# Minimal property

of the partial sums of an orthogonal expansion

For any function $f \in L _ {2} [ a , b ]$, any orthonormal system $\{ \phi _ {k} \} _ {k=} 1 ^ \infty$ on $[ a , b ]$ and for any $n$, the equality

$$\inf _ {\{ a _ {k} \} _ {k=} 1 ^ {n} } \ \int\limits _ { a } ^ { b } \left | f ( x) - \sum _ { k= } 1 ^ { n } a _ {k} \phi _ {k} ( x) \ \right | ^ {2} d x =$$

$$= \ \int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x$$

holds, where

$$S _ {n} ( f , x ) = \ \sum _ { k= } 1 ^ { n } c _ {k} ( f ) \phi _ {k} ( x)$$

is the $n$- th partial sum of the expansion of $f$ with respect to the system $\{ \phi _ {k} \}$, that is,

$$c _ {k} ( f ) = \ \int\limits _ { a } ^ { b } f ( x) \phi _ {k} ( x) d x .$$

The minimum is attained precisely at the sum $S _ {n} ( f , x )$ and

$$\int\limits _ { a } ^ { b } | f ( x) - S _ {n} ( f , x ) | ^ {2} d x =$$

$$= \ \int\limits _ { a } ^ { b } f ^ { 2 } ( x) d x - \sum _ { k= } 1 ^ { n } | c _ {k} ( f ) | ^ {2} ,\ n = 1 , 2 ,\dots .$$

Bessel's inequality, Parseval's equality for complete systems and also certain other basic properties of orthogonal expansions essentially are corollaries of this equality (cf. Bessel inequality; Parseval equality; Complete system of functions; Orthogonal series; Orthogonal system).

How to Cite This Entry:
Minimal property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_property&oldid=47842
This article was adapted from an original article by A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
{"ft_lang_label":"__label__en","ft_lang_prob":0.5521238,"math_prob":0.9998309,"size":1786,"snap":"2022-27-2022-33","text_gpt3_token_len":660,"char_repetition_ratio":0.116722785,"word_repetition_ratio":0.3231132,"special_character_ratio":0.45856664,"punctuation_ratio":0.14409222,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000082,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-28T18:57:58Z\",\"WARC-Record-ID\":\"<urn:uuid:7eaf81df-b517-461b-afb0-3eabcfbd95e4>\",\"Content-Length\":\"16203\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bf57c891-5df8-485f-ad4a-6ebd39a38f04>\",\"WARC-Concurrent-To\":\"<urn:uuid:c7827ce2-a5bd-4785-8919-2b50a06b598c>\",\"WARC-IP-Address\":\"34.96.94.55\",\"WARC-Target-URI\":\"https://encyclopediaofmath.org/wiki/Minimal_property\",\"WARC-Payload-Digest\":\"sha1:ED6PJGKESFZAQD4EA6FSG25E42CTEHYP\",\"WARC-Block-Digest\":\"sha1:6WGXS6YW3WG4TM457RWWBEDSMPXYFDYX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103573995.30_warc_CC-MAIN-20220628173131-20220628203131-00040.warc.gz\"}"}