I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 A. Fourier series in L2.
 B. Fourier transform.
 C. Fourier transform of delta function.
 D. Poisson formula for delta function and Whittaker sampling theorem.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Poisson formula for delta function and Whittaker sampling theorem.

hat happens if one takes Fourier series of non-periodic function? To put the question in exact terms, suppose the function is such that the restriction of it to the interval belongs to . We form the function with the functions being the Fourier exponentials normalized to . Such function will be the -periodic repetition of the restriction to the intervals , :

This is so because the function is -periodic and the sum is the Fourier series of . Then by uniqueness of Fourier series for -periodic functions we must have

Proposition

(Poisson formula for delta function) For any we have in -sense. See the section ( Weak derivative section ) for notation.

Proof

We use the above remark to formally take Fourier series of the function on the interval . We have Thus

Proposition

(Whittaker sampling theorem) For a continuous function and we form Then

Proof

We have Then, by the proposition ( Basic properties of Fourier transform )-6, where and by the proposition ( Poisson formula for delta function ) Therefore

Proposition

(Fourier transform of projection on span of translates) Let be a function . We introduce the notation Suppose is such that the operation , is well defined. (For example, the is well defined if are orthonormal or if has compact support.)

Then

Proof

We introduce the notation Let be such that Then where . We have We apply the proposition ( Whittaker sampling theorem ). where

 Notation. Index. Contents.