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1: 1.1 Special Notation
x , y real variables.
L 2 ( X , d α ) the space of all Lebesgue–Stieltjes measurable functions on X which are square integrable with respect to d α .
2: 28.30 Expansions in Series of Eigenfunctions
Then every continuous 2 π -periodic function f ( x ) whose second derivative is square-integrable over the interval [ 0 , 2 π ] can be expanded in a uniformly and absolutely convergent series …
3: 1.8 Fourier Series
1.8.5 1 π π π | f ( x ) | 2 d x = 1 2 | a 0 | 2 + n = 1 ( | a n | 2 + | b n | 2 ) ,
1.8.6 1 2 π π π | f ( x ) | 2 d x = n = | c n | 2 ,
where f ( x ) is square-integrable on [ π , π ] and a n , b n , c n are given by (1.8.2), (1.8.4). If g ( x ) is also square-integrable with Fourier coefficients a n , b n or c n then …
4: 1.4 Calculus of One Variable
Square-Integrable Functions
A function f ( x ) is square-integrable if …
5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
For a Lebesgue–Stieltjes measure d α on X let L 2 ( X , d α ) be the space of all Lebesgue–Stieltjes measurable complex-valued functions on X which are square integrable with respect to d α , …
1.18.64 f ( x ) = 𝝈 c f ^ ( λ ) ϕ λ ( x ) d λ + 𝝈 p f ^ ( λ n ) ϕ λ n ( x ) , f ( x ) C ( X ) L 2 ( X ) .
6: 30.4 Functions of the First Kind
If f ( x ) is mean-square integrable on [ 1 , 1 ] , then formally …
7: 33.22 Particle Scattering and Atomic and Molecular Spectra
  • Eigenstates using complex-rotated coordinates r r e i θ , so that resonances have square-integrable eigenfunctions. See for example Halley et al. (1993).

  • 8: 1.14 Integral Transforms
    Suppose f ( t ) and g ( t ) are absolutely and square integrable on ( , ) , then … Suppose f ( t ) and g ( t ) are absolutely and square integrable on [ 0 , ) , then …
    9: Errata
  • Equations (1.8.5), (1.8.6)
    1.8.5 1 π π π | f ( x ) | 2 d x = 1 2 | a 0 | 2 + n = 1 ( | a n | 2 + | b n | 2 )
    1.8.6 1 2 π π π | f ( x ) | 2 d x = n = | c n | 2

    Previously these equations were given as inequalities. For square integrable functions the inequality can be sharpened to = .