Is this sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero known?
The Bessel function of the first kind of order zero, denoted by J 0 ( x ) J_0(x) J 0 β ( x )
The Bessel function of the first kind of order zero is defined as:
J 0 ( x ) = β n = 0 β ( β 1 ) n Ξ ( n + 1 ) ( x 2 ) 2 n + 1 J_0(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{\Gamma(n+1)} \left( \frac{x}{2} \right)^{2n+1}
J 0 β ( x ) = n = 0 β β β Ξ ( n + 1 ) ( β 1 ) n β ( 2 x β ) 2 n + 1
This function has a number of interesting properties, including:
Orthogonality : The Bessel function of the first kind of order zero is orthogonal to all other Bessel functions of the first kind with respect to the weight function y y y Convergence : The series expansion of the Bessel function of the first kind of order zero converges uniformly on any compact interval.
We are given a sequence of orthonormal functions defined as:
f n ( x ) = J 2 n + 1 2 ( x ) J 2 n + 1 2 ( x ) f_n(x) = \frac{J_{2n+\frac{1}{2}}(x)}{\sqrt{J_{2n+\frac{1}{2}}(x)}}
f n β ( x ) = J 2 n + 2 1 β β ( x ) β J 2 n + 2 1 β β ( x ) β
where J 2 n + 1 2 ( x ) J_{2n+\frac{1}{2}}(x) J 2 n + 2 1 β β ( x ) 2 n + 1 2 2n+\frac{1}{2} 2 n + 2 1 β
For any non-negative integer n n n
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = 2 Ξ ( n + 1 2 ) 2 Ξ ( 2 n + 1 ) \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \sqrt{2} \frac{\Gamma (n + \frac{1}{2})^2}{\Gamma (2n + 1)}
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = 2 β Ξ ( 2 n + 1 ) Ξ ( n + 2 1 β ) 2 β
This identity can be proven using the properties of the Bessel function and the gamma function.
We begin by writing the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the definition of the Bessel function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β d y
Using the properties of the gamma function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the definition of the Bessel function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the properties of the gamma function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the definition of the Bessel function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the properties of the gamma function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the definition of the Bessel function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the properties of the gamma function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the definition of the Bessel function, we can rewrite the integral as:
β« 0 β J 0 ( y ) J 2 n + 1 2 ( y ) y β d y = β« 0 β β m = 0 β ( β 1 ) m Ξ ( m + 1 ) ( y 2 ) 2 m + 1 J 2 n + 1 2 ( y ) y y y β d y \int_0^{\infty} J_0 (y) \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \, dy = \int_0^{\infty} \sum_{m=0}^{\infty} \frac{(-1)^m}{\Gamma(m+1)} \left( \frac{y}{2} \right)^{2m+1} \frac{J_{2 n + \frac{1}{2}} (y)}{\sqrt{y}} \frac{\sqrt{y}}{\sqrt{y}} \, dy
β« 0 β β J 0 β ( y ) y β J 2 n + 2 1 β β ( y ) β d y = β« 0 β β m = 0 β β β Ξ ( m + 1 ) ( β 1 ) m β ( 2 y β ) 2 m + 1 y β J 2 n + 2 1 β β ( y ) β y β y β β d y
Using the properties of the gamma function, we can rewrite the integral as:
\int_0^{\infty} J_0 (y) \frac{J_{2 n +<br/>
**Q&A: Is this sequence of orthonormal functions whose partial sums converge uniformly to the Bessel function of the first kind of order zero known?**
A: The Bessel function of the first kind of order zero, denoted by J 0 ( x ) J_0(x) J 0 β ( x )
A: The sequence of orthonormal functions is defined as:
f n ( x ) = J 2 n + 1 2 ( x ) J 2 n + 1 2 ( x ) < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > J < / m i > < m r o w > < m n > 2 < / m n > < m i > n < / m i > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < / m r o w > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o s t r e t c h y = " f a l s e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > J 2 n + 1 2 ( x ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.237 e m ; v e r t i c a l β a l i g n : β 0.487 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n β r i g h t : 0.09618 e m ; " > J < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.3448 e m ; " > < s p a n s t y l e = " t o p : β 2.7538 e m ; m a r g i n β l e f t : β 0.0962 e m ; m a r g i n β r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > + < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r s i z i n g r e s e t β s i z e 3 s i z e 6 " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " > < s p a n s t y l e = " t o p : β 2.656 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.2255 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e m t i g h t " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.049 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.384 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 