The first four Hermite polynomials are 1, 2x, 4x² - 2, 8x³ - 12x. These polynomials arise in many fields: signal processing, physics, numerical analysis, probability, combinatorics, etc. Show that the first four Hermite polynomials form a basis of P3.
The first four Hermite polynomials are 1, 2x, 4x² - 2, 8x³ - 12x. These polynomials arise in many fields: signal processing, physics, numerical analysis, probability, combinatorics, etc. Show that the first four Hermite polynomials form a basis of P3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Text for Educational Website**
1. **Hermite Polynomials Basis**
The first four Hermite polynomials are \(1\), \(2x\), \(4x^2 - 2\), and \(8x^3 - 12x\). These polynomials are prevalent in numerous applications across diverse fields such as signal processing, physics, numerical analysis, probability, and combinatorics. The objective is to demonstrate that these first four Hermite polynomials form a basis for the polynomial space \(\mathbb{P}_3\).
2. **Linear Combination in \(\mathbb{P}_3\)**
With the first four Hermite polynomials acting as a basis for \(\mathbb{P}_3\), determine the linear combination for the polynomial:
\[
p(x) = 7 - 12x - 8x^2 + 12x^3
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Transcribed Image Text:**Text for Educational Website**
1. **Hermite Polynomials Basis**
The first four Hermite polynomials are \(1\), \(2x\), \(4x^2 - 2\), and \(8x^3 - 12x\). These polynomials are prevalent in numerous applications across diverse fields such as signal processing, physics, numerical analysis, probability, and combinatorics. The objective is to demonstrate that these first four Hermite polynomials form a basis for the polynomial space \(\mathbb{P}_3\).
2. **Linear Combination in \(\mathbb{P}_3\)**
With the first four Hermite polynomials acting as a basis for \(\mathbb{P}_3\), determine the linear combination for the polynomial:
\[
p(x) = 7 - 12x - 8x^2 + 12x^3
\]
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