Generalized Fourier series

\[g(x)= \sum_{j=0}^\infty c_j \varphi_j(x)\]

\(\{\varphi_j\}\) – orthonormal basis; \(c_j=\int g(x)\varphi_j(x)dx\)

Parseval’s identity: \(\int g(x)^2 dx =\sum_{j=0}^\infty c_j^2\)

Jean Baptiste Joseph Fourier

21 March 1768 – 16 May 1830

Who was he?

Mathematician, physicist, teacher, archeologist, director of the Statistical Bureau of the Seine, prefect of Isere & Rhone, secret policeman, political prisoner, governor of Egypt, friend of Napoleon and secretary of the Academie des Sciences.

had a crazy idea (1807):

Any periodic function can be written as a weighted sum of sines and cosines of different frequencies

Do not believe it?

– neither did Lagrange, Laplace, Poisson and others

– not translated to English untill 1878

But… it’s true!

– called Fourier series

– possibly the main tool in signal processing, engineering, time series, etc.

Legendre polynomials

Approximantion of various functions (solid) by polynomial series: J=3 (dotted), J=5 (short-dashed), J=10 (long-dashed)

Fourier sine and cosine series

Approximantion of various functions (solid) by Fourier sine and cosinse series: J=3 (dotted), J=5 (short-dashed), J=10 (long-dashed

Cosine series

Approximantion of various functions (solid) by cosine series: J=3 (dotted), J=5 (short-dashed), J=10 (long-dashed

Various nonparametric estimators as Fourier linear shrinkage estimators

truncated Fourier series: \(w_j=I\{j \leq J\}\)

moving average: \(w_j=\frac{\sin(\pi \lambda j/(2n))}{\lambda \sin(\pi j/(2n))}\)

spline smoothing: \(w_j=\frac{1}{1+\lambda (2\pi j)^4}\)

Haar series

Approximantion of various functions (solid) by Haar series: J=3 (dotted) and J=5 (short-dashed)