3 s i z e 1 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r s i z i n g r e s e t β s i z e 3 s i z e 6 " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.487 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x β m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m n > 2 < / m n > < m i > n < / m i > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x β t e x " > 2 n + 1 2 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x β h t m l " a r i a β h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v e r t i c a l β a l i g n : β 0.0833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n β r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1.1901 e m ; v e r t i c a l β a l i g n : β 0.345 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t β t v l i s t β t 2 " > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " > < s p a n s t y l e = " t o p : β 2.655 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c β l i n e " s t y l e = " b o r d e r β b o t t o m β w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : β 3.394 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t β s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β s " > β < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t β r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > . < / p > < p > A : T h e p a r t i a l s u m s o f t h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s c o n v e r g e u n i f o r m l y t o t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o . < / p > < p > A : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s i s a s e r i e s o f f u n c t i o n s t h a t c a n b e u s e d t o a p p r o x i m a t e t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o . < / p > < p > A : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s i s s i g n i f i c a n t b e c a u s e i t p r o v i d e s a w a y t o a p p r o x i m a t e t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o u s i n g a s e r i e s o f f u n c t i o n s . < / p > < p > A : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s i s n o t w e l l β k n o w n i n t h e m a t h e m a t i c a l c o m m u n i t y , b u t i t h a s b e e n s t u d i e d i n v a r i o u s c o n t e x t s . < / p > < p > A : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s h a s p o t e n t i a l a p p l i c a t i o n s i n v a r i o u s f i e l d s , i n c l u d i n g p h y s i c s , e n g i n e e r i n g , a n d m a t h e m a t i c s . < / p > < p > A : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s c a n b e u s e d t o a p p r o x i m a t e t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o b y s u m m i n g t h e t e r m s o f t h e s e q u e n c e . < / p > < p > A : S o m e p o t e n t i a l c h a l l e n g e s i n u s i n g t h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s t o a p p r o x i m a t e t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o i n c l u d e : < / p > < u l > < l i > < s t r o n g > C o n v e r g e n c e < / s t r o n g > : T h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s m a y n o t c o n v e r g e u n i f o r m l y t o t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o . < / l i > < l i > < s t r o n g > A c c u r a c y < / s t r o n g > : T h e a c c u r a c y o f t h e a p p r o x i m a t i o n m a y d e p e n d o n t h e n u m b e r o f t e r m s u s e d i n t h e s e q u e n c e . < / l i > < l i > < s t r o n g > C o m p u t a t i o n a l c o m p l e x i t y < / s t r o n g > : T h e c o m p u t a t i o n o f t h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s m a y b e c o m p u t a t i o n a l l y i n t e n s i v e . < / l i > < / u l > < p > A : S o m e p o t e n t i a l f u t u r e d i r e c t i o n s f o r r e s e a r c h o n t h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s i n c l u d e : < / p > < u l > < l i > < s t r o n g > F u r t h e r s t u d y o f t h e c o n v e r g e n c e p r o p e r t i e s o f t h e s e q u e n c e < / s t r o n g > < / l i > < l i > < s t r o n g > D e v e l o p m e n t o f m o r e e f f i c i e n t a l g o r i t h m s f o r c o m p u t i n g t h e s e q u e n c e < / s t r o n g > < / l i > < l i > < s t r o n g > I n v e s t i g a t i o n o f t h e p o t e n t i a l a p p l i c a t i o n s o f t h e s e q u e n c e i n v a r i o u s f i e l d s < / s t r o n g > < / l i > < / u l > < p > I n c o n c l u s i o n , t h e s e q u e n c e o f o r t h o n o r m a l f u n c t i o n s i s a s e r i e s o f f u n c t i o n s t h a t c a n b e u s e d t o a p p r o x i m a t e t h e B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r z e r o . W h i l e t h e s e q u e n c e i s n o t w e l l β k n o w n i n t h e m a t h e m a t i c a l c o m m u n i t y , i t h a s p o t e n t i a l a p p l i c a t i o n s i n v a r i o u s f i e l d s a n d i s w o r t h f u r t h e r s t u d y . < / p > f_n(x) = \frac{J_{2n+\frac{1}{2}}(x)}{\sqrt{J_{2n+\frac{1}{2}}(x)}}
