Page 2
Semester 1: Real Analysis I
Functions of bounded variation - Properties of monotonic functions - Total variation
Functions of bounded variation
Definition and Examples
A function is said to have bounded variation on an interval if the total variation over that interval is finite. For example, the function f(x) = x is of bounded variation on any bounded interval. In contrast, the function f(x) = sin(1/x) for x > 0 is of unbounded variation on any interval containing 0.
Properties of Functions of Bounded Variation
Functions of bounded variation can be expressed as the difference of two increasing functions. They are also continuous and can have a finite number of discontinuities. Bounded variation is an important property as it implies integrability over a certain interval.
Total Variation
The total variation of a function f on an interval [a, b] is defined as the supremum of the sums of absolute differences of the function at finitely many points in the interval. Formally, V(f, [a, b]) = sup { Σ |f(x_{i+1}) - f(x_i)| : a ≤ x_1 < x_2 < ... < x_n ≤ b, n ∈ ℕ }.
Monotonic Functions
A function is monotonic if it is either entirely non-increasing or non-decreasing over its domain. Monotonic functions are always of bounded variation, and thus their total variation coincides with the absolute difference between the values at the endpoints of the interval.
Implications in Real Analysis
The study of functions with bounded variation is crucial in real analysis as it connects to various concepts such as the Riemann-Stieltjes integral. Additionally, functions of bounded variation play a role in different theorems, such as the Jordan decomposition theorem.
The Riemann - Stieltjes Integral - Linear properties - Integration by parts
The Riemann - Stieltjes Integral - Linear properties - Integration by parts
Introduction to Riemann-Stieltjes Integral
The Riemann-Stieltjes integral is an extension of the classical Riemann integral. It is defined for functions f and g over a closed interval [a, b], where g is a function of bounded variation. The notation for the integral is given by ∫_a^b f(x) dg(x).
Existence of Riemann-Stieltjes Integral
Conditions under which the Riemann-Stieltjes integral exists include the continuity of f and the bounded variation of g. If f is continuous and g is of bounded variation, then the integral exists.
Linear Properties of the Riemann-Stieltjes Integral
The Riemann-Stieltjes integral enjoys linearity, meaning that ∫_a^b [αf(x) + βh(x)] dg(x) = α∫_a^b f(x) dg(x) + β∫_a^b h(x) dg(x), where α and β are constants.
Integration by Parts
The integration by parts formula for the Riemann-Stieltjes integral is given by ∫_a^b f(x) dg(x) = f(b)g(b) - f(a)g(a) - ∫_a^b g(x) df(x). This is useful for evaluating integrals where direct integration is difficult.
Applications
The Riemann-Stieltjes integral is used in various fields, including probability theory, statistics, and economics. It provides a general framework for integration that includes cases where traditional methods are inadequate.
Conclusion
The Riemann-Stieltjes integral is an important concept in real analysis, allowing for generalizations of integration techniques. Understanding its properties and applications is crucial for deeper studies in mathematics.
Integrators of bounded variation - Mean value theorems - Differentiation under integral sign
Integrators of bounded variation
Definition of Bounded Variation
A function is said to be of bounded variation on an interval if the total variation, which is the supremum of the sums of absolute differences of the function values at partition points, is finite.
Properties of Functions of Bounded Variation
Functions of bounded variation can be expressed as the difference of two monotonic functions. They are also integrable and have well-defined integrals.
Integrals of Functions of Bounded Variation
The integral of a function of bounded variation can be evaluated using the Riemann-Stieltjes integral, which allows for accommodating discontinuities while preserving integration properties.
Mean Value Theorems for Integrators
Mean value theorems provide conditions under which a function attains a value equal to the average of its outputs over an interval. This is crucial for understanding integrals of bounded variation.
Differentiation under the Integral Sign
This concept allows for differentiating an integral with respect to a parameter. If the integrand is of bounded variation, the interchange of integration and differentiation is justified.
Applications and Examples
Understanding integrals of bounded variation is essential in real analysis, where applications include solving differential equations and analyzing the behavior of functions.
Infinite Series and infinite Products - Double sequences - Power series
Introduction to Infinite Series
An infinite series is the sum of an infinite sequence of terms. It can be expressed as the limit of partial sums of a sequence. The general form is S = a1 + a2 + a3 + ... + an, where n approaches infinity.
Convergence of Infinite Series
A series converges if the limit of its partial sums exists and is finite. Various tests for convergence include the comparison test, ratio test, root test, and the integral test.
Infinite Products
An infinite product is a product of an infinite number of factors. It can converge under certain conditions and is often used in various mathematical applications, including representation of functions.
Double Sequences
A double sequence is a function defined on the Cartesian product of two indices. It can be represented as a two-dimensional array and is useful in studying convergence in multiple dimensions.
Power Series
A power series is an infinite series of the form S(x) = a0 + a1(x - c) + a2(x - c)^2 + ..., where c is the center of the series. Power series can converge within a certain radius of convergence.
Applications of Infinite Series and Products
Infinite series and products are used in various fields, including calculus, number theory, and combinatorics. They play a crucial role in developing series expansions of functions.
Sequences of Functions - Pointwise and uniform convergence - Riemann-Stieltjes integration
Sequences of Functions - Pointwise and Uniform Convergence - Riemann-Stieltjes Integration
Definitions
Pointwise convergence occurs when a sequence of functions converges at each point in its domain. Uniform convergence means the convergence is uniform across the entire domain, allowing the interchange of limit and integration or differentiation. Riemann-Stieltjes integration extends the Riemann integral by integrating with respect to a function that may not be monotonic.
Pointwise Convergence
A sequence of functions {f_n} converges pointwise to a function f if for every point x in the domain, the limit of f_n(x) as n approaches infinity equals f(x). This form of convergence does not guarantee the preservation of properties such as continuity.
Uniform Convergence
A sequence {f_n} converges uniformly to a function f if, for every epsilon > 0, there exists an N such that for all n ≥ N and all x in the domain, |f_n(x) - f(x)| < epsilon. Uniform convergence preserves continuity, integrability, and differentiation.
Comparison of Convergences
Pointwise convergence can occur without uniform convergence. Uniform convergence implies pointwise convergence, but the converse is not true. Examples illustrate that uniform convergence produces better analytical properties.
Riemann-Stieltjes Integration
This integration is defined as the limit of Riemann sums involving an integrator function g. It is particularly useful when dealing with non-monotonic functions and is applied in contexts involving probability and stochastic processes. Uniform convergence of sequences of functions ensures the validity of swapping limits and integrals.
Applications
Understanding convergence types is crucial for proving theorems concerning the interchange of limit and integral. In Riemann-Stieltjes integration, uniform convergence aids in establishing results like the Dominated Convergence Theorem and integration of limits.
Conclusion
Studying pointwise and uniform convergence helps in grasping the behavior of sequences of functions. The Riemann-Stieltjes integral enriches integration techniques and is foundational in advanced analysis topics.
