexpansive mappingsNonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappingsAlmost statistical convergence of order \(\beta \) of sequences of fuzzy numbersOn strong almost convergenceTheorems of Toeplitz-Silverman type by applying a reduction methodOn some spaces of almost lacunary convergent sequences derived by Riesz mean and weighted almost lacunary statistical convergence in a real \(n\)-normed spaceBanach limits: invariance and functional characteristicsSome sequence spaces of invariant means and lacunary defined by a Musielak-Orlicz function over \(n\)-normed spacesSummation Process by Max-Product OperatorsInvariant Banach limits and their convex subsetsA new view at Dixmier traces on \(\mathfrak{L}_{1,\infty} (H)\)On the difference spaces of almost convergent and strongly almost convergent double sequencesBanach limits and uniform almost summabilityFurther remarks on uniform statistical convergence of order \(\alpha\)Order and geometric properties of the set of Banach limitsSome properties of the sequence spaceEffective results on nonlinear ergodic averages in CAT spacesUnnamed ItemApproximation theorems for functions of two variables via \(\sigma\)-convergenceUnnamed ItemNew type of generalized difference sequence of fuzyy numbers involving lacunary sequencesAlmost-convergent sequences with respect to polynomial hypergroupsAlmost convergence and triple band matrixSome properties of a function connecting to exponent of convergence for double sequencesVector-valued almost convergence and classical properties in normed spacesMatrix summability and positive linear operatorsThe method of almost convergence with operator of the form fractional order and applicationsViscosity Approximation Methods for Generalized Multi-Valued Nonexpansive Mappings with ApplicationsAdvances in Dixmier traces and applicationsSome topological properties of the spaces of almost null andalmost convergent double sequencesStrong convergence of approximated sequences for nonexpansive mappings in Banach spacesCharacterization of ω -almost convergent sequences in Lp(T2)The Nonlinear Ergodic Theorem for Asymptotically Nonexpansive Mappings in Banach SpacesOn some `duality' of the Nikodym property and the Hahn propertyAlmost convergence of double sequences and strong regularity of summability matricesThe space of almost convergent sequencesUnnamed ItemNonlinear evolution equations and nonlinear ergodic theoremsUnnamed ItemOn the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell _{\infty}\). IIThe variances of a sequenceInfinite matrices and almost bounded sequencesDual Convergences of Iteration Processes for Nonexpansive Mappings in Banach SpacesDixmier traces and extrapolation description of noncommutative Lorentz spacesLacunary Strong (A σ , p)-ConvergenceA new type of convergenceOn some generalizations of Lorentz's almost convergenceAlmost-convergent double sequencesConsistency Theorems for Almost ConvergenceSome new observations on Catalan almost convergent sequence spaces and the Catalan coreA strengthened form of a theorem of WienerCoefficient properties for certain Fourier seriesSummability properties for the distribution of sequencesBanach limits, infinite matrices and sublinear functionalsOn \(\sigma\)-finite invariant measures for Markov processesAlmost summable Taylor seriesMatrix transformations of almost convergent sequencesFactor sequences for summability matricesSome Topics in Summability TheoryDegree of Approximation of Functions Through Summability MethodsNegative Resultate über Tauber-BedingungenStrongly almost (ω, λ, q)-summable sequencesWeak mixing implies weak mixing of higher orders along tempered functionsA Class of Regular MatricesAn Application of Banach LimitsInfinite matrices and almost convergenceToeplitz matrices, ergodicity and Fréchet spacesAlmost convergent vector sequencesMean Values and Banach LimitsOn the Hausdorff-limitability of \(\{ z^n\}\)A study of the matrix method of summability of the trigonometrical seriesAlmost convergence in Banach spacesA note on the \(\varepsilon\)-entropy of monotone functions in the Levy normOn the lacunary partial sums of Walsh Fourier seriesThe Spectral Theory of Bounded