Page 242 - DMTH401_REAL ANALYSIS

Real Analysis

Notes
Example: Let f(x) = x on [0, 1] and {f } ¥ . We get Fourier coefficients a = x, f  =
n n 1 n n
=
2 n
1 æ 1 ö 2 1 . Therefore, we can compute Fourier series for f(x) = x which is
f
0 ò x (x)dx = - ç è 2 ø 2 = - 2 n 1
n ÷
+
1
-å ¥ n 1 f n (x) .
=
+
2 n 1
Self Assessment
Fill in the blanks:
f
1. For any f: [0, 1]   define its .................................. B (x) such that
n
n æ k ö
f
f
B (x) = å ç ÷ p (x).
n kn
k 0 è ø
n
=
2. Let f be a real-valued function on [0, 1]. If f is continuous then ............................ .
3. Define the two—norm ||· || on f Î[a, b] such that ......................... .
2
¥
4. A collection of Riemman integrable functions {f n n 1 on [a, b] is called an .........................
}
=
b
f
f ,f = ò f (x) (x)dx = 0, " m ¹ n.
m n a m n
19.3 Summary
 Let I   and f be a real-valued function on I. We say that f is uniformly continuous on I if
"  > 0  > 0 " x, y Î I : |x – y| <   |f(x) – f(y)| < .
Also remind, that a continuous function on [a, b] is always uniformly continuous.
n
n
2
 " n Î : å n k 0 p (x) = 1, (b) " n Î : å k 0 k pkn (x) = nx, (c) " n Î : å k 0 (k – nx) p (x) =
=
=
=
kn
kn
nx(1 – x).
If x Î [0, 1], we can define a random variable Y the number of heads observed on unfair
n,x
x-coin tossed n-times. Then
æ n ö
k
(Y = k) = ç ÷ x (1 – x) n – k = p (x).
n,x è k ø kn
Moreover, we find the following relation with (b)
n
[Y ] = å k (x) = nx.
n,x p kn
k 0
=
 For any f, g Î [a, b] define inner product of f and g .,. such that
b
f, g = ò f(x)g(x)dx.
a
 Recall from Linear Algebra that the inner product space is finite-dimensional vector space
V equipped with mapping ·,·: V  V  (or ).
19.4 Keyword
Fourier Series: For any f, g Î [a, b] define inner product of f and g .,. such that
b
f, g = ò f(x)g(x)dx.
a

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