What is monotone sequence?
Monotone Sequences. Definition : We say that a sequence (xn) is increasing if xn ≤ xn+1 for all n and strictly increasing if xn < xn+1 for all n. Similarly, we define decreasing and strictly decreasing sequences. Sequences which are either increasing or decreasing are called monotone.
Are polynomials dense in LP?
Since, by the domi- nated convergence theorem, uniform convergence implies Lp(µ) — convergence, it follows from the Weierstrass approximation theorem (see Theorem 8.34 and Corollary 8.36 or Theorem 12.31 and Corollary 12.32) that polynomials are also dense in Lp(µ).
What is monotonic sequence theorem?
In real analysis, the monotone convergence theorem states that if a sequence increases and is bounded above by a supremum, it will converge to the supremum; similarly, if a sequence decreases and is bounded below by an infimum, it will converge to the infimum.
Is convergent sequence monotonic?
A bounded monotonic increasing sequence is convergent. We will prove that the sequence converges to its least upper bound (whose existence is guaranteed by the Completeness axiom). So let α be the least upper bound of the sequence.
Which of the following function is nowhere differentiable?
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
Are step functions dense in LP?
For every p ∈ [1,∞), (a) step functions are dense in Lp(R); (b) the same holds for Rd with arbitrary locally finite measure. Every function (of class Lp) is nearly a step function.
What is a monotonic series?
We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value.
What is a monotonic sequence example?
A sequence is said to be monotone if it is either increasing or decreasing. The sequence n2 : 1, 4, 9, 16, 25, 36, 49, is increasing. The sequence 1/2n : 1/2, 1/4, 1/8, 1/16, 1/32, is decreasing. The sequence (1)n1/n : 1, 1/2, 1/3, 1/4, 1/5, 1/6, is not monotone.
Is Weierstrass function differentiable?
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve.
Why is Weierstrass function not differentiable?
The higher-order terms create the smaller oscillations. With b carefully chosen as in the theorem, the graph becomes so jagged that there is no reasonable choice for a tangent line at any point; that is, the function is nowhere differentiable.
Are step functions dense in L1?
D = {step functions} is dense in L1. any measurable set A with finite measure, and any ϵ > 0, there exists some g ∈ D such that ∥χA − g∥L1 < ϵ.
Is LP space a Banach space?
(Riesz-Fisher) The space Lp for 1 ≤ p < ∞ is a Banach space.
What is Weierstrass’s theorem?
The Weierstrass extreme value theorem, which states that a continuous function on a closed and bounded set obtains its extreme values The Weierstrass–Casorati theorem describes the behavior of holomorphic functions near essential singularities The Weierstrass preparation theorem describes the behavior of analytic functions near a specified point
Can Weierstrass’infinite product theorem be generalized to an arbitrary domain?
Weierstrass’ infinite product theorem can be generalized to the case of an arbitrary domain $ D \\subset \\mathbf C $: Whatever a sequence of points $ \\ { \\alpha _ {k} \\} \\subset D $ without limit points in $ D $, there exists a holomorphic function $ f $ in $ D $ with zeros at the points $ \\alpha _ {k} $ and only at these points.
Is the Stone-Weierstrass theorem valid for all real-valued functions?
The theorem is also valid for real-valued continuous $ 2 \\pi $- periodic functions and trigonometric polynomials, e.g. for real-valued functions which are continuous on a bounded closed domain in an $ m $- dimensional space, or for polynomials in $ m $ variables. For generalizations, see Stone–Weierstrass theorem.
What is the significance of the Lindemann-Weierstrass theorem?
The Lindemann–Weierstrass theorem concerning the transcendental numbers The Weierstrass factorization theorem asserts that entire functions can be represented by a product involving their zeroes The Sokhatsky–Weierstrass theorem which helps evaluate certain Cauchy-type integrals