Fourier Series and Harmonic Analysis

Periodic Functions

• A function f(x) is said to be periodic if it satisfies the condition f(x + T) = f(x) for all x, where T is the period.
• Periodic functions can be represented by a Fourier expansion.

Fourier Expansion of Periodic Functions

• The Fourier expansion of a periodic function f(x) in the interval (C, C + 2L) is given by: f(x) = a₀/2 + ∑[aₙ cos(nπx/L) + bₙ sin(nπx/L)]
• The coefficients a₀, aₙ, and bₙ can be determined using the formulas: a₀ = (1/L) ∫[C,C+2L] f(x) dx aₙ = (1/L) ∫[C,C+2L] f(x) cos(nπx/L) dx bₙ = (1/L) ∫[C,C+2L] f(x) sin(nπx/L) dx

Half Range Fourier Series

• The half-range Fourier series is used to represent a function f(x) defined in the interval (0, L) or (C, C + L).
• The half-range Fourier series is given by: f(x) = a₀/2 + ∑[aₙ cos(nπx/L) + bₙ sin(nπx/L)]

Parseval's Theorem

• Parseval's theorem states that the energy of a function f(x) is equal to the sum of the energies of its harmonics.
• The theorem is given by: ∫[C,C+2L] |f(x)|² dx = ∑[aₙ² + bₙ²]

Harmonic Analysis

• Harmonic analysis is the study of the representation of functions as sums of harmonics.
• Harmonic analysis is used in many fields, including physics, engineering, and signal processing.

Inverse of a Matrix by Partitioning

• The inverse of a matrix A can be found by partitioning A into smaller submatrices.
• The inverse of A is given by: A⁻¹ = [A₁¹⁻¹ A₂¹⁻¹]
• Where A₁ and A₂ are the submatrices of A.

Rank of a Matrix

• The rank of a matrix A is the maximum number of linearly independent rows or columns of A.
• The rank of A is denoted by ρ(A).

Rank-Nullity Theorem

• The rank-nullity theorem states that the rank of a matrix A plus the nullity of A is equal to the number of columns of A.
• The theorem is given by: ρ(A) + η(A) = n

System of Linear Equations

• A system of linear equations is a set of equations in which the variables are raised to the power of 1.
• The system of linear equations can be represented in matrix form as: Ax = b

Eigenvalues and Eigenvectors

• Eigenvalues and eigenvectors are used to diagonalize a matrix.
• The eigenvalue λ of a matrix A satisfies the equation: Ax = λx
• The eigenvector x is a non-zero vector that satisfies the equation.

Cayley-Hamilton Theorem

• The Cayley-Hamilton theorem states that every matrix A satisfies its own characteristic equation.
• The theorem is given by: |A - λI| = 0

Periodic Functions

• A function is said to be periodic if it has a repetitive pattern over a fixed interval.
• The Fourier expansion of a periodic function in (C, C+2L) is a way to represent the function as a sum of sine and cosine waves.

Fourier Series

• The Fourier series is a way to represent a periodic function as a sum of sine and cosine waves.
• The half-range Fourier series is used for functions that have a period of 2L, but are defined only on the interval [C, C+L].

Parseval's Theorem

• Parseval's Theorem states that the sum of the squares of the Fourier coefficients of a function is equal to the integral of the square of the function over one period.

Harmonic Analysis

• Harmonic analysis is the study of the representation of functions as a sum of sine and cosine waves.
• Inverse by Partitioning is a method used to find the inverse of a matrix.

Matrices

• The rank of a matrix is the maximum number of linearly independent rows or columns.
• The Rank-nullity theorem states that the sum of the rank and the nullity of a matrix is equal to the number of columns.

System of Linear Equations

• A system of linear equations is a set of equations in which the unknowns are raised to the power of 1.
• Eigen values and Eigen Vectors are used to diagonalize a matrix.

Eigen Values and Eigen Vectors

• Eigen values are scalar values that represent how much a linear transformation changes a vector.
• Eigen Vectors are non-zero vectors that, when transformed by a linear transformation, result in a scaled version of the same vector.

Cayley-Hamilton Theorem

• The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic equation.