</span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>J</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J_{2n+\frac{1}{2}}(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.237em;vertical-align:-0.487em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.7538em;margin-left:-0.0962em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.487em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> is the Bessel function of the first kind of order <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">2n+\frac{1}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">β</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.</p>
<p>A: The partial sums of the sequence of orthonormal functions converge uniformly to the Bessel function of the first kind of order zero.</p>
<p>A: The sequence of orthonormal functions is a series of functions that can be used to approximate the Bessel function of the first kind of order zero.</p>
<p>A: The sequence of orthonormal functions is significant because it provides a way to approximate the Bessel function of the first kind of order zero using a series of functions.</p>
<p>A: The sequence of orthonormal functions is not well-known in the mathematical community, but it has been studied in various contexts.</p>
<p>A: The sequence of orthonormal functions has potential applications in various fields, including physics, engineering, and mathematics.</p>
<p>A: The sequence of orthonormal functions can be used to approximate the Bessel function of the first kind of order zero by summing the terms of the sequence.</p>
<p>A: Some potential challenges in using the sequence of orthonormal functions to approximate the Bessel function of the first kind of order zero include:</p>
<ul>
<li><strong>Convergence</strong>: The sequence of orthonormal functions may not converge uniformly to the Bessel function of the first kind of order zero.</li>
<li><strong>Accuracy</strong>: The accuracy of the approximation may depend on the number of terms used in the sequence.</li>
<li><strong>Computational complexity</strong>: The computation of the sequence of orthonormal functions may be computationally intensive.</li>
</ul>
<p>A: Some potential future directions for research on the sequence of orthonormal functions include:</p>
<ul>
<li><strong>Further study of the convergence properties of the sequence</strong></li>
<li><strong>Development of more efficient algorithms for computing the sequence</strong></li>
<li><strong>Investigation of the potential applications of the sequence in various fields</strong></li>
</ul>
<p>In conclusion, the sequence of orthonormal functions is a series of functions that can be used to approximate the Bessel function of the first kind of order zero. While the sequence is not well-known in the mathematical community, it has potential applications in various fields and is worth further study.</p>
f n β ( x ) = J 2 n + 2 1 β β ( x ) β J 2 n + 2 1 β β ( x ) β < / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > J < / mi >< m ro w >< mn > 2 < / mn >< mi > n < / mi >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< / m ro w >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m os t re t c h y = " f a l se " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > J 2 n + 2 1 β β ( x ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.237 e m ; v er t i c a l β a l i g n : β 0.487 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in β r i g h t : 0.09618 e m ; " > J < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.3448 e m ; " >< s p an s t y l e = " t o p : β 2.7538 e m ; ma r g in β l e f t : β 0.0962 e m ; ma r g in β r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< s p an c l a ss = " mbinm t i g h t " > + < / s p an >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m o p e nn u ll d e l imi t ers i z in g rese t β s i ze 3 s i ze 6" >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8443 e m ; " >< s p an s t y l e = " t o p : β 2.656 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.2255 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e m t i g h t " s t y l e = " b or d er β b o tt o m β w i d t h : 0.049 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.384 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 3 s i ze 1 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.344 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t ers i z in g rese t β s i ze 3 s i ze 6" >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.487 