Mathematical Physics Vol 1

6.2 The fundamental convergence theorem for Fourier series

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6.2 The fundamental convergence theorem for Fourier series

Before formulating this theorem, let us first define some terms that are going to be used in this formulation. Partially smooth functions

Definition The function f ( x ) is said to be partially continuous on the interval [ a , b ] , if it is continuous in all points of the interval, except for a finite number of points, in which it has first-order discontinuities.

Definition A partially continuous function f ( x ) , defined on the interval [ a , b ] , is said to be partially smooth , if its first derivative f ′ ( x ) exists, and this derivative is a continuous function in all points of the interval [ a , b ] , except for a finite number of points, in which the left and right limit values f ′ ( x + 0 )= lim t → + 0 f ′ ( x + t ) , f ′ ( x − 0 )= lim t → + 0 f ′ ( x − t ) , (6.14) exist. It is also assumed that finite limit values f ′ ( a + 0 ) and f ′ ( b − 0 ) exist in the end points of the interval [ a , b ] .

R Note that besides the term "partially" the term "part-by-part" is also used.

Let us demonstrate a partially smooth function on an example (graphically).

Figure 6.2: Partially smooth function.

Note that this function has tangents at points x 1 ,..., x 4 , but the tangents on the left and right side are not equal. Thus, the function is smooth in intervals ( a , x 1 ) ∪ ( x 1 , x 2 ) ∪ ( x 2 , x 3 ) ∪ ( x 3 , x 4 ) . The fundamental convergence theorem for Fourier series