FunctionsThe main classes of invariant Banach limitsOn the degree of approximation of signals Lip(α,r), (r≥ 1) class by almost Riesz means of its Fourier seriesAbsolute summability functions for Hausdorff methodsUnnamed ItemAlmost Convergence and the Buck-Pollard PropertyWeighted means, generalised Hausdorff matrices and the borel propertyA Note on the Ergodic Properties of HomeomorphismsA Sufficient Condition for a Regular Matrix to Sum a Bounded Divergent SequenceMatrix transformations and compact operators on the binomial sequence spacesOn the almost summability of a trigonometric sequenceOn the ergodic theorem for positive operators Iξ -Double Strongly Summable Sequences of order θUnnamed ItemOn the summability of a sequence of Walsh functionsBanach and Statistical Cores of Bounded SequencesOn the Spaces of $\lambda _{r}$-almost Convergent and $\lambda _{r}$-almost Bounded SequencesPositive linear operators and summabilityOn the Ergodic Mixing TheoremConstant Functions and Left Invariant Means on SemigroupsOn the Gamma Spaces Including the Spaces of Absolutely p-Summable, Null, Convergent and Bounded SequencesSome new sequence spaces defined by bounded variation in 2-normed spacesLacunary almost convergence and some new sequence spacesAlmost Summable SequencesOn the K_{a}-continuity of real functionsApproximation by sampling type discrete operatorsMean ergodic type theoremsOn strongly almost convergence of doubles equences via complex uncertain variableUnnamed ItemHausdorff dimension of the set of almost convergent sequencesBanach limits: extreme properties, invariance and the Fubini theoremOn the Range of an Invariant MeanAlmost Convergent and Almost Summable Sequences𝒜 $$\mathcal{A}$$ -Summability of Sequences of Linear Conservative OperatorsMultivariate approximation in φ\varphi-variation for nonlinear integral operators via summability methodsRegular summability methods in the approximation by max-min operatorsOn uniform statistical convergenceConvergence of a conjugate function in Zygmund space by almost Nörlund transformHow large is the space of almost convergent sequences?Hausdorff dimension of sets defined by almost convergent binary expansion sequencesSymmetric stable processes on amenable groupsA new paranormed sequence space and invariant meansAlmost convergent sequence spaces derived by the domain of quadruple band matrixAsymptotic behavior of a periodic sequence of nonexpansive mappings with applicationsAlmost convergent \(0\)-\(1\)-sequences and primesDomain of the Cesàro mean of order α in Maddox’s space ℓ(p)Riesz I-convergent sequence spacesA General Korovkin Result Under Generalized ConvergenceOn a mixing property of operators in Lp spacesOn invariant measures for operatorsSummability in Amenable SemigroupsGeometry of Banach limits and their applicationsA relation of Banach limit and difference matrix to generate some Orlicz sequence spacesSummability on Mellin-type nonlinear integral operatorsInvariant Banach limits and their extreme pointsOn Nonstrongly Regular MatricesUnnamed ItemUnnamed ItemConsistency theory for SM-methodsErgodic properties of operators in some semi-Hilbertian spacesUnnamed ItemUnnamed ItemWeighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation ResultsOn a new I-convergent sequence spaceUnnamed ItemConsistency theory for SM-methodsCharacterization of Absolute and Uniform ContinuityUnnamed ItemUnnamed ItemWeak almost-convergence theorem without Opial's conditionFinitely additive measures on \(\mathbb{N}\)Infinite Matrices and Invariant MeansOn some sequence spacesSome new sequence spaces defined by lacunary sequencesOrder Summability and Almost ConvergenceCharacterization of Cesàro andL-asymptotic limits of matricesMultipliers for the Space of Almost-Convergent Functions on a SemigroupThe existence of non-trivial bounded functionals implies the Hahn-Banach extension theoremStrong asymptotic behavior of almost-orbits of nonlinear semigroupsAn ergodic theorem for nonlinear semigroups of Lipschitzian mappings in Banach spacesUnnamed ItemA direct approach to positive normalised traces on simply generated idealsVariational principles for spectral radius of weighted endomorphisms of 𝐶(𝑋,𝐷)Unnamed ItemUnnamed ItemComplementarity of subspaces of ℓ_∞ revisitedI2-lacunary strongly summability for multidimensional measurable functionsNonexpansive retractions and nonlinear ergodic theorems in Banach spacesUnnamed ItemOn the asymptotic behaviour of almost-orbits of nonlinear contraction semigroups in Banach spacesQuadratic inequalities for functionals in l^{∞}A survey on some paranormed sequence spacesUnnamed ItemMultidimensional Strongly Deferred Invariant ConvergenceOn the ergodic theorem for positive operators IStatistical σ-convergence of double sequences with applicationLacunary sequences of complex uncertain variables defined by Orlicz functionsA-statistical convergence of order α via φ - functionAlmost λ-Statistical Convergence of Complex Uncertain SequencesStrongly \((V_{\sigma},\lambda,q)\)-summable sequences defined by Orlicz functionsThe \(\sigma\)-convergence and \(\sigma\)-core of double sequencesOn lacunary \(\sigma\)-statistical convergenceReproducing kernels and skew-dilation of Hilbert space operatorsOn lacunary invariant sequence spaces defined by a sequence of modulus functionsDistribution of values of bounded generalized polynomialsOn domain of Nörlund matrixExtreme points of the set of Banach limitsConnes-Dixmier traces, singular symmetric functionals, and the notion of Connes measurable elementApproximation processes with respect to positive multiplication operatorsAlmost lacunary statistical and strongly almost lacunary convergence of sequences of fuzzy numbersUniform distribution, invariant means, and Riemann integralsGeneralized difference strongly summable sequence spaces of fuzzy real numbers defined by ideal convergence and Orlicz function\(r\)-convex sequences and matrix transformationsApplications of Orlicz-Pettis theorem in vector valued multiplier spaces of generalized weighted mean fractional difference operators\(K_a\)-convergence and Korovkin type approximationAsymptotic behavior of solutions to the semidiscrete diffusion equationA summability process on Baskakov-type approximationGeneralized matrix summability of a conjugate derived Fourier seriesStatistical convergence of double sequences in locally solid Riesz spacesSpaces of strongly lacunary invariant summable sequencesDouble almost lacunary statistical convergence of order \(\alpha\)On some new sequence spaces defined by infinite matrix and modulusGeneralized weighted invariant mean based on fractional difference operator with applications to approximation theorems for functions of two variablesA note on double lacunary statistical \(\sigma\)-convergence of fuzzy numbersGeneralized limits and sequence of matricesOn almost \(B\)-summable double sequence spacesWeighted almost convergence and related infinite matricesDomain of the Nörlund matrix in some of Maddox's spacesDouble almost statistical convergence of order \(\alpha\)Dilation invariant Banach limitsA generalization of almost convergenceExtreme points of the set of dilation-invariant Banach limitsSequential definitions of continuity for real functions.Almost periodicity and ergodic theorems for nonexpansive mappings and semigroups in Hadamard spacesConvergence in phi-variation and rate of approximation for nonlinear integral operators using summability processGeneralized almost convergence and core theorems of double sequences\(\sigma\)-core and \(\mathcal I\)-core of bounded sequences.Almost conservative four-dimensional matrices through de la Vallée-Poussin meanCompact operators for almost conservative and strongly conservative matricesSome new classes of generalized difference strongly summable \(n\)-normed sequence spaces defined by ideal convergence and Orlicz functionVector valued Banach limits and generalizations applied to the inhomogeneous Cauchy equationGeneralized statistically almost convergence based on the difference operator which includes the \((p,q)\)-gamma function and related approximation theoremsContinuity conditions for finite-dimensional locally bounded representations of connected locally compact groupsDomain of Riesz mean in some spaces of double sequencesSome seminormed difference sequence spaces over \(n\)-normed spaces defined by a Musielak-Orlicz function of order \((\alpha, \beta)\)Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach spaceStatistical \(\mathcal{A}\)-summation process and Korovkin type approximation theorem on modular spacesOn generalized sequence spaces via modulus functionAlmost convergence and generalized weighted mean. IIVector valued multiplier spaces of \(f_\lambda \)-summability, completeness through \(c_0(X)\)-multiplier convergence and continuity and compactness of summing operatorsOn paranorm \(\mathrm{BV}_\sigma\) \(I\)-convergent double sequence spaces defined by an Orlicz functionOn the domain of the triangle \(A(\lambda)\) on the spaces of null, convergent, and bounded sequencesSummability process by Mastroianni operators and their generalizationsThe spaces of \(B(r,s,t,u)\) strongly almost convergent double sequences and matrix transformations\( \lambda \)-almost summable spacesStatistical relative \(\mathcal{A}\)-summation process and Korovkin-type approximation theorem on modular spacesNonlinear approximation in \(N\)-dimension with the help of summability methodsOperator valued series, almost summability of vector valued multipliers and (weak) compactness of summing operatorA new RH-regular matrix derived by Jordan's function and its domains on some double sequence spacesOn subsets of the space of bounded sequencesBanach limits: some new thoughts and perspectivesTauberian conditions for almost convergence in a geodesic metric spaceThe order of approximation by positive linear operatorsBanach limits and a measure on the set of 0-1-sequencesWijsman lacunary \(\mathcal{I}\)-invariant convergence of sequences of setsOn the degree of approximation of functions in a weighted Lipschitz class by almost matrix summability methodAn axiomatic approach to complete patience and time invarianceOn the factorable spaces of absolutely \(p\)-summable, null, convergent, and bounded sequencesOn the space of almost convergent sequences\(f\)-asymptotically lacunary ideal equivalence of double sequencesWeak almost periodicity and the strong ergodic limit theorem for contraction semigroupsRelative weighted almost convergence based on fractional-order difference operator in multivariate modular function spacesStrongly almost lacunary \(I\)-convergent sequencesAlmost sequence spaces derived by the domain of the matrix \(A^r\)Optimal simultaneous approximation via \(\mathcal{A}\)-summabilityPower boundedness and similarity to contractions for some perturbations of isometriesAlmost strongly regular matrices and a core theorem for double sequencesAlmost convergence and a core theorem for double sequencesGeneral methods of convergence and summabilityAlmost absolute weighted summability with index \(k\) and matrix transformationsIndependent sequences, monotonicity of Banach limits and applications to noncommutative geometryInvariant linear functionals on \(L^\infty(\mathbb{R}_+)\)A mean ergodic theorem for nonlinear semigroups which are asymptotically nonexpansive in the intermediate senseSummation of almost convergent divergent sequences, topological methodsA generalization of almost convergence, completeness of some normed spaces with wuC series and a version of Orlicz-Pettis theoremTranslation-invariant linear forms on C\(_0\)(G), C(G), L\(^p\)(G) for noncompact groups\(P\)-statistical summation process of sequences of convolution operatorsOn the characterization of some classes of four-dimensional matrices and almost \(B\)-summable double sequencesOn Fourier-Haar coefficients of the function from the Marcinkiewicz spaceCriteria of regularity for norm-sequencesWeak convergence theorems for asymptotically nonexpansive mappings and semigroupsConvergences of nonexpansive iteration processes in Banach spacesSpaces of \(A_\lambda\)-almost null and \(A_\lambda\)-almost convergent sequencesApproximation of functions by the sequence of integral operatorsOn some classical properties of normed spaces via generalized vector valued almost convergenceComments on the paper Weak almost-convergence theorem without Opial's conditionOrder properties of the set of Banach limitsOn Banach-Mazur limitsBanach limit, fixed points and Ulam stability