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< / s p an >< / s p an >< / s p an > i s t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d er < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x β ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mn > 2 < / mn >< mi > n < / mi >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x β t e x " > 2 n + 2 1 β < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x β h t m l " a r ia β hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.7278 e m ; v er t i c a l β a l i g n : β 0.0833 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in β r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1.1901 e m ; v er t i c a l β a l i g n : β 0.345 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t β t v l i s t β t 2" >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8451 e m ; " >< s p an s t y l e = " t o p : β 2.655 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c β l in e " s t y l e = " b or d er β b o tt o m β w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : β 3.394 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t β s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t β s " > β < / s p an >< / s p an >< s p an c l a ss = " v l i s t β r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.345 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > . < / p >< p > A : T h e p a r t ia l s u m so f t h ese q u e n ceo f or t h o n or ma l f u n c t i o n sco n v er g e u ni f or m l y t o t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero . < / p >< p > A : T h ese q u e n ceo f or t h o n or ma l f u n c t i o n s i s a ser i eso ff u n c t i o n s t ha t c anb e u se d t o a pp ro x ima t e t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero . < / p >< p > A : T h ese q u e n ceo f or t h o n or ma l f u n c t i o n s i ss i g ni f i c an t b ec a u se i tp ro v i d es a w a y t o a pp ro x ima t e t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero u s in g a ser i eso ff u n c t i o n s . < / p >< p > A : T h ese q u e n ceo f or t h o n or ma l f u n c t i o n s i s n o tw e ll β kn o w nin t h e ma t h e ma t i c a l co mm u ni t y , b u t i t ha s b ee n s t u d i e d in v a r i o u sco n t e x t s . < / p >< p > A : T h ese q u e n ceo f or t h o n or ma l f u n c t i o n s ha s p o t e n t ia l a ppl i c a t i o n s in v a r i o u s f i e l d s , in c l u d in g p h ys i cs , e n g in eer in g , an d ma t h e ma t i cs . < / p >< p > A : T h ese q u e n ceo f or t h o n or ma l f u n c t i o n sc anb e u se d t o a pp ro x ima t e t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero b ys u mmin g t h e t er m so f t h ese q u e n ce . < / p >< p > A : S o m e p o t e n t ia l c ha ll e n g es in u s in g t h ese q u e n ceo f or t h o n or ma l f u n c t i o n s t o a pp ro x ima t e t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero in c l u d e :< / p >< u l >< l i >< s t ro n g > C o n v er g e n ce < / s t ro n g >: T h ese q u e n ceo f or t h o n or ma l f u n c t i o n s ma y n o t co n v er g e u ni f or m l y t o t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero . < / l i >< l i >< s t ro n g > A cc u r a cy < / s t ro n g >: T h e a cc u r a cyo f t h e a pp ro x ima t i o nma y d e p e n d o n t h e n u mb ero f t er m s u se d in t h ese q u e n ce . < / l i >< l i >< s t ro n g > C o m p u t a t i o na l co m pl e x i t y < / s t ro n g >: T h eco m p u t a t i o n o f t h ese q u e n ceo f or t h o n or ma l f u n c t i o n s ma y b eco m p u t a t i o na ll y in t e n s i v e . < / l i >< / u l >< p > A : S o m e p o t e n t ia l f u t u re d i rec t i o n s f orrese a rc h o n t h ese q u e n ceo f or t h o n or ma l f u n c t i o n s in c l u d e :< / p >< u l >< l i >< s t ro n g > F u r t h ers t u d yo f t h eco n v er g e n ce p ro p er t i eso f t h ese q u e n ce < / s t ro n g >< / l i >< l i >< s t ro n g > De v e l o p m e n t o f m oree ff i c i e n t a l g or i t hm s f orco m p u t in g t h ese q u e n ce < / s t ro n g >< / l i >< l i >< s t ro n g > I n v es t i g a t i o n o f t h e p o t e n t ia l a ppl i c a t i o n so f t h ese q u e n ce in v a r i o u s f i e l d s < / s t ro n g >< / l i >< / u l >< p > I n co n c l u s i o n , t h ese q u e n ceo f or t h o n or ma l f u n c t i o n s i s a ser i eso ff u n c t i o n s t ha t c anb e u se d t o a pp ro x ima t e t h e B esse l f u n c t i o n o f t h e f i rs t kin d o f or d erzero . Whi l e t h ese q u e n ce i s n o tw e ll β kn o w nin t h e ma t h e ma t i c a l co mm u ni t y , i t ha s p o t e n t ia l a ppl i c a t i o n s in v a r i o u s f i e l d s an d i s w or t h f u r t h ers t u d y . < / p >