On Convergence of Some Infinite Products(Distribution of values of arithmetic functions)
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I Foundations.- 1 Sets and Functions.- 1.1 Sets.- 1.2 Ordered pairs.- 1.3 Functions.- 2 Real and Complex Numbers.- 2.1 Algebraic properties of real numbers.- 2.2 Order.- 2.3 Upper and lower bounds.- 2.4 Complex numbers.- 2.5 Notation.- II Limits.- 3 Limits.- 3.1 Introduction.- 3.2 Directed sets.- 3.3 The definition of a limit.- 3.4 Examples of limits.- 3.5 Sums, products, and quotients of limits.- 3.6 Limits and inequalities.- 3.7 Functions tending to infinity.- 4 Bisection Arguments.- 4.1 Nested intervals.- 4.2 The Intermediate Value Therem.- 4.3 The Mean Value Inequality.- 4.4 The Cauchy Criterion.- 5 Infinite Series.- 5.1 Infinite series.- 5.2 Unordered sums.- 5.3 Absolute convergence and rearrangements.- 5.4 The Cauchy Product.- 5.5 Iterated sums.- 6 Periodic Functions.- 6.1 The exponential function.- 6.2 The trigonometric functions.- 6.3 Periodicity and ?.- 6.4 The argument of a complex number.- 6.5 The logarithm.- III Analysis.- 7 Sequences.- 7.1 Convergent sequences.- 7.2 Some important examples.- 7.3 Bounded sequences.- 7.4 The Fundamental Theorem of Algebra.- 7.5 Unbounded sequences.- 7.6 Upper and lower limits.- 8 Continuous Functions.- 8.1 Continuous functions.- 8.2 Functions continuous on an interval.- 8.3 Monotonic functions.- 8.4 Uniform continuity.- 8.5 Uniform convergence.- 9 Derivatives.- 9.1 The derivative.- 9.2 The Chain Rule.- 9.3 The Mean Value Theorem.- 9.4 Inverse functions.- 9.5 Power series.- 9.6 Taylor series.- 10 Integration.- 10.1 The integral.- 10.2 Upper and lower integrals.- 10.3 Integrable functions.- 10.4 Integration and differentiation.- 10.5 Improper integrals.- 10.6 Integration and differentiation of series.- 11 ?, ?, e, and n!.- 11.1 The number e.- 11.2 The number ?.- 11.3 Euler's constant ?.- 11.4 Stirling's formula for n!.- 11.5 A series and an integral for ?.- Appendix: Mathematical Induction.- References.
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This paper investigates rational approximations of a Markov function that have the highest order of contact with it at infinity, and whose denominators are invariant under multiplication of their argument by a root of unity of some fixed degree (such approximations are used in many number-theoretical problems). The approximations converge under mild restrictions on the measure. Moreover, the denominators of the approximants and the corresponding functions of the second kind have logarithmic asymptotics expressible in terms of a certain extremal measure which, in the simplest case, is the Tchebycheff measure. An explicit form is found for the extremal measure in the general case; in fact, the inverse of the distribution function is expressed in terms of elementary functions, the power moments are calculated, and the Markov function of the extremal measure is connected with algebraic equations and generalized hypergeometric functions. Bibliography: 10 titles.
Padé approximant
Elementary function
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A major portion of approximation theory is concerned with the approximation of functions by polynomials or by sequences {Tn} of linear operators, more specifically with the connections between the structural properties of the function f being approximated and the convergence per se and/or rate of convergence of ‖Tn (f) — f ‖ to zero for n → ∞. in particular, the wide area of approximation theory and its applications is devoted to the convergence per se and the rate of vonvergence of, for example, (a) the best trigonometric approximation of a given function, (b) the partial sums of the Fourier series of a function to the function. itself, (c) the solution of Dirichlet's problem for the unit disk to the given boundary value, (d) the Whittaker — Shannon sampling series expansion of a duration-limited function to the functioninquestion, (e) the sums occuring in the weak law of large numbers in probability theory.
Zero (linguistics)
Trigonometric series
Approximation Theory
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These are classified by the direction of approximation (from above or below), the set family types (partition or covering) of simple functions, the coefficient signature (non-negative or signed), and cardinal number of terms of simple functions(finite or countable infinite). We will compare these integrals considering the monotone increasing/decreasing convergence theorems.
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Part 1 The Real Numbers: The Real Number System Upper and Lower Bounds. Part 2 Sequences and Series: Algebraic Operations on Limits Monotone Sequences Infinite Series. Part 3 Continuous Functions: Functions, Limits and Continuity The Intermediate Value Property for Continuous Functions Uniform Continuity Increasing Functions. Part 4 Differentiable Functions: Repeated Differentiation Mean Value Theorems Local Maxima and Minima Taylor's Theorem. Part 5 Further Results on Infinite Series: Tests for Convergence Series of Complex Terms Power Series Multiplication of Series. Part 6 Special Functions: The Exponential Function The Logarithm Trigonometric Functions Inverse Trigonometric Functions. Part 7 The Riemann Integral: Integral Forms of the Mean Value Theorems Integration Over Unbounded Intervals Integration of Unbounded Functions. Part 8 The Number Pi.
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The paper discusses both theoretical properties and practical implementation of product integration rules of the form \[ ∫ − ∞ ∞ k ( x ) f ( x ) d x ≈ ∑ i = 1 n w n i f ( x n i ) , \int _{ - \infty }^\infty {k(x)f(x)\,dx \approx \sum \limits _{i = 1}^n {{w_{ni}}f({x_{ni}}),} } \] where f is continuous, k is absolutely integrable, the nodes { x n i } \{ {x_{ni}}\} are roots of the Hermite polynomials H n ( x ) {H_n}(x) , and the weights { w n i } \{ {w_{ni}}\} are chosen so that the rule is exact if f is any polynomial of degree > n > n . Convergence of the rule to the exact integral as n → ∞ n \to \infty is proved for a wide class of functions f and k (including singular or oscillatory functions k ), and rates of convergence are estimated. The rules are shown to have the property of asymptotic positivity, and as a consequence exhibit good numerical stability. Numerical calculations for some practical cases are presented, which show the method to be computationally effective for integrands (including highly oscillatory ones) that decay suitably at infinity. Applications of the method to integration over [ 0 , ∞ ) [0,\infty ) are also discussed.
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Let and be entire functions of order less than 1 with and transcendental. We prove that every solution of the equation , , being has zeros with infinite exponent of convergence.
Exponent
Transcendental equation
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This note is concerned with arithmetic properties of power series with integral coefficients that are lacunary in the following sense. There are two infinite sequences of integers {r n } and {s n }, satisfying such that It is also assumed that f ( z ) has a positive radius of convergence, R f say, where naturally . A power series with these properties will be called admissible .
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Radius of convergence
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Power function
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In the first part we have shown that, for $L_2$-approximation of functions from a separable Hilbert space in the worst-case setting, linear algorithms based on function values are almost as powerful as arbitrary linear algorithms if the approximation numbers are square-summable. That is, they achieve the same polynomial rate of convergence. In this sequel, we prove a similar result for separable Banach spaces and other classes of functions.
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