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Chapter 7

Main Definitions and Formulas.

Integral Transforms

7.1. Some Definitions, Remarks, and Formulas

7.1-1. Some Definitions
2
A function f(x) is said to be square integrable on an interval [a, b]if f (x) is integrable on [a, b].
The set of all square integrable functions is denoted by L 2 (a, b) or, briefly, L 2 .* Likewise, the set of
all integrable functions on [a, b] is denoted by L 1 (a, b) or, briefly, L 1 .
Let us list the main properties of functions from L 2 .
◦
1 . The sum of two square integrable functions is a square integrable function.
2 . The product of a square integrable function by a constant is a square integrable function.
◦
3 . The product of two square integrable functions is an integrable function.
◦
◦
4 .If f(x) ∈ L 2 and g(x) ∈ L 2 , then the following Cauchy–Schwarz–Bunyakovsky inequality
holds:
2
2
2
(f, g) ≤ f g ,
b b
2
2
(f, g)= f(x)g(x) dx, f =(f, f)= f (x) dx.
a a
The number (f, g) is called the inner product of the functions f(x) and g(x) and the number f is
called the L 2 -norm of f(x).
◦
5 .For f(x) ∈ L 2 and g(x) ∈ L 2 , the following triangle inequality holds:
f + g ≤ f + g .

◦
6 . Let functions f(x) and f 1 (x), f 2 (x), ... , f n (x), ... be square integrable on an interval [a, b]. If

b
2
lim f n (x)– f(x) dx =0,
n→∞
a
then the sequence f 1 (x), f 2 (x), ... is said to be mean-square convergent to f(x).
Note that if a sequence of functions {f n (x)} from L 2 converges uniformly to f(x), then f(x)∈L 2
and {f n (x)} is mean-square convergent to f(x).
* In the most general case the integral is understood as the Lebesgue integral of measurable functions. As usual, two equivalent
functions (i.e., equal everywhere, or distinct on a negligible set (of zero measure)) are regarded as one and the same element
of L 2 .